### Coupon Structures: 1. Zero-coupon bonds & 2. Accrual Bonds

Coupon structures:

1. Zero-coupon bonds - do not pay periodic interest or carry coupons

- sold at a deep discount from their par values.

- market convention dictates that semi-annual compounding should be used when pricing zero-coupon bonds.

2. Accrual Bonds - similar to zero-coupon bonds, BUT are sold originally @ Par Value.

- have a stated coupon rate

- but the coupon builds up at a compounded rate until maturity

### Coupon Structures: 3. Step-up Notes & 4. Deferred Coupon Bonds

3. Step-up notes have coupon rates that increase over time at a specified rate.

4. Deferred-coupon bonds carry coupons, but the initial coupon payments are deferred for some period.

- at the end of deferral period, the accrued (compound) interest is paid, and the bonds then make regular coupon payments until maturity.

### Floating rate securities

Floating (variable) rate securities - make varying coupon interest pmts which are set based on a specified interest rate or index (i.e. LIBOR) using the specified coupon formula

new coupon rate = reference rate ± quoted margin

### Limits on Floating rate securities

Upper and Lower limits on Floating rate securities:

Upper limit = cap = max interest rate paid by the borrower; limits upside potential so disadvantage to bondholder

Lower limit = floor = min interest rate received by the lender; limits how low the rate could go, so disadvantage to the issuer.

Bond with both upper and lower limit → "collar"

### Bond Valuation Process: 3 steps

Bond Valuation Process:

1. Estimate the CFs over life of security (both coupon pmts & principal)

2. Determine appropriate discount rate based on risk associated with estimated CFs

3. Calculate the PV of estimated CF

### 3 Difficulties in estimating Expected CFs

3 Difficulties in estimating expected CFs:

1. Timing of principal repayments is NOT known with certainty

2. Coupon pmts are NOT known with certainty

3. Bond = covertible or exchangeable into another security

### Bond Prices can be expressed as..

Bond prices, established in the market, can be expressed as:

1. % of par value

OR

2. Yield (YTM)

### YTM for annual pay bonds =

Bonds that make annual payments:

Yield to maturity (YTM) is the ANNUAL discount rate that will make the PV of bond's promised ANNUAL CFs = Market Price

YTM (annual pay) = [ 1 + (YTM semi-annual / 2) ]² - 1

### YTM for semi-annual pay bonds =

Bonds that make semi-annual payments:

Yield to maturity = 2 x Semiannual discount rate that will make the PV of semiannual coupon pmts = Market Price

YTM for a semi-annual pay bond = BEY

YTM (semi-annual pay) = 2 x [ (1 + YTM semi-annual /2)² - 1 ]

### Relationship between semi-annual YTM (or a BEY) and Price for a bond with N years to maturity =

Bond Price = CPN₁ / (1 + YTM/2)

+ CPN₂ / (1 + YTM/2)²

+ CPN₃ / (1 + YTM/2)³ ......

+ CPN + Par Value / (1 + YTM/2)²ⁿ

### Zero-coupon Bond Price & YTM =

Zero-coupon Bond Price = Face Value / [ 1 + (YTM/2) ]²ⁿ

Zero-coupon YTM = 2 x [ (face value/Price)¹/²ⁿ - 1 ]

Price-Yield relationship for a coupon-bond with N years to maturity is based on semi-annual YTM (or BEY by convention)

### Bondholder will actually realize the YTM on his initial investment only if..

Bondholder will actually realize the YTM on his initial investment only if

(1) all payments are made as scheduled and

(2) bond is held to maturity and

(3) all interim CFs are reinvested @ YTM

### Par-Yield Curve

Par Yield curve plots YTM vs. Bond Maturity

and is constructed with the YTMs for bonds trading @ par value

### Spot Rates

Spot Rates are market discount rates for single payments to be received in the future and can be thought of, theoretically, as discount rates ≈ to market yields on zero-coupon bonds.

Given the spot-rate yield curve, we can discount each of a bond's promised CF @ its appropriate spot rate. Sum the resulting PVs = market value (Price) of bond.

### Stripping the bond

Govt bond dealers can separate T bonds into their "pieces", with the individual coupon and principal CFs.

- individual pieces are a series of zero-coupon bonds with different maturity dates.

- each piece can be valued by discounting @ the spot rate for the appropriate maturity

### Recombining Bonds

Since bond dealers can also recombine a bond's individual cash flows into a bond, arbitrage prevents the market P of the bond from being more or less than the value of the individual cash flows discounted at spot rates.

Spot rate = no-arbitrage values

### Spot rate > Market P

If the spot rate (no-arbitrage value) > market P:

a bond dealer can BUY the bond, strip it and sell the pieces to earn an arbitrage profit.

### Market P > Spot rate

If Market P of bond > Spot rate (no-arbitrage value),

a bond dealer can BUY the pieces, combine them into a bond, and sell the bond to make a profit.

### Nominal Yield spreads measure

Nominal yield spreads measure the difference between the market yields on 2 bonds.

### Yield spreads are caused by:

Yield spreads are caused by differences in:

1. Credit quality

2. Call features

3. Tax treatments

4. Maturity

### Absolute Yield Spread =

Absolute Yield Spread = Higher Yield - Lower Yield

Absolute yield spread quantifies the difference between nominal yields on two bonds or two types of bonds.

- typically this yield spread is calculated between a non-Treasury security and a benchmark Treasury security.

### Relative Yield Spread =

Relative Yield Spread = [ Higher Yield / Lower Yield ] - 1

Quantifies the absolute spread as a % of the lower yield

### Credit Spread refers to the difference...

Credit spread refers to the difference in yield between two issues that are identical in ALL respects EXCEPT their credit ratings.

- credit spreads are a function of the state of the economy

Economic expansions → credit spreads ↓ as corporations are expected to have stronger cash flows

Recessions/Contractions → credit spreads ↑ since cash flows are pressured and there is a ↑ probability of default.

### Sources of Bond Return

Sources of bond return -

(1) periodic coupon interest payments

(2) recovery of principal, (and any capital gain and loss)

(3) Reinvestment income

### Current Yield =

Current Yield = Annual Coupon Payment / Bond Price

CY is based on actual cash received during the investment horizon and is typically composed of dividends and interest.

### YTC computation

YTC

- calculated similarly to YTM. Both are essentially internal rate of return measures.

N = # of semi-annual periods until the call date under consideration

FV = call Price (replaces maturity value)

Key to YTC computations is using the right # of periods (to first call) and the appropriate terminal value (the call price)

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### Accrued Interest

Accrued interest is the interest earned since the last coupon payment date and is paid by a bond buyer to a bond seller.

### Clean Price vs. Full Price

Clean price is the quoted price of the bond without accrued interest.

Full price refers to the quoted price + accrued interest

### Callable but non-refundable bonds

Callable but non-refundable bonds can be called prior to maturity, but their redemption cannot be funded by the issuance of bonds with a lower coupon rate.

### Affirmative vs. Negative Covenants in Bond Indentures

Affirmative covenants - what borrower/issuer MUST perform.

Negative covenants - what borrower/issuer CANNOT do. .

A bond's indenture contains the obligations, rights and any options available to the issuer or buyer of a bond.

###
Which statement is LEAST accurate?

a) With zero-coupon bonds, investors have NO reinvestment risk

b) Graph of current corporate bond yields = spot yield curve

c) YTM on a zero-coupon bond is called spot interest rate

Least accurate: b) Graph of current corporate yields ≠ spot yield curve.

- The graph of yields on zero-coupon bonds (spot rates) is called the spot yield curve.

- Note: Return on zero-coupon bonds is based entirely on price appreciation.

- An investor in a default-free zero-coupon bond will NOT have to worry about reinvesting coupons to realize the YTM.

### When a bond is trading "ex-coupon", what price does the buyer pay to the seller?

When a bond is trading "ex-coupon", the buyer will pay the CLEAN PRICE to the seller (or the price without accrued interest ~ the quoted price).

When a bond is trading "cum-coupon", the buyer will pay the DIRTY PRICE or full price to the seller.

### If the issuer of a bond is in default, the bond will be trading...

If an issuer of a bond is in default (i.e. it has not been making periodic contractual coupon payments), the bond is traded WITHOUT accrued interest and is said to trade "FLAT".

### Non-refundable bonds may be called, as along as...

Non-refundable bonds may be called as long as the firm does NOT use less expensive debt to do so. They may be refunded without outside capital (new debt or equity issues), just NOT cheaper debt

### Regular redemption prices vs. Special redemption prices for callable bonds

Regular redemption price (can be at premium or at par) refers to bonds being called according to the provisions specified in the bond indenture.

When bonds are redeemed to comply with a sinking fund provision or because of a property sale mandated by govt authority, the redemption prices (typically par value) are referred to as "special redemption prices".

There is NO such thing as "specific redemption price".

###
The refunding provision found in nonrefundable bonds allows bonds to be retired UNLESS:

a) funds comes from a lower cost bond issue

b) funds come from the sale of a new common stock

c) market interest rates have increased substantially

(A) Refunding from a new debt issue at a higher interest rate is not prohibited, however their purchase cannot be funded by the simultaneous issuance of lower coupon bonds.

### True/False: An investor concerned about premature redemption is indifferent between a noncallable bond and a nonrefundable bond.

False! The term "refunding" specifically means redeeming a bond with funds raised from a new bond issued at a lower coupon rate.

Even a sinking fund is a type of redemption, which refers to the retirement of bonds.

An investor concerned with "premature redemption" would prefer a noncallable bond because a noncallable bond cannot be called for ANY reason.

A bond that is callable, but nonrefundable can be called for any reason OTHER than refunding.

A nonrefundable bond can be redeemed with funds from operations or a new equity issue.

### Retiring funds via sinking fund vs. accelerated sinking fund

A sinking fund actually retires the bonds based on a schedule.

- accomplished through either a) payment of cash or b) delivery of securities

An accelerated sinking fund provision allows the company to retire more than is stipulated in the indenture.

###
Which of the following is LEAST LIKELY an amortizing security:

a) Coupon Treasury Bonds

b) Bonds with sinking fund provisions

c) Mortgage-backed securities (MBS)

A) Coupon Treasury Bonds ≠ amortizing security

Coupon Treasury bonds and most corporate bonds are non-amortizing securities because they pay only interest until maturity. At maturity, these bonds repay the entire par value or face value.

MBS is backed by pools of loans that generally have a schedule or partial principal payments, making these "amortizing securities".

A sinking fund provision is another example of an amortizing feature of a bond. This feature is designed to pay a part of the entire total of the issue by the maturity date.

### Characteristics and examples of embedded options that benefit the issuer

Embedded options that benefit the issuer

- ↓ bond value

- ↑ bond yield

to a bond buyer.

Examples:

- Call Provisions

- Accelerated Sinking Fund Provisions

- Caps (maximum interest rates) on floating rate bonds

### Characteristics and examples of embedded options that benefit the bondholders

Embedded options that benefit the bondholder:

- ↑ bond value

- ↓ bond yield

to a bond buyer.

Examples:

- Conversion options (the option of bondholders to convert their bonds into shares of the bond issuer's common stock)

- Put options (option of bondholders to return their bonds to the issuer at a predetermined price). Even if the put is out of money, it still has value to the bondholder.

- Floors (minimum interest rates) on floating rate bonds.

### What are two ways institutions can finance secondary market bond purchases?

Institutions can finance secondary market bond purchases by

a) margin buying (borrowing some of the purchase price, using the securities as collateral) OR

b) repurchase agreements (repo) - an arrangement in which the institution sells a security with a promise to buy it back at an agreed-upon higher price @ a specified date in the future.

- repo agreements are more commonly used.

### Risks associated with fixed income securities: Interest rate risk

Interest rate risk - uncertainty about bond prices due to changes in market interest rates

- probability of an ↑ in interest rates causing a bond's price ↓

### Risks associated with fixed income securities: Call Risk

Call Risk - the risk that a bond will be called (redeemed) prior to maturity under the terms of the call provision and that the funds must then be reinvested at the then-current (lower) yield

### Risks associated with fixed income securities: Prepayment Risk

Prepayment risk - the uncertainty about the amount of bond principal that will be repaid prior to maturity.

### Risks associated with fixed income securities: Yield Curve Risk

Yield curve risk - the risk that changes in the shape of the yield curve will reduce bond values

### Risks associated with fixed income securities: Credit Risk

Credit Risk - includes

- the risk of default

- the risk of a ↓ in bond value due to a ratings downgrade

- the risk that the credit spread ↑ for a particular rating

### Risks associated with fixed income securities: Liquidity Risk

Liquidity risk - the risk that an immediate sale will result in a price < fair value price (or the prevailing market price)

### Risks associated with fixed income securities: Exchange Rate Risk

Exchange Rate Risk - the risk that the domestic currency value of bond payments in a foreign currency will ↓ due to exchange rate changes

### Risks associated with fixed income securities: Volatility Risk

Volatility Risk - the risk that ∆ in expected interest rate volatility will affect the values of bonds with embedded options

### Risks associated with fixed income securities: Inflation Risk

Inflation risk - the risk that actual inflation > expected inflation

- this erodes the purchasing power of the cash flows from a fixed income security.

### Risks associated with fixed income securities: Event Risk

Event Risk - the risk of ↓ in a security's value from disasters, corporate restructurings, or regulatory changes that negatively affect the firm.

- anything that ↓ issuer's earnings or ↓ asset values.

- takeovers or restructurings that can have negative effects on the priority of bondholder's claims.

- regulatory changes that can ↓ issuer's earnings or narrow the market for a particular class of bonds.

### Risks associated with fixed income securities: Sovereign Risk

Sovereign Risk - the risk that governments may refuse to pay or repudiate debt or not be able to make debt payments in the future.

- possibly due to poor economic conditions and government deficit spending.

###
Interest rate risk for a bond refers to the fact that when interest rates:

a) ↓...the realized yield on the bond < YTM

b) ↑...the bond's value ↓

c) ↑...prepayments of principal will ↓

(B) Interest rate risk is the risk that the bond's value will ↓ because interest rates ↑.

Reinvestment risk is the risk that a bond's CFs will be reinvested at lower-than-expected rates.

Prepayment risk refers to the fact that prepayments of a mortgage-backed security's principal may differ from the expected rate.

### When a bond's coupon rate < market yield, how does this affect the bond value?

When a bond's coupon rate < market yield, the bond will trade at a DISCOUNT to its par value

### When a bond's coupon rate > market yield, how does this affect the bond value?

When a bond's coupon rate > market yield, the bond will trade at a PREMIUM to its par value

### The level of a bond's INTEREST RATE RISK or duration is positively and negatively related to...

The level of a bond's interest rate risk (=duration) is:

- ↑ Maturity, ↑ Interest Rate Risk

- ↓ Coupon rate, ↑ Interest Rate Risk

- ↓ Market YTM, ↑ Interest Rate Risk

Less over some ranges for bonds with embedded options

### What type of bonds have the highest interest rate risk?

Zero-coupon bonds have the highest interest rate risk because they deliver all their cash flows at maturity.

- another way to think of this: a zero-coupon bond has the lowest coupon (0.00%), so it has the highest price volatility, since the coupon rate is inversely related to price volatility.

###
For a given change in yield, which will experience a smaller change in price?

a) higher coupon bond

b) lower coupon bond

(A) In addition to market yields, the timing and magnitude of cash flows affect price volatility.

For a given change in yield, a higher coupon bond will experience a SMALLER change in price than a lower coupon bond.

### True/False Price sensitivity is lower when the level of interest rates is higher.

True! The price sensitivity is lower when the level of interest rates is higher. Put another way, a bond that has the lowest duration is therefore the least sensitive to changes in interest rates.

###
All else equal, the lower the bond's YTM, the:

a) longer the duration and higher the interest rate risk

b) shorter the duration and lower the interest rate risk

c) shorter the duration and higher the interest rate risk

(A) A lower yield to maturity would result in a longer duration and higher interest rate risk.

### Price of a callable bond =

Price of a callable bond = Price of an identical option-free bond - value of the embedded call.

### As interest rates decrease, how does the issuer value the call option?

As interest rates ↓, the issuer values the call option more because the company has the potential to "call" the bond and replace existing debt with lower-coupon (and thus lower cost) debt.

### Who does the call option benefit: issuer or the investor? Is the call price the ceiling or the floor on the value of a callable bond?

The call option benefits the issuer - not the investor. The call price acts as a ceiling on the value of a callable bond.

Value of a callable bond will always be ≤ otherwise identical non-callable bond.

### Difference between the curves of callable and noncallable bonds:

The noncallable bond has the traditional PY shape.

The callable bond bends backwards.

The difference between the two curves = value of the option

### What kind of risk can floating-rate bonds have between reset dates?

Floating-rate bonds have INTEREST RATE RISK between reset dates.

- their prices differ from their par values, even at reset dates, due to changes in liquidity or in credit risk after they have been issued.

### Does holding a floating-rate bond eliminate price fluctuations?

Holding floating-rate bonds MINIMIZES, but does not eliminate price fluctuations.

- with a perfect, continuously resetting coupon rate, a floating-rate bond's value would always = par value

- a cap rate can ↑ price volatility of a floating-rate bond.

### In general, the longer the time until the next reset date of a floating-rate security affects interest rate risk by...

In general, the longer the time until the next reset, the greater the interest rate risk of the floating-rate security.

- the interest rate risk of a floating-rate security ↓ as the reset date approaches because the coupon reset will return the price to par, as long as the margin above the reference rate accurately reflects the bond's risk.

### True/False The greater the time lag between reset dates, the greater the amount of price fluctuation in floating-rate securities.

True. The more frequent the reset dates, the less the time lag that causes volatility. So, the greater the gap between reset dates, the greater the amount of price fluctuation.

### True/False A fixed margin coupon exposes the bond to more price fluctuations than an adjustable margin (as is the case with an extendible reset bond)

True! A fixed margin coupon exposes the bond to more price fluctuations than an adjustable margin (as is the case with an extendible reset bond)

### What is cap risk in a floating-rate bond?

Cap risk refers to when market interest rates rise to the point that the coupon on a floating rate security hits the cap and the bond begins to behave like a fixed coupon bond, which has more price fluctuations.

- this is a risk to the holder of a floating-rate bond.

### Duration of a bond vs. Dollar Duration of a bond

Duration of a bond = approximate %Price∆ for a 1% change in yield

$ Duration of a bond = approximate $Price∆ for a 1% change in yield.

### For a zero-coupon bond, duration is approximately equal to...?

For a zero-coupon bond, duration is approximately equal to the # of years to maturity. We use the term "approximately" because it ignores the curvature of the price/yield curve.

### True/False A bond's percentage change in price and dollar change in price are both tied to the underlying price volatility.

True! A bond's %∆ in price and $∆ in price are both tied to the underlying price volatility.

### Effective duration & basis points

The effective duration formula result is for 1.00% change in interest rates.

100 basis points = 1.00% = 0.01 (in decimal form).

1 basis point = 0.0001

### True/False If a bond has an effective duration of 7.5, it means that a 1% change in rates will result in a 7.5% change in price.

True! Because of convexity, it will be approximately 7.5% change in price, not an actual 7.5% change in price. The readings are very explicit about this distinction.

### How is duration related to maturity and coupon rate and YTM?

Duration ↑

...Maturity ↑

A longer term bond pays its cash flows later than a shorter term bond, increasing the duration.

...Coupon Rate ↓

A lower coupon bond pays lower annual cash flows than a higher coupon bond and thus has less influence on duration.

...YTM ↓

Because the bond's price (or present value) is inversely related to interest rates. When market yields ↓, the value (or cash flows) of a bond ↑ WITHOUT increasing the time to maturity.

### How is duration related to a bond's cash flows?

Duration is approximately equal to the point in years where the investor receives half of the PV of the bond's cash flows. Therefore, the later the cash flows are received, the ↑ duration. This rationale is similar to price volatility.

### What is the indirect relationship between coupon rate and duration?

The higher the coupon rate, the shorter the duration. Why? A higher coupon bond pays higher annual cash flows than a lower coupon bond and thus has more influence on duration.

### What type of bonds have the highest price volatility and the longest duration?

Zero-coupon bonds have the highest price volatility and the longest duration (at approximately ≈ maturity). This is because zero-coupon bonds pay all cash flows in one lump sum at maturity.

### In what way is duration a function of volatility or risk?

Duration is also a function of volatility (risk). Higher volatility (risk) = higher duration. A ↑ coupon bond has a ↓ duration relative to similar bond with a lower coupon, because the bond holder is getting more of their cash value sooner (because of the higher coupon). This ↓ the overall risk of the bond resulting in ↓ duration.

### Duration of zero-coupon bonds vs. coupon paying bonds

The duration of a zero-coupon bond is approximately equal to its time to maturity.

For coupon-paying bonds, duration < maturity. Any coupon amount will shorten duration because some cash flow is received prior to maturity.

### Yield curve risk of a bond portfolio -

Yield curve risk of a bond portfolio is the risk (in addition to interest rate risk) that the portfolio's value may ↓ due to a non-parallel shift in the yield curve (change in its shape).

### What happens to the duration of a bond portfolio when a yield curve has a non-parallel shift?

When yield curve shifts are not parallel - the duration of a bond portfolio does NOT capture the true price effects because the yield on various bonds in the portfolio may change by different amounts.

### TRUE/FALSE If long-term rates are low, the PV of cash flows far into the future will be high, and the bond's value will be high.

True! If long-term rates are low, the PV of cash flows far into the future will be high and the bond's value will be high. The value of a bond is comprised of discounted cash flows and a lower discount rate translates to higher cash flows.

### What does the yield curve plot?

The yield curve plots

Term to maturity vs. Yield to Maturity (<< YTM, not coupon rate!)

### Why does the yield curve usually have a non-zero slope?

The yield curve usually has a nonzero slope because the rates change by different basis points across maturities.

### Portfolio duration -

Portfolio duration is a measure of a portfolio's interest rate risk.

- measures the sensitivity of the portfolio's value to an equal change in yield for all the bonds in the portfolio.

- calculated as the weighted average of the individual bond durations using the proportions of the total portfolio value represented by each of the bonds.

- does not capture the effect of changes in the yield curve (term structure)

### What are the disadvantages to an investor of a callable or prepayable security:

- Timing of cash flows = uncertain

- Principal is most likely to be returned early when interest rates are available for reinvestment are low.

- Potential price appreciation is < than that of option-free bonds.

### Compared to a callable bond, the yield on a non-callable bond is... more or less?

When compared to a callable bond, the yield on a noncallable bond is LESS. With a noncallable bond, the issuer does not have to compensate the investor for call risk/cash flow uncertainty with any premium.

### Call Risk is a combination of...

Call risk is a combination of cash flow uncertainty and reinvestment risk.

When a bond is called, the investor faces a disruption in cash flow and a reduced rate of return.

### When yield volatility increases, what happens to both (puts and calls) embedded options?

Embedded options (puts and calls) increase in value when volatility increases. In the formula, The value of a putable bond is calculated by ADDING the option to a straight bond. The value of a callable bond is calculated by SUBTRACTING the option from a straight bond.

### What characteristics cause a bond to have more reinvestment risk?

A security has more reinvestment risk when it has a...

- HIGHER coupon

- is callable

- is an amortizing security

- has a prepayment option

### Why does a prepayable amortizing security have greater reinvestment risk?

A prepayable amortizing security has greater reinvestment risk because of the probability of accelerated principal payments when interest rates (including reinvestment rates) ↓

### True/False Zero-coupon bonds have NO reinvestment risk over their term.

Zero-coupon bonds have NO reinvestment risk over their term ONLY if they are held till maturity. Reinvestment risk means that a bond investor risks having to reinvest bond cash flows (both coupon and principal) at a rate lower than the promised yield.

### How do maturities and coupon amounts impact reinvestment risk?

Reinvestment risk ↑ with

longer maturities and higher coupons.

### Can an investor ever eliminate reinvestment risk?

While a bond investor can eliminate price risk by holding a bond until maturity, he usually cannot eliminate bond reinvestment risk.

ONE EXCEPTION: zero-coupon bonds have no reinvestment risk as long as they are held to maturity.

### True/Falso Short-term bonds should be purchased if the investor anticipates higher reinvestment rates.

True! If an investor expects interest rates to rise, he would want a bond with a shorter maturity so that he receives his CFs sooner and could reinvest at the higher rate. Also, there is less prepayment risk with shorter maturities.

### Why is reinvestment risk important to consider? And when does reinvestment risk become the least important?

Reinvestment risk is important because the yield-to-maturity (YTM) calculation for a bond assumes that the investor can reinvest cash flows at exactly the coupon rate. It is the risk that if the rates fall, cash flows will be reinvested at lower rates, resulting in a holding return < expected return (at the time of bond purchase).

Reinvestment risk is least important with the combination of shorter maturity and lower coupon rate.

### Do mortgage-backed securities face reinvestment risk?

Yes! Mortgage backed and other asset backed securities have HIGH reinvestment risk because in addition to cash flows from periodic interest payments (like bond coupons), these securities have repayment of principal. The ↓ interest rates, the ↑ chances of prepayments.

###
What kind of security has

a) high interest rate risk AND

b) low reinvestment risk?

A zero-coupon bond can have high interest rate risk (because its single cash flows subjects it to the full amount of discounting when interest rates change) and low reinvestment risk (the single cash flow minimizes prepayment risk).

### Order the following securities from least reinvestment risk to most: callable bonds, straight coupon bonds, mortgage-backed securities, and zero-coupon bonds

Least to Most Reinvestment risk:

1) Zero-coupon bonds (noncallable & held to maturity)

2) Straight coupon bonds (no prepayment risk, but do have periodic coupon payments).

3) Callable Bonds (the right to prepay principal compounds reinvestment risk. A call option is one form of prepayment right that benefits the issuer (and not the bondholder).

4) Mortgage-backed securities - most reinvestment risk because of high prepayment risk, periodic interest payments, and repayment of principal.

### Credit Risk includes what 3 types of risk:

Credit risk includes:

1) Default Risk - probability of default

2) Downgrade Risk - probability of a reduction in the bond rating

3) Credit Spread Risk - uncertainty about the bond's yield spread to T-bonds ("Treasuries") based on its bond rating.

### Credit Ratings

Credit ratings are designed to indicate to investors a bond's relative probability of default.

Lowest probability of default - AAA

### Credit ratings for Investment grade vs. Speculative or high yield bonds...

Investment grade Bonds: AA, A, and BBB

Speculative or high yield bonds: BB or lower

### TRUE/FALSE When a rating agency downgrades a security, the bond's price usually falls.

TRUE! The market will likely demand a higher yield from the downgraded bond (the risk premium has increased) and thus the price will likely fall.

### Technical Default -

Technical default refers to an issuer's violation of bond covenants, such as debt ratios, rather than the failure to pay interest or principal. In the event of a default, the holder (lender) may recover some or all of the investment through legal action or negotiation. The percentage recovered = "recovery rate"

### Credit spread Risk (or the yield on a risky asset) formula:

Yield(risky) = Yield (risk-free) + RP

RF = default - free rate

### The yield differential above the return on a benchmark security measures...

The yield differential above the return on a benchmark security measures the CREDIT SPREAD risk.

Credit Spread risk = Risk Premium = "Spread"

### Call Risk is composed of what three components:

Call Risk is composed of:

1) unpredictability of the cash flows

2) compression of the bond's price

3) reinvestment risk (or the high prob that when the bond is called, the investor will be faced with less attractive investment ops)

### How does a lack of liquidity affect portfolio values?

Lack of liquidity can have ADVERSE effects on calculated portfolio values and therefore, on performances measures for a portfolio.

- this makes liquidity a concern for a manager even though sale of the bonds is not anticipated.

### How is liquidity important to institutional investors and dealers?

Liquidity is important to institutional investors that must determine market values for NAVs and to dealers in the repurchase market for collateral valuation.

### Bonds bought with cash flows denominated in a foreign currency...

An investor who buys a bond with CFs denominated in a foreign currency (not $$) will see the value of the bond ↓...

- if the foreign currency depreciates OR

- the exchange value of the foreign currency declines

relative to the investor's home currency.

### True/False Appreciation of the Swiss Franc benefits the US investor holding Swiss bonds.

True! The appreciation of the foreign currency (Swiss Franc) benefits domestic investors (US citizens) who own foreign (Swiss) bonds.

- when the Swiss Franc appreciates, each Swiss franc buys more of the US dollar than before.

- So, the US investor gains by owning foreign bonds because the investor realizes (a) return from the bond and (b) gain from the foreign currency appreciation.

### What happens if the inflation increases unexpectedly?

If inflation increases unexpectedly, the purchasing power of a bond's future cash flows is decreased and bond values ↓

### True/False Treasury securities are considered immune to inflation and liquidity risk

False! The statement that T securities are considered immune to inflation and liquidity risk is partially true - T securities are immune to liquidity risk, but fixed-coupon T securities have high inflation risk and generally low real returns.

### True/False The short term inflation premium < long term premium

The ST inflation premium < LT inflation premium because investors are better able to predict inflation in the short term. Inflation risk ↑ as time ↑ and so investors want to be compensated for this uncertainty.

### True/False The real return on a fixed coupon bond is variable.

True! The real return on a fixed coupon is variable. An investor's real return is not fixed - even though an investor may hold a fixed-rate coupon bond, the real return depends on a variable - inflation.

↑ inflation rates result in a ↓ purchasing power of bond payments.

### How does ↑ Yield volatility impact put and call options embedded in bonds?

↑ Yield volatility, ↑ value of put and call options embedded in bonds...thus:

- ↓ value of a callable bond (cuz the bondholder is short the call)

- ↑ value of a putable bond (cuz the bondholder is long the put)

### Value of a callable bond =

Value of callable bond = Value of a Straight bond - Call Option

Remember: the call option is subtracted from the bond value because the call option is of value to the issuer and not the bondholder.

↑ Yield volatility, ↑ Call OPTION value, ↓ Callable BOND value → Bondholder loses.

### Value of a putable bond =

Value of a putable bond = Value of a Straight Bond + Put Option Value

Remember: the put option is added to the bond value because it is of value to the bondholder and not the issuer.

↓ Yield Volatility, ↓ Put OPTION value, ↓ Putable BOND value

### US Treasuries are the ______ debt of the US Government

US Treasury securities are SOVEREIGN debt of the US government and are considered free of credit risk.

- Sovereign debt refers to the debt obligations of governments.

- Sovereign debt of other countries has varying degrees of credit risk.

### Sovereign debt is typically issued using 4 methods:

Sovereign debt is typically issued using one of 4 methods:

1. Regular auction cycle with the entire issue sold at a single price

2. Regular auction cycle with bonds issued at multiple prices.

3. Ad Hoc auction system with no regular cycle

4. Tap system, auctioning new bonds identical to previously issued bonds.

### Ad Hoc auction system

The Ad Hoc auction system is a method in which a central government distributes new government securities via auction when it determines that market conditions are advantageous.

### True/False When a central government issues securities, those securities can only be denominated in the local currency regardless of where the bonds are issued.

False! When a central government issues securities, those securities are generally denominated in the currency of the issuing country, but a government can issue bonds denominated in any currency.

- Also, Sovereign bonds are not necessarily free from default risk.

### 3 types of US Treasury securities:

Securities issued by the US Treasury include:

1. Bills - pure discount securities maturing in 4 weeks, 3 months, or 6 months

2. Notes - coupon securities maturing in 2, 5, and 10 years

3. Bonds - coupon securities maturing in 20 or 30 years

### TIPS -

Treasury Inflation Protected Securities (TIPS) are US Treasury issues in which:

- Coupon rate = FIXED

- Par Value = adjusted periodically for inflation, based on changes in the CPI.

### On the run vs. Off the run issues

US Treasuries from the most recent auction are referred to as "on the run" issues, while Treasuries from previous auctions are referred to as "off the run" issues.

### Impact of deflation on a TIPS

If deflation (inflation ↓) occurs over the life of a TIPS, the Treasury will redeem the security for its initial face value of $1000 at maturity.

Remember: The principal of a TIPS adjusts semi-annually and then the FIXED coupon rate x (new principal value) = new inflation-adjusted interest payment.

### How are T-bond prices quoted?

Bond prices are quoted in 32nds. Ex: a quote for 94 10/32.

Divide 10/32 = 0.3125. Now you have 94.3125%.

So, the bond price = $943.125 for a $1000 T-bond.

Ex: a quote for 96 27/32 = 96.84. So, bond price = $9684.38 for a $10,000 bond.

### If a T-bond is quoted at 92-16, the price of the bond is:

Bond prices are quoted in 32nds.

Quote: 92-16, which means 92 16/32 = 92.5% of par value

0.925 x $1000 (par value) = $925

### Treasury strips are traded in 2 forms

Treasury strips are traded in 2 forms:

1. Coupon strips

2. Principal Strips

and are taxed by the IRS on the basis of accrued interest, like other zero-coupon securities.

### True/False Taxable investors holding zero-coupon bonds can have negative cash flow prior to maturity

True! It is possimpible that taxable investors will have (-) cash flows from holding zero-coupon securities, since there is no cash income, but taxes must be paid at least annually on the implicit interest.

### Securities issued by the US Federal agencies

Agencies of the US government, inc federally related institutions and govt sponsored enterprises, issue bonds that are not obligations of the US Treasury but are considered to be almost default risk free.

### This type of security redistributes the prepayment risk among the investors through tranches.

A collaterized mortgage obligation (CMO), and NOT a passthrough security, redistributes the prepayment risk among the investors through tranches.

- CMOs are split into tranches, with each tranche having a different claim and risk structure on the pool of CFs.

### This type of security may be retired before maturity at face value with no penalty.

Because the mortgage holders may prepay the mortgage, a passthrough security may indeed be retired before maturity at face value with no penalty.

### True/False Government sponsored enterprises (GSE) are owned by the US government and therefore have essentially no credit risk.

False! Government sponsored enterprises (GSE) are privately owned, and therefore investors assume some credit risk.

###
Which of the following is NOT a GSE (government-sponsored enterprise)?

a) Federal Farm Credit System

b) Government National Mortgage Association

c) Student Loan Marketing Association

(B) Government National Mortgage Association (Ginnie Mae) is NOT a GSE.

Federally related (or government-owned) agencies are arms of the federal government. Both the "Federal Farm Credit System" & "Student Loan Marketing Assoc" are GSE's.

### Mortgage Passthrough Security

Mortgage Passthrough security

- backed by a pool of amortizing mortgage loans (the "collateral")

- has monthly CFs that include:

a) interest payments

b) scheduled principal payments

c) prepayments of principal.

### Is prepayment risk significant for Investors in passthrough securities?

Yes, Prepayment risk is significant for investors in passthrough securities because most mortgage loans contain a prepayment option which allows the issuer (borrower - you) to make additional principal payments at any time.

### Collaterized Mortgage Obligations (CMOs)

Collaterized Mortgage Obligations are customized claims to the principal and/or interest payments of mortgage passthrough securities

- redistribute the prepayment risk and/or maturity risk of the securities.

### Sequential Tranches

Sequential tranches issued as a CMO do not have proportionate claims on the CFs from the pool

- Have sequential claims

- shortest term tranche receives principal and interest payments until it is paid off.

- CFs then go to the second tranche until it is paid off and so on.

- This structure allows securities with different timings and risk profiles to be issued from the same pool of certificates.

### How is prepayment of a mortgage like an embedded call option feature?

A borrower who prepays a mortgage is in effect exercising a call option, similar to a corporate bond issuer who calls a bond and prepays the principal.

- therefore the pool of mortgages and the securities created from it behave as if they had an embedded call feature.

### Curtailment

If a prepayment of principal is for an amount < full outstanding loan balance...then it is known as "curtailment".

### What are CMOs created to do?

CMOs are created to ↓ borrowing costs by:

a) redistributing prepayment risk or

b) altering the maturity structure

...to better suit investor preferences.

CMOs redistribute the risk between tranches on an UNEQUAL basis, and not on a random basis.

### Munis and Interest Payments

Interest payments on state and local government securities ("munis") are usually exempt from US Federal taxes, and from state taxes in the state of issuance.

###
Municipal bonds include:

1) Tax-backed bonds

2) Revenue bonds

Municipal bonds include:

1) Tax-backed (general obligation) bonds backed by the taxing authority of the governmental unit issuing the security.

2) Revenue bonds, backed only by the NET revenues from the project specifically financed by the bond issue.

### Are GSE's backed by the full faith and credit for the treasury?

Government sponsored enterprises (GSE) are NOT backed by the full faith and credit of the Treasury.

### True/False Before tax-yields on municipal bonds are usually < before tax-yields on Treasury bonds.

True! Treasury bonds are considered default free and have the least amount of risk.

### True/False The vast majority of muni bonds sell at lower yields because their bond interest is exempt from federal income tax.

True! After tax yields are highest for individuals in the highest tax bracket who benefit from the muni bond's tax-exempt status.

- before tax yields on muni bonds are lower due to their tax shield.

### True/False All interest on muni securities is tax-exempt at the federal level.

False! Some interest on muni bonds, such as muni bond issues to build stadiums/arenas, is taxable at the federal level. Note: most muni bonds are tax-exempt - taxable munis tend to be the exception rather than the rule.

###
3 muni bonds; rank in order of highest to lowest market yields:

Series W - backed by District's authority to levy property tax.

Series X - fees from the plant are the only source of the interest and principal payments on the bonds

Series Y - carry a 3rd party guarantee of principal and interest payments

Highest to lowest market yield: X, W, Y

Series X - revenue bond. Because they pay only int and prin only if the revenues from the project they finance are sufficient, revenue bonds are typically riskier and therefore have higher mkt yields than general obligation bonds.

Series Y - insured bond. Muni bond insurance typically results in a higher rating, and therefore lower mkt yield, than an equivalent bond from the same muni issuer.

### Types of Corporate Debt

Corporate debt securities include:

1. bonds

2. medium-term notes, and

3. commercial paper.

### How do bond rating agencies rate corporate bonds?

Bond rating agencies rate corporate bonds on:

1. capacity to repay (liquid assets and cash flow)

2. mgmt quality

3. industry prospects

4. corporate strategy

5. financial policies

6. credit history

7. overall debt levels

8. collateral for the issue

9. nature of the covenants

### Secured and unsecured Corporate Bonds

Corporate bonds may be secured or unsecured (called "DEBENTURES").

Security can be in the form of

1. Real property

2. Financial assets or

3. Personal property/equipment

### Medium-Term Notes

Medium-Term Notes (MTN) are issued periodically by corporations under a shelf registration.

- sold by agents on a "best-efforts basis"

- have maturities: 9 months → + 30 yrs

### Structured Notes

Structured Notes combine a bond with a derivative to create a security that fills a need for particular institutional investors.

### Commerical Paper

Commercial Paper is a short-term corporate financing vehicle.

- according to Securities Act of 1933, CP must be registered with the SEC

BUT!

- does NOT require registration with the SEC IF it's maturity < 270 days

### 2 forms of Commerical Paper

CP comes in 2 forms:

1. Directly-placed paper sold directly by the issuer

2. Dealer-placed paper sold to investors through agents/brokers

Expense of Direct paper < Dealer paper

### Negotiable CDs

Negotiable CDs are issued in a wide-range of maturities by banks.

- trade in a secondary market

- backed by bank assets

- termed "Eurodollar CDs" when denominated in US $ and issued OUTSIDE of the US.

### Bankers' Acceptances

Bankers' Acceptances are issued by BANKS to guarantee a future payment for goods shipped

- sold at a discount to the future payment they promise

- Short-Term

- have limited liquidity

### Asset-backed securities (ABS)

Asset-backed securities (ABS) are debt that is supported by the CFs from an underlying pool of

- mortgages

- auto loans

- credit card receivables

- commercial loans

- or other financial assets

### Special Purpose Vehicle

A special purpose vehicle is an entity to which teh assets that back an ABS are legally transferred.

- if the corporation transferring these assets goes bankrupt, the assets are NOT subject to claims from its creditors.

Result → ABS can receive a ↑ credit rating than the corporation and ↓ corporation's funding costs.

### External credit enhancements for an ABS

External credit enhancements for an ABS can include:

1. Corporate guarantees

2. Letters of credit

3. 3rd party bond insurance

### How does decreasing credit enhancements impact cost of borrowing?

↓ Credit enhancements → Cost of borrowing ↑

### How do SPV's work?

1. SPV's, or "Special purpose vehicles" buy the assets from the corporation.

2. The SPV then separates the assets used as collateral from the corporation that is seeking financing.

Result: shields the assets from other creditors.

### True/False Residential mortgages represent the largest type of asset that has been securitized.

True! Residential mortgages represent the largest type of asset that has been securitized.

### True/False The credit rating of an ABS must the same as that of the issuer.

False! The credit rating of an ABS (Asset-backed security) is a function of its credit enhancements.

↑ credit enhancements, ↑ credit ratings

###
A corporation may issue asset-backed securities because:

a) it wants to change the structure of its balance sheet

b) it wants to reduce the cost of borrowing

c) both a and b

Both A and B! A corporation may issue asset-backed securities because it wants to change the structure of its balance sheet and because it wants to reduce the cost of borrowing.

### Collateralized Debt Obligations (CDOs) vs. CMO

Collateralized Debt Obligations (CDOs) are backed by an underlying diversified pool of debt securities which may be any one of a number of types:

1. Corporate bonds

2. Loans

3. Emerging markets debt

4. Mortgage-backed securities (as in Asset-backed securities)

5. Other CDOs

6. Mortgages

7. Non-performing loans

CMO - Collaterized Mortgage Obligation is a debt obligation that is backed by mortgages.

### Arbitrage CDO

A Collateralized Debt Obligation (CDO) issued to profit on the spread between the return on the underlying assets and the return paid to investors is referred to as an arbitrage CDO.

### Balance Sheet CDO

Balance Sheet Collaterized Debt Obligation is created by a bank or insurance company to reduce their loan exposure on the balance sheet.

### Primary Bond Markets

Primary markets in bonds include:

1. underwritten public offerings

2. best-efforts public offerings

3. private placements

### Secondary Bond Markets

Secondary bond markets:

- include some trading on exchanges

- much larger volume of trading in a dealer market

- electronic trading networks which are an increasingly important part of the secondary market for bonds.

### Rule 144A offering

When bonds are sold in a Rule 144A offering, they are sold privately to a small number of investors or institutions.

- this offering does not require SEC registration and this is valuable to the issuer.

- the investor will require a slightly higher yield because the bonds cannot be resold to the public unless they are registered with the SEC.

### True/False Bonds sold from a dealer's inventory do NOT represent a primary market offering.

True! When bonds are sold from a dealer's inventory, the bonds have already been sold once and the transaction takes place on the secondary market.

### Best-efforts basis

When bonds are sold on a best-efforts basis, the investment banker does NOT take ownership of the securities and agrees to sell all they can.

### True/False When bonds are sold in a BOUGHT deal, the transaction takes place on the secondary market.

False! When bonds are sold in a bought deal, the transaction takes place on the PRIMARY markets.

- In a bought deal, the investment banker buys the issue of bonds from the issuer and then resells them (i.e. they have underwritten the offer and the arrangement is termed a firm commitment)

### True/False A best-efforts offering, which is a form of negotiated offering, occurs when an investment banker purchases an entire issue to resell.

False! A firm commitment (NOT a best-efforts offering) is an arrangement where the investment banker purchases the entire issue and resells it.

### How does the Fed affect short-term interest rates?

The Federal Reserve Board's tools for affecting short-term interest rates are:

1. Discount rate

2. Open-market operations

3. Reserve requirement

4. Persuasion to influence banks' lending policies

### What are the two most important tools available to the Fed Reserve?

The two most important tools available to the Fed are:

- changing the discount rate (the rate at which banks can borrow from the Fed's discount window)

- open market operations (Fed's activity of buying and selling T-securities)

### What do yield curves plot and what are the most general shapes?

Yield curves plot:

Yield vs. Maturity

General Yield curve shapes:

- upward sloping

- downward sloping

- flat

- humped

### The slope of the yield curve is a function of...?

Since the yield curve depicts the yield on securities with different maturities, the slope of the curve between 2 maturities is a function of the maturity spread.

###
A normal sloped yield curve has a:

a) positive slope

b) zero slope

c) negative slope

A normally shaped yield curve is one in which the long-term rates > short-term rates.

Result: Yield curve = Positive slope

### Inverted yield curve

An inverted yield curve reflects the condition:

long term rates < short term rates

Result: Yield curve = downward (negative) slope

###
If investors expect future rates will be higher than current rates, the yield curve should be:

a) vertical

b) upward sloping

c) downward sloping

(B) When interest rates are expected to go up in the future, the yield curve will be upward sloping because:

x-axis: time (or maturity)

y-axis: interest rates

thus forming an upward sloping curve.

### Normal yield curve shape vs. Humped yield curve shape

Normal yield curve: long term rates > short term rates

Humped yield curve: rates in the middle of the maturity spectrum are HIGHER OR LOWER than those for both bonds with a short and long-term matuirty

###
A downward sloping yield curve generally implies:

a) shorter term bonds are less risky than longer term bonds

b) interest rates are expected to increase in future

c) interest rates are expected to decline in future

(C) A downward sloping yield curve implies that interest rates are expected to ↓ in the future.

Since a yield curve has

x-axis: time

y-axis: interest rates,

when the yield curve is downward sloping, it means that rates are expected to ↓

### Yield curve Theories:

1. Pure expectations theory

2. Liquidity preference theory

3. Market Segmentation theory

### 3 Theories of the yield curve and their implications for the shape of the yield curve: 1. Pure-expectations theory

Pure-expectations theory: rates at ↑ maturities depend only on expectations of future short-term rates.

- is consistent with ANY yield curve shape

### 3 Theories of the yield curve and their implications for the shape of the yield curve: 2. Liquidity Preference theory

Liquidity preference theory of the term structure states that longer-term rates reflect investors' expectations about future short-term rates.

- an ↑ liquidity premium is required to compensate investors for exposure to ↑ interest rate risk at longer maturities.

- theory can be consistent with a DOWNWARD sloping curve, IF an expected ↓ in short term interest rates outweighs the liquidity premium.

### 3 Theories of the yield curve and their implications for the shape of the yield curve: 3. Market Segmentation Theory

Market Segmentation Theory argues that lenders and borrowers have preferred maturity ranges.

- the shape of the yield curve is determined by the supply and demand for securities within each maturity range and is INDEPENDENT of the yield in other maturity ranges.

- consistent with any yield curve shape

- this theory in a somewhat weaker form is called "preferred habitat theory".

###
According to the expectations hypothesis, investors' expectations of ↓ inflation will result in:

a) a flat yield curve

b) upward-sloping yield curve

c) downward-sloping yield curve

(C) According to the expectations hypothesis, investors' expectations of ↓ inflation will result in a downward sloping yield curve.

- expectations hypothesis holds that the shape of the yield curve reflects investor expectations about the future behavior of inflation and market interest rates.

thus: if investor believe ↓ inflation, the yield curve = downward sloping

###
If the slope of the yield curve begins to ↑ sharply, it is usually an indication that:

a) stocks are offering abnormally high rates of return

b) rate of inflation is starting to ↑ or is expected to do so in near future

c) Fed has been aggressively driving up short-term interest rates

(B) According to the expectations hypothesis, ↑ long-term interest rates and, therefore, upward-sloping yield curves will occur if inflation rate starts to heat up or expected to do so in the near future.

### Correct interpretation of forward rates under the pure expectations hypothesis: Forward rates = expected future ____?

The pure expectations theory states that

forward rates are solely a function of EXPECTED future spot rates

###
Liquidity preference theory of the term structure of interest rates implies that the shape of the yield curve should be:

a) upward sloping

b) variable

c) flat or humped

(A) Liquidity preference theory definitely puts upward pressure on the long end of the term structure and by itself, would lead to an upward sloping yield curve.

###
This theory states that

(a) the yield curve is determined by the supply an demand for securities in particular maturity ranges and

(b) the yield curve has no specific shape

Theory: Market Segmentation Theory

The shape of the yield curve under this theory is determined by the supply and demand for securities within a given maturity range.

- no specific shape of the yield curve is implied by this theory.

###
According to the pure expectations theory, an upward sloping yield curve implies:

a) interest rates are expected to ↑ in future

b) longer term bonds are riskier than short term bonds

c) interest rates are expected to ↓ in future

(A) According to the pure expectations theory, shape of the yield curve results from the interest rate expectations of market participants.

- more specifically, it holds than any long-term interest rate simply represents the geometric mean of current and future 1-year interest rates expected to prevail over the maturity of the issue.

- expectations theory can explain any shape of the yield curve.

- expectations for RISING short-term rates in the future cause a RISING (upward-sloping) yield curve.

- expectations for FALLING short-term rates in the future will cause long-term rates < current short term rates, and yield curve will DECLINE (downward-sloping).

Thus, upward sloping yield curve implies that interest rates are expected to ↑ in future.

### True/False The preferred habitat theory suggests that investors prefer to stay within a particular maturity range of the yield curve regardless of yields in other maturity ranges.

False! The preferred habitat theory states that investors prefer to stay within a particular maturity range, but WILL MOVE from their preferred range to another area on the curve to achieve higher yields.

### True/False An upward sloping yield curve can be consistent with the liquidity preference theory EVEN with expectations of declining short term interest rates.

True! With the liquidity preference theory, the yield curve can remain upward sloping EVEN if the short term rates are predicted to ↓ AS LONG AS liquidity premium is sufficiently large.

### True/False The liquidity preference theory holds that the yield curve has an upward-sloping bias.

True! The liquidity preference theory suggests an upward-sloping yield bias with regard to the shape of the yield curve because:

- investors generally prefer the ↑ liquidity and ↓ risk that accompanies short-term securities

- so as a result: require premium (↑ yields) to get them to invest in longer-term securities.

- however, yield curve can still be downward-sloping even with the liquidity premium, for example, if short-term interest rates are expected to ↓ sharply in the future.

### 6-mo spot rate = 4% and 1-year annualized spot rate = 9% (4.5% on a semi-annual basis). Based on pure-expectations theory of interest rates, the implied 6-mo rate six months from now is:

₁r₁ = [(1 + R₂)² / (1 + R₁)¹] - 1 = [(1.045)² / (1.04)¹] - 1

[1.092 / 1.04] - 1 = 0.05

the implied 6-mo rate six months from now is 5%

### Treasury spot rates

Treasury spot rates are the appropriate discount rates for single cash flows (coupon or principal payments) from a US Treasury security, given the time until the payment is to be received.

###
The Treasury spot rate yield curve is closest to which of the following curves?

a) Par bond yield curve

b) Zero-coupon bond yield curve

c) Forward yield curve rate

(B) The spot rate yield curve shows the appropriate rates for discounting single CFs occurring at different times in the future.

- conceptually, these rates ≈ yields on zero-coupon bonds

a) Par bond yield curve shows the YTMs on coupon bonds by maturity.

c) Forward rates = expected future short-term rates

### True/False The spot rates used for different time periods that produce a value equal to the market price of a Treasury bond are called "forward rates" or "future expected spot rates".

False! The spot rates used for different time periods that produce a value ≈ market price of a T-bond are called "arbitrage-free Treasury spot rates".

###
3 Types of yield spreads:

1. Absolute Yield Spread

Absolute Yield Spread:

= [Yield on a particular security/sector] - [Yield on a reference (benchmark) security/sector]

- The "benchmark" is often on-the-run Treasury securities of like maturity.

###
3 Types of yield spreads:

2. Relative Yield Spread

Relative Yield Spread:

- is the "absolute yield spread" expressed as a % of the Benchmark Yield.

- arguably a superior measure to the absolute spread because it will reflect changes in the level of interest rates even when the absolute spread ≈ constant.

###
3 Types of Yield Spreads:

3. Yield Ratio

Yield Ratio

= Yield on a security/sector /

Yield on a benchmark security/sector

= 1 + Relative Yield Spread

### Yield Spread =

Yield Spread = Yield to Maturity( of the bond) - Yield to Maturity (of the reference)

### Credit Spread =

Credit Spread = (Yield of Bond A) - (Yield of Bond B) due to differences in their credit ratings.

Credit spreads ↓ when the economy = healthy and expanding.

Credit spreads ↑ when economy = contractions or recessions

- reflects a flight to ↑ quality by investors.

### How is default risk and credit spreads related? How is greater uncertainty and credit spreads related?

The probability that corporations may default ↑ and causes credit spreads to widen.

With greater uncertainty, investors require a higher return for taking on more risk, and thus credit spreads widen.

### What options increase yields and yield spreads and what options decrease yield and yield spreads as compared to option-free bonds?

Options that ↑ yields & yield spreads:

1. Call options

2. Prepayment options

Options that ↓ yields & yield spreads:

1. Put options

2. Conversion options

### As compared to an equivalent non-callable bond, a callable bond's yield should be: a) higher b) lower

Higher! A callable bond favors the issuer.

- value of the bond is discounted by the value of the option, which means the yield will be ↑

### As compared to an equivalent nonputable bond, a putable bond's yield should be: a) higher b) lower

(B) Lower!

A putable bond favors the buyer (investor).

- premium will be paid for the option → yield ↓

### Bonds with less liquidity must offer a ↑ or ↓ yield?

Bonds with less liquidity are less desirable and must offer a HIGHER yield.

Larger bond issues are more liquid and thus will have LOWER yield spreads.

### True/False As compared to a bond sold as part of a large issue, an otherwise equivalent bond sold as part of a SMALLER issue will be sold for a lower price and a higher yield to maturity.

True! Bonds that are sold as part of a smaller issue have higher liquidity risk than bonds that are sold in a large issue.

- investors will demand a higher yield to maturity to cover the liquidity risk

- Bond P from smaller issues < bond P from larger issues.

###
To compare a tax-exempt bond with a taxable issue, use one of 2 methods:

1. After-tax yield

2. Taxable-equivalent yield

1. After-tax yield

= taxable yield × (1 − marginal tax rate)

...and compare it to tax-EXEMPT yield.

2. Taxable-equivalent yield

= Tax-free yield / (1 - marginal tax rate)

...and compare it to the a TAXABLE yield

### LIBOR

LIBOR for various currencies is determined from rates at which large London banks loan money to each other.

- is the most important reference rate globally for

1. floating-rate debt and

2. short-term loans

...of various maturities.

### Fed funds rate vs. LIBOR

The Fed funds rate = rate paid on interbank loans within the US.

LIBOR = interest paid on negotiable CDs by the banks in London.

- determined every day by the British Bankers Association.

###
The most important LIBOR rate for funded investors is the:

a) 20 year rate

b) 10 year rate

c) 1 year or less rate

(C) A funded investor is "one who borrows to invest".

- investors typically borrow short-term and the interest rate on their loan is typically the short-term LIBOR + margin (e.g. LIBOR plus 30 bp)

### Steps to value a bond:

To value a bond:

1. Estimate the amount and timing of the bond's future pmts of int and principal

2. Determine the appropriate discount rate(s).

3. Calculate the sum of the present values of the bond's CFs.

### What adds complexity to the estimation of bond values?

Certain bond features:

- embedded options (call/put features or sinking fund provisions)

- convertibility (conversion or exchange privilege)

- floating rates (variable rather than fixed rates)

- increased credit risk

...can make the estimation of future CFs uncertain, which adds complexity to the estimation of bond values.

### Compute the value of an option-free coupon bond =

To compute the value of an option-free coupon bond...

1. Value the coupon pmts as an annuity

2. Add the PV of the principal repayment at maturity

### Compute the value of a zero-coupon bond:

Value of a zero-coupon bond is calculated using the semi-annual discount rate, i ≈ 1/2(annual YTM)

Bond Value = Maturity Value / (1 + i) ^ (# of yrs x 2)

###
If current interest rates < coupon rate of the bond, the bond will be issued at a... a) discount

b) premium

Since coupon rate > current interest rates → Premium bond (current value > Par Value)!

###
Annual coupon rate = 6%

Compute the PV of a security by using a discount rate = 7%

a) Premium Bond

b) Discount Bond

If the annual coupon rate < discount rate used on the bond → Discount Bond (current value < Par Value)!

###
A coupon bond has a YTM = 10%

What is the value of the bond if the coupon rate = 12%?

A) Discount Bond

B) Premium Bond

Since coupon rate > YTM → Premium bond (current value > Par Value)!

### An investor buys a 25 year, 10% annual pay bond for $900 and will sell the bond in 5 yrs when estimates the yield will be 9%. Will the price for which the investor expects to sell the bond be lesser or greater than the Par Value?

Since coupon rate (10%) > expected yield @ selling time (9%) → bond selling price > Par Value → "sell at a premium"

### A coupon bond that pays interest semi-annually has a par value = $1000, matures in 5 yrs, and has a YTM = 10%. What is the value of the bond today if the coupon rate = 8%? Is the value greater or lower than the Par Value?

If the coupon rate (8%) < YTM (10%) → bond's value today is "at a discount" to Par Value.

###
Using the following spot rates for pricing the bond, what is the present value of a three-year security that pays a fixed annual coupon of 6%?

Year 1: 5.0%

Year 2: 5.5%

Year 3: 6.0%

Present Value = 6/1.05 + [6/1.055]² + [106/1.06]³ = 100.10

Note: the "106" at the end includes the principal payment and coupon payment at maturity.

### When interest rates do not change, what happens to a bond's price over time?

When interest rates (yields) remain ↔, a bond's price → → par value as time passes and the maturity date approaches.

- to compute the change in value that is attributable to the passage of time, revalue the bond with a smaller number of periods to maturity.

### How can a change in bond value attributable to a change in the discount rate be calculated?

The change in value that is attributable to a change in the discount rate can be calculated: as the ∆ in the bond's present value based on the new discount rate (yield).

### When is the Treasury spot yield curve considered to be "arbitrage-free"?

A Treasury spot yield is considered "arbitrage-free" if the present values of Treasury securities calculated using these spot rates ≈ equilibrium market prices

### If bond prices ≠ arbitrage-free values...how can dealers make arbitrage profits?

If bond prices ≠ arbitrage-free values...dealers can generate arbitrage profits by

1. buying the lower-priced alternative → either

a) the bond or

b) the individual cash flows

2. selling the higher-priced alternative → either

a) the individual cash flows or

b) a package of the individual CFs ≈ bond

### What are the 3 sources of return from investing in a bond?

3 sources of return to a coupon bond:

1. Coupon interest payments

2 Reinvestment income on the coupon cash flows

3. Capital gain or loss on the principal value

### Yield to Maturity (YTM)

YTM for a semi-annual pay coupon bond:

= 2 x semiannual discount rate

...that makes the PV of the bond's promised CFs = Market Price + Accrued Interest

YTM for an annual pay coupon bond:

= annual discount rate

...that makes the PV of the bond's promised CFs = Market Price + Accrued interest

### Yield to Call (or put)

Yield to call (or put) is calculated as a YTM

...BUT with the

n = # of periods until the call (or put)

Maturity value or FV = call (or put) Price

### Cash Flow Yield

Cash Flow Yield is a monthly internal rate of return based on:

1. a presumed prepayment rate and

2. the current mkt Price of a mortgage backed or asset-backed security.

### What are 4 traditional yield measures for fixed-rate bonds and their common assumptions?

Traditional yield measures for fixed-rate bonds:

1. Yield to Maturity

2. Current Yield

3. Yield to Call (or Put)

4. Cash Flow Yield

These yield measures are limited by their common assumptions that:

1. all CFs can be discounted at the same rate

2. the bond will be held to maturity w/ all coupons reinvested to maturity at a rate of return = bond's YTM

3. all coupon payments will be made as scheduled.

### When is YTM not the realized yield on an investment?

YTM (Yield to Maturity) is NOT the realized yield on an investment unless the reinvestment rate = YTM

### What is the amount of reinvestment income required to generate the YTM over a bond's life?

Amt of reinvestment income required to generate the YTM over a bond's life is:

(Purchase price of the bond compounded at the YTM until maturity) - (∑ bond's interest and principal CFs)

"the difference between the purchase price of the bond, compounded at the YTM until maturity, and the sum of the bond's interest and principal cash flows."

### What factors affect reinvestment risk?

Reinvestment risk ↑ when

...coupon rate ↑ (and maturity is held ↔)

...bond has a longer maturity (and coupon rate is held ↔)

### Bond Equivalent Yield of an annual pay bond

BEY of an annual-pay bond is:

BEY = [ (√1 + annual-pay YTM) - 1 ] x 2

### "Annual-pay yield" can be calculated from the YTM of a semi-annual pay bond

"Annual-pay yield" can be calculated from the YTM of a semi-annual pay bond as:

EAY = [ 1 + (semi-annual pay YTM / 2) ]² - 1

### Theoretical Treasury spot rate curve -

The theoretical Treasury spot rate curve is derived by calculating the spot rate for each successive period N based on...

1. the spot rate for period N - 1 and

2. market price of a bond with N coupon payments

### How to compute the value of a bond using spot rates:

To compute the value of a bond using spot rates:

1. Discount each separate CF using the spot rate corresponding to the N (# of periods) until the CF is to be received.

### 3 commonly used yield spread measures:

1. Nominal spread: bond YTM - Treasury YTM

2. Zero-volatility spread (Z-spread or static spread): the equal amount of additional yield that must be added to each Treasury spot rate to get:

→ spot rates that will produce a PV for a bond = market Price

3. Option-adjusted spread (OAS): spread to the yield curve after adjusting for the effects of embedded options.

- OAS reflects the spread for CREDIT risk and LIQUIDITY risk primarily.

### When is there no difference & when is there the greatest difference between the nominal and Z-spread?

There is no difference between nominal spread and Z spread when the yield curve = flat

Difference between the spreads:

- the steeper the spot yield curve

& the earlier bond principal is paid (amortizing securities),

the greater the difference in nominal spread and Z-spread.

### Option cost of a bond with an embedded option, especially callable and putable bonds.

Option cost for a bond w/ an embedded option is...

Option Cost = Z-spread - OAS

CALLABLE bonds: z-spread > OAS

and option cost > 0

PUTABLE bonds: z-spread < OAS

and option cost < 0

### Define forward rates

Forward rates are current lending/borrowing rates for SHORT-term loans to be made in future periods

### Calculate spot rates from forward rates

For a maturity of N periods

spot rate = geometric mean of forward rates over the N periods.

Note: same relation can be used to solve for a forward rate, given spot rates for 2 different periods

### Calculate the value of a bond using forward rate

To value a bond using forward rates:

1. Discount the CFs at times 1 → N by he product of (1 + each forward rate) for periods 1 → N.

2. ∑ the above

### What is the probable change in price of a 30-year semiannual 6.5% coupon, $1000 par value bond yielding 8% when the nominal risk-free rate changes from 5% to 4%?

Price at 8% is N = 60, FV = $1,000, I = 4%, PMT = $32.50, Pmt = (6.5%/2)/100 x $1000 = $32.50

CPT PV = $830.32;

Price at 7% is N = 60, FV = $1,000, I = 3.5%, FV = $1,000, CPT PV = $937.64.

[PV = $937.64] - [PV = $830.32] = $107.31.

### What is the value of a zero-coupon bond if the term structure of interest rates is flat at 6% and the bond has two years remaining to maturity?

Zero-Coupon Bond Price = 100/1.03⁴ = 88.85.

### Assume that there are no transaction costs and that securities are infinitely divisible. If an 8% coupon paying Treasury bond that has six months left to maturity trades at 97.54, and there is a Treasury bill with six months remaining to maturity that is correctly priced using a discount rate of 9%, is there an arbitrage opportunity?

Yes, there is an arbitrage opp - the coupon bond price is too low.

The coupon bond has a cash flow at maturity of 104, which discounted at 9% results in a bond price of 99.52. Therefore, the bond is underpriced.

An arbitrage trade can be set up by short-selling 1.04 units of the T-bill at 99.52 and then using the proceeds to buy 1.02 units of the coupon bond.

###
The arbitrage-free bond valuation approach can best be described as the:

a) use of series of spot interest rates that reflect the current term structure

b) use of a single discount factor

c) geometric average of the spot interest rates

(A) The use of multiple discount rates (i.e., a series of spot rates that reflect the current term structure) will result:

- more accurate bond pricing

- also, will eliminate any meaningful arbitrage opportunities. That is why the use of a series of spot rates to discount bond cash flows is considered to be an arbitrage-free valuation procedure.

###
Which of the following packages of securities is equivalent to a three-year 8% coupon bond with semi-annual coupon payments and a par value of 100? A three-year zero-coupon bond:

A) with a par of 100 and six zero-coupon bonds with a par value of 8 and maturities equal to the time to each coupon payment of the coupon bond.

B) with a par of 100 and six zero-coupon bonds with a par value of 4 and maturities equal to the time to each coupon payment of the coupon bond.

(B)This combination of zero-coupon bonds has exactly the same cash flows as the above coupon bond and therefore it is equivalent to it.

### A 2-year option-free bond (par value of $10,000) has an annual coupon of 15%. An investor determines that the spot rate of year 1 is 16% and the year 2 spot rate is 17%. Using the arbitrage-free valuation approach, the bond price is:

We can calculate the price of the bond by discounting each of the annual payments by the appropriate spot rate and finding the sum of the present values.

Price = [1,500/(1.16)] + [11,500/(1.17)2] = $9,694. Or, in keeping with the notion that each cash flow is a separate bond, sum on calculator:

N=1, I/Y=16.0, PMT=0, FV=1,500, CPT PV=1,293

N=2, I/Y=17.0, PMT=0, FV=11,500, CPT PV=8,401

Price = 1,293 + 8,401 = $9,694.

### To value non-Treasury securities, what is added to each of the treasury spot yields?

For valuing non-Treasury securities, a credit-spread is added to each treasury spot yield.

### The credit spread is a function of what 2 factors?

The credit spread is a function of

1. Default risk

2. Term to Maturity

### True/False The Yield to Maturity is the bond's internal rate of return.

True! The bond's YTM ≈ bond's IRR

that equates ∑ all CFs = Bond Price

Note: Current spot rates have nothing to do with the bond's YTM

### A bond has a par value of $1,000, a time to maturity of 20 years, a coupon rate of 10% with interest paid annually, a current price of $850, and a yield to maturity (YTM) of 12%. If the interest payments are reinvested at 10%, the realized compounded yield on this bond is:

Ans = 10.9% The realized yield would have to be between the reinvested rate of 10% and the yield to maturity of 12%.

Calculating the realized yield: The value of the reinvested coupons at the maturity date is: N = 20; I/Y = 10; PMT = 100; PV = 0; CPT FV = 5,727.50. Adding the principal repayment, total cash at maturity is $6,727.50.

Realized yield: N = 20; PMT = 0; PV = -850; FV = 6727.5; CPT I/Y = 10.8975.

### An investor purchased a 6-year annual interest coupon bond 1 yr ago. The coupon interest rate was 10% and the par value was $1,000. At the time he purchased the bond, YTM was 8%. If he sold the bond after receiving the first interest payment and the YTM continued to be 8%, his annual total rate of return on holding the bond for that year would have been:

Ans = 8.0%

Purchase price N = 6, PMT = 100, FV = 1,000, I = 8

compute PV = 1,092.46

Sale price N = 5, PMT = 100, FV = 1,000, I = 8

compute PV = 1,079.85

% return = [(1,079.85 - 1,092.46 + 100) / 1,092.46] x 100 = 8%

### A 6-year annual interest coupon bond was purchased 1 yr ago. The coupon rate is 10% and par value is $1,000. At the time the bond was bought, YTM = 8%. If the bond is sold after receiving the first interest payment and the bond's changed YTM = 7%, the annual total rate of return on holding the bond for that year would have been:

Ans = 11.95%

Price 1 year ago N = 6, PMT = 100, FV = 1,000, I = 8, Compute PV = 1,092

Price now N = 5, PMT = 100, FV = 1,000, I = 7, Compute PV = 1,123

% Return = (1,123.00 + 100 − 1,092.46)/1,092.46 x 100 = 11.95%

###
If an investor holds a bond for a period less than the life of the bond, the rate of return the investor can expect to earn is called:

a) expected return, horizon return

b) approximate yield

c) bond equivalent yield

(A) "Expected return, or horizon return"

The horizon return is the total return of a given horizon such as 5 years on a ten year bond.

###
A bond is selling at a discount relative to its par value. So, which of the following is true:

A) yield to maturity < coupon rate < current yield.

B) current yield < coupon rate < yield to maturity.

C) coupon rate < current yield < yield to maturity.

C) coupon rate < current yield < yield to maturity.

When a bond is selling at a discount, it means that the bond has a larger YTM (discount rate that will equate the PV of the bond's cash flows to its current price)

...than its current yield (coupon payment/current market bond price) and coupon payment.

### Does current yield take into consideration gains/losses and reinvestment income?

The current yield of a bond only considers INTEREST INCOME.

Current Yield = Coupon Pmt / Bond Price

Does NOT consider:

- gains/losses

- reinvestment income

### True/False YTM is based on the assumption that any payments received are reinvested at the current yield.

False!

Reinvestments occur at the YTM (and NOT at the current yield).

- the YTM will find the PV of a future value and associated payments.

- YTM is the discount rate that will set the PV of the pmts = bond Price

###
If interest rates and risk factors remain constant over the remainder of a coupon bond's life, and the bond is trading at a discount today, it will have a:

A) negative current yield and a capital gain.

B) positive current yield, only.

C) positive current yield and a capital gain.

(C)

A coupon bond will have a (+) current yield. If it is trading at a discount, it will have a capital gain because its value at maturity > its price today.

### Define YTM

The Yield to Maturity (YTM) is the interest rate that will make

PV of CFs from a bond = Market Price + Accrued Interest

- YTM is the most popular of all yield measures used in the bond marketplace.

### Total Dollar Return is made up of 3 sources

Total dollar return is made up of 3 sources:

1. Coupons

2. Principal

3. Reinvestment income

Note: all coupons and principal payments must be reinvested at a specific rate of return (i.e. YTM) for the total dollar return % = YTM %

### At what rate must the coupons be reinvested, for the realized yield = YTM?

For the realized yield = YTM,

coupon reinvestment must occur @ that YTM

- whether reinvesting the coupons at the coupon rate will result in a realized yield ↑ or ↓ than the YTM depends on whether the bond is a:

- discount (coupon < YTM) or

- premium (coupon > YTM)

### Calculate BEY given a monthly yield =

BEY = 2 × [(1 + monthly yield)⁶ − 1]

Note: always convert all yield % into decimal form.

### Annual Equivalent Yield =

Annual equivalent yield =

[1 + (nominal yield/# of pmts per yr)]^(# of pmts per yr)

- 1

### What is the semiannual-pay bond equivalent yield on an annual-pay bond with a yield to maturity of 12.51%?

The semiannual-pay bond equivalent yield of an annual-pay bond = 2 × [ √(1 + YTM on the annual-pay bond) - 1] = 12.14%.

###
Yr: 0.5 → Spot: 5.227%; Yr: 1 → Spot: 5.5537%; Yr: 1.5 → 5.5444%

Compute the price of a Treasury note with a coupon rate = 5.375% semi-annually and 1.5 yrs left to maturity.

The semiannual coupon payment as a percentage of par is 5.375 / 2 = 2.6875. The price is:

2.6875 / [1 + (0.055227 / 2)]

+ 2.6875 / [1 + (0.055537 / 2)]²

+ 102.6875 / [1 + (0.055444 / 2)]³

= 2.6153 + 2.5442 + 94.5999 = 99.7594

###
3-year, annual-pay bond:

Par value of $1,000

Coupon of 8%

Current price of $1,100

1-year spot interest rate is 5%

2-year spot interest rate is 6%

The 3-year spot rate is:

The value of the bond = PV of discounted future cash flows, using the appropriate spot rate as the discount rate for each CF.

- coupon payment of the bond is $80 (0.08 × 1,000) and the face value is $1,000.

Hence, bond price of 1,100= 80/(1.05)+ 80/(1.06)² + 1,080/(1 + 3-year spot rate)³.

Using the yx key on calculator, we can solve for the 3-year spot rate of 4.27%

###
Maturity, Coupon & Price:

6 mo → 3.0% → $100

1 yr → 4.0% → $100

18 mo → 5.0% → $100

Using bootstrapping, what are closest to the theoretical Treasury spot rate curve?

The bond with 6 mo. left to maturity has a semiannual discount rate of 0.03/2 = 0.015 or 3.0% on an annual bond equivalent yield (BEY) basis.

Since the bond will only make a single payment of 101.50 in 6 mo, the YTM = spot rate for CFs to be received 6 mo. from now.

The 1 yr bond will make 2 payments, (1) in 6 mo. of 2 and (2) in 1 1 yr of 102. We can solve for the one-year spot rate in the equation:

(2/1.015) + 102 / (1 + {spot rate₁ / 2})² = 100 where spot rate₁ = annualized 1-yr spo rate = 4.01005%

Using the 6-mo and 1-yr spot rates, we can use the same approach to find the 18-mo spot rate from the equation:

2.5/1.015 + 2.5/(1.02005025)²

+ 102.5 / ( 1 + {S₁₈/2} )³ = 100

S₁₈ = annualized 18 mo spot rate = 5.0338823%.

###
2-year, annual-pay bond:

Par value of $1,000

Coupon of 4%

1-year spot interest rate is 2%

2-year spot interest rate is 5%

Current price of bond = ?

The value of the bond is simply the present value of discounted future cash flows, using the appropriate spot rate as the discount rate for each cash flow.

Coupon pmt = $40 (0.04 × 1,000).

Bond price = 40/(1.02) + [1000 + 40]/(1.05)² = $982.53.

### Can spot interest rates be used to value a callable bond?

Yes! Any complex debt instruments (call & put bonds, mortgage-backed securities) can be seen as the ∑ PV of individual CFs, with each CF discounted @ appropriate zero-coupon bond spot rate.

Note: While the appropriate spot interest rate can be used to discount each CF, determining the actual pattern of CFs is uncertain due to the possibility of the bond being called away.

### True/False To calculate a theoretical Treasury spot rate curve from the yields on coupon bonds, we must know at least 2 actual Treasury spot rates.

False! If we know one actual spot rate, we can calc the theoretical spot rate for the next longer period. Now, with the two spot rates, calc the next theoretical sport rate and so on up the coupon curve.

### What are some common limitations of the nominal spread?

Nominal spread: bond YTM - Treasury YTM

Because nom spread is based on YTM, it suffers the same shortcomings as YTM.

- assumes that all CFs can be discounted at the same rate (i.e. assumes a flat yield curve)

- assumes all coupon pmts will be received in prompt and timely fashion, and reinvested at maturity, at a rate of return ≈ bond's YTM or its BEY.

### True/False The option-adjusted spread (OAS) for a putable bond is the Z-spread - cost of option

False! Since the buyer of a putable bond must pay extra for the put option...

OAS spread for a putable bond = Z-spread + put option cost (in %)

### True/False OAS is the spread added to the Treasury spot rate curve that the bond would have it it were option-free.

True!

###
True/False OAS spread is the spread that accounts for non-option characteristics like

- credit risk

- liquidity risk and

- interest rate risk

True!

### Zero-volatility spread vs. Nominal Spread

Zero-volatility spread (z-spread) - is the interest rate that is added to each zero-coupon bond spot rate so that → PV of risky bond's CFs = Market Value

Z-spread = OAS + Cost of call option %

Nominal spread - added to the YTM of a similar maturity Treasury bond that will then → (YTM of T-bond + nom spread) ≈ YTM of the risky bond

### True/False Z-spread may be used for bonds that contain call options.

False! Z-spread is used for risky bonds that do NOT contain call options in an attempt to improve the shortcomings of the nominal spread.

### The six-year [R₆] spot rate is 7% and the five-year [R₅] spot rate is 6%. The implied one-year forward rate five years [₁r₅] from now is closest to:

₁r₅= [(1 + R₆)⁶ / (1 + R₅)⁵] - 1 = [(1.07)⁶/(1.06)⁵] - 1 = [1.5 / 1.338] - 1 = 0.12

### The one-year spot rate is 6% and the one-year forward rates starting in one, two and three years respectively are 6.5%, 6.8% and 7%. What is the four-year spot rate?

For a maturity of N periods

spot rate = geometric mean of forward rates over the N periods. So, the four-year spot rate is computed from forward rate as follows:

Four-year spot rate = [(1 + 0.06)(1 + 0.065)(1 + 0.068)(1 + 0.07) ]¹/⁴ - 1 = 6.57%

### Suppose that the six-month spot rate is equal to 7% and the two-year spot rate is 6%. Which of the following is the best answer concerning the level of the one-and a half-year forward rate starting six months from now? The forward rate has to:

The following relationship has to hold:

(1 + spot rate₀,₀.₅ / 2 )¹ x (1 + forw rate₀.₅,₂ / 2 )³ = (1 + spot rate₀,₂ / 2)⁴.

For this relationship to hold the forward rate has to be less than 6%.

### Given the implied forward rates of: R₁ = 0.04; ₁r₁ = 0.04300; ₁r₂ = 0.05098; ₁r₃ = 0.051005, what is the theoretical 4-period spot rate?

For a maturity of N periods

spot rate = geometric mean of forward rates over the N periods.

4-period spot rate calculated from forward rates:

[(1.04)(1.043)(1.05098)(1.051005)]¹/⁴ − 1

### Given that the two-year spot rate is 5.89% and the one-year forward rate one-year from now is 6.05%, assuming annual compounding what is the one year spot rate?

Spot rate =

Spot rate₀,₁ = [ (1 + spot rate₀,₂)² / (1 + forward rate₁,₂)¹ ] - 1

= (1 + 0.0589)² / (1 + 0.0605)¹ - 1 = 5.73%

### Given that the one-year spot rate is 6.05% and the two-year spot rate is 7.32%, assuming annual compounding what is the one-year forward rate starting one year from now?

Forward rate₁,₂ = (1 + spot rate₀,₂)² / (1 + spot rate₀,₁)¹ - 1 = (1 + 0.0732)² / (1 + 0.0605)¹ - 1 = 8.61%

### The one-year spot rate is 5% and the two-year spot rate is 6.5%. What is the one-year forward rate starting one year from now?

One year Forward rate = ( 1 + spot rate)² / (1 + spot rate)¹ = 1.065² / 1.05 = 8.02%

### True/False Forward rates do not account for the market's tolerance for risk.

False! Spot rates are the result of:

1) the market participant's tolerance for risk and

2) their collective view regarding the future path of interest rates

If we assume that these results are purely a function of expectations, we can use spot rates to estimate the market's consensus on forward interest rates.

### Full valuation approach (~scenario analysis approach) vs. Duration/convexity approach for measuring interest rate risk

Full valuation approach - measuring interest rate risk involves using a price model to value individual bonds

- can be used to find the price impact of any scenario of interest rate/yield curve changes

- advantages: flexibility & preciision

Duration/convexity approach - summary measures of interest rate risk

- simpler to use for a portfolio of bonds than the full-valuation approach.

- theoretically correct ONLY for parallel shifts in the yield curve.

### 4 steps in the Full Valuation approach:

4 steps in the full-valuation approach:

1. Begin with the current market yield and price

2. Estimate the hypothetical changes in required yields

3. Recompute the bond prices using the new required yields

4. Compare the resulting price changes.

###
Holding other factors constant, a coupon bond's interest rate risk ↑ when:

a) current yield is higher

b) yield to maturity is lower

c) coupon rate is higher

(B)

A coupon's bond's interest rate risk ↑ when Yield to Maturity is ↓

The higher interest rate risk is caused is from having a low yield to maturity ("initial yield"). A higher coupon yield and a higher current yield will cause a ↓ interest rate risk

### 3 features that determine the magnitude of Bond Price Volatility:

↑ Bond Price Volatility...when:

↓ Coupon

↑ Maturity term

↓ Initial Yield

### Price Volatility in Callable & Putable bonds

Callable bonds & Prepayable securities:

- have ↓ price volatility (↓ duration) at ↓ yields

...as compared to option-free bonds

Putable bonds:

- have ↓ price volatility at ↑ yields

...as compared to option-free bonds

###
A $1,000 face, 10-year, 8.00% semi-annual coupon, option-free bond is issued at par (market rates are thus 8.00%). Given that the bond price decreased 10.03% when market rates increased 150 basis points (bp), which of the following statements is CORRECT? If market yields:

a) decrease by 150bp, the bond's price will increase by more than 10.03%.

b) decrease by 150bp, the bond's price will increase by 10.03%.

(A)

(B) is false because of positive convexity - bond prices ↑ faster than they fall. Positive convexity applies to both dollar∆ and % Price∆. For any given absolute change in yield, the ↑ in price > ↓ in price for a fixed-coupon, noncallable bond.

As yields ↑, bond prices ↓, and the price curve gets flatter, and changes in yield have a smaller effect on bond prices.

As yields ↓, bond prices ↑, and the price curve gets steeper, and changes in yield have a larger effect on bond prices.

Here, for an absolute 150bp change, the price increase would be more than the price decrease. For a 100bp increase, the price decrease would be less than that for a 150bp increase.

### Option-free bonds & convexity

Option-free bonds:

- have price-yield relationship that is curved

- curved ~ "convex towards the origin"

- exhibit POSITIVE convexity

- Bond Prices ↓ less in response to an ↑ in yield < Bond Prices ↑ more in response to an equal-sized ↓ yield

...bond prices fall less than they rise in response to ∆yield

### Callable bonds & convexity

Callable bonds:

- exhibit NEGATIVE convexity at LOW yield levels (option-free bonds have (+) convexity)

- Bond prices ↑ less in response to a ↓ in yield < Bond Prices ↓ more in response to an equal-sized ↑ yield

...bond prices rise less than they fall in response to ∆yield

### What is the only type of fixed income security have a negative convexity?

The only type of fixed income security with a negative convexity → callable bonds

### How does ∆ yields affect bond prices and the shape of the yield curve?

As yields ↑, bond prices ↓, price curve ≈ flatter

...and changes in yield have a smaller effect on bond prices.

As yields ↓, bond prices ↑, price curve ≈ steeper

...and changes in yield have a larger effect on bond prices.

### Modified duration

Modified duration is a good approx of price changes for an option-free bond BUT only for relatively small changes in interest rates.

- as rate changes get larger, the curvature of the bond price/yield relationship becomes more prevalent → the linear estimate becomes increasingly inaccurate.

### What is the error in the modified duration estimate of the price/yield relationship?

The modified duration estimate = linear estimate

- ∆price is the same for each ∆bp of yield

- error in the estimate is due to the curvature of the actual price path..."degree of convexity".

### Do mortgage-backed securities have negative convexity?

Mortgage-backed securities (MBS) may have negative convexity because when interest rates fall, owners will refinance for lower rates, thus prepaying the outstanding principal and increasing the interest rate risk that investors of MBS may incur.

### True/False The price/yield relationship is concave for low yields for the callable bond and always convex for the option-free bond.

True! Since the issuer of a callable bond has an incentive to call the bond when interest rates ↓ in order to get cheaper financing, this puts an upper limit on the bond prices for low interest rates.

→ introduces the concave relationship between yields and prices.

### Positive convexity vs. Negative convexity

As interest rates change...

When bond prices go up faster than they go down → positive convexity

- bond price sensitivity is lowest when market yields are high, and highest when mkt yields are low.

When bond prices go down faster than they go up → negative convexity

###
If a put feature expires on a bond so that it becomes option-free, then the curve depicting the price and yield relationship of the bond will become:

a) more convex

b) inversely convex

c) less convex

Less convex!

When the option expires, the prices at the lower end of the curve will become lower. This will make the curve less convex.

### Effective Duration =

Effective Duration = (V-) - (V+) / 2V₀(∆y)

V- = estimated price if yield ↓ by a given amt

V+ = estimated price if yield ↑ by a given amt

V₀ = initial observed price

∆y = change in required yield (in decimals)

- calculated as the ratio of the avg % price ∆ for an equal-sized increase and decrease in yield, to the change in yield.

### A bond's yield to maturity decreases from 8% to 7% and its price increases by 6%, from $675.00 to $715.50. The bond's effective duration is closest to:

Effective duration = % change in price for a 1% change in yield, which is given as 6.

### An investor finds that for every 1% increase in interest rates, a bond's price decreases by 4.21% compared to a 4.45% increase in value for every 1% decline in interest rates. If the bond is currently trading at par value, the bond's duration is closest to:

Duration is a measure of a bond's sensitivity to changes in interest rates.

Effective Duration = (V-) - (V+) / 2V₀(∆y)

= (104.45 - 95.79) / (2 x $100 x 0.01) = 4.33

### What's the trick to converting basis points into decimal form?

Divide the basis point by 10,000

1 basis point = 0.0001

50 basis points = 0.005

100 basis points = 0.01

### A 30-year semi-annual coupon bond issued today with market rates at 6.75% pays a 6.75% coupon. If the market yield declines by 30 basis points, the price increases to $1,039.59. If the market yield rises by 30 basis points, the price decreases to $962.77. What is the approx percentage change in price for a 100 basis point change in the market interest rate?

Rework the effective duration formula. Effective Duration = (V-) - (V+) / 2V₀(∆y)

Approximate % change in price =

(price if yield down − price if yield up) / (2 × initial price × yield change expressed as a decimal).

Here, the initial price is par, or $1,000 because we are told the bond was issued today at par. So, the calculation is: (1039.59 − 962.77) / (2 × 1000 × 0.003) = 76.82 / 6.00 = 12.80.

### If bond prices fall 5% in response to a 0.5% increase in interest rates, what is the bond's effective duration?

Approximate percentage price change of a bond = - (duration) (∆y) =

-5 = - (duration) (0.5) = 10.

### Approximate % change in bond price =

Approximate percentage change in bond price = −duration x ∆y

∆y = change in yield in percent.

###
As coupon rates ↑, duration: ↑ or ↓?

As maturities increase, duration: ↑ or ↓?

As coupon rates ↑, duration ↓ because investors are receiving more cash flow sooner

As maturity ↑, duration ↑ because payments are spread out over a longer period of time.

### Most intuitive interpretation of duration:

The most intuitive interpretation of duration:

%∆ in bond price for a 1%∆ in YTM

### Macaulay duration & Modified duration are both based on:

Macaulay & Modified duration are based on a bond's promised cash flows.

- Macaulay duration does NOT take the current YTM into account as modified duration does.

### True/False Effective duration is the most appropriate measure of interest rate risk for bonds with embedded options.

True!

Effective duration is appropriate for estimating ∆Price in bonds with embedded options because:

- takes into account the effect of embedded options on a bond's cash flows

### Assumptions of modified vs. effective duration

Modified duration assumes that the CFs on the bond will NOT change (i.e. like "non-callable" bonds).

Effective duration considers expected changes in CFs that may occur for bonds with embedded options.

### Why is effective duration only an approximation?

Effective duration is an "approximation" because the duration calculation ignores the curvature of the price/yield graph.

### For option-free bonds, will there be a difference between modified duration and effective duration?

For option-free bonds, modified duration ≈ effective duration.

- both duration measures are based on the value impact of a parallel shift in a flat yield curve.

### Duration of a bond portfolio (given the duration of the bonds comprising the portfolio) =

Duration of a bond portfolio =

weighted avg of the individual bond durations

- weights are the proportions of total portfolio value in each bond position.

### Limitations of Portfolio Duration -

Portfolio Duration is limited because it gives the sensitivity of portfolio value only to yield changes that are equal for all bonds in the portfolio.

- assumes the yield change is equal for all bonds in the portfolio → parallel shift in the yield curve

- this is an unlikely scenario in most portfolios.

### How does convexity impact the duration measure?

Because of convexity...

- the duration measure ≈ poor approximation

of price sensitivity for yield changes that are NOT absolutely tiny.

Convexity adjustment accounts for the curvature of the price-yield relationship

###
Bond's percentage change in price =

given:

a) bond's duration

b) convexity

c) ∆ interest rates or yield

Incorporating both duration & convexity...

%∆P in response to a ∆yield =

{ [ (−Duration)(∆y) ] + [ (Convexity)(∆y)² ] } x 100

- if it's a ↓ yield → "−"∆y in both parts of the equation

### True/False Why is convexity a good thing for a bonholder: because when compare to a low convexity bond, a high convexity bond has better price changes, regardless of the direction of the yield change.

True! Relative to bonds with low convexity, the price of a bond with high convexity will ↑ MORE when rates decline and ↓ LESS when rates rise.

###
With respect to an option-free bond, when interest-rate changes are large, the duration measure will overestimate the:

A) fall in a bond's price from a given increase in interest rates.

B) increase in a bond's price from a given increase in interest rates.

C) final bond price from a given increase in interest rates.

(A)

When interest rates increase by 50 - 100 basis points or more...

→ duration measure overestimates the ↓ in the bond's price

###
For a given change in yields, the difference between the actual change in a bond's price and that predicted using the duration measure will be greater for:

A) a bond with less convexity.

B) a bond with greater convexity.

(B)

Duration is a linear measure of the relationship between a bond's price and yield.

- true relationship is not linear as measured by the convexity. When convexity is ↑, duration will be ↓ accurate in predicting a bond's price for a given change in interest rates.

###
How does the convexity of a bond influence the yield on the bond? All else the same, for a bond with high convexity investors will require:

a) higher yield

b) lower yield

(B) lower yield

Convexity is to the advantage of the BONDHOLDER because a high-convexity bond's price will decrease less when rates increase and will increase more when rates decrease than a low-convexity bond's price.

### True/False Convexity is more important when rates are unstable.

True! Since interest rates and price of bonds are inversely related, unstable interest rates will lead to larger price fluctuations.

- larger the ∆Price, the ↑ error will be introduced in determining the new price of the bond...especially if only duration is used.

...cuz duration assumes price-yield relationship is linear, when it is in fact a CURVED CONVEX line.

If duration alone is used to price the bond, the curvature of the line magnifies the error introduced by ∆y, and makes the convexity adjustment that much more important.

### Modified convexity vs. Effective Convexity

Effective convexity - considers expected changes in CFs that may occur for bonds with embedded options.

Modified convexity - does not consider the above.

### True/False For an option-free bond, modified convexity ≈ effective convexity.

True! For an option-free bond, modified convexity and effective convexity should be very near equal.

### Effective convexity calculation -

Calculation of effective convexity→ requires an adjustment in the estimated bond values to reflect any change in estimated cash flows due to the presence of embedded options.

Note: this is the same process used to calc effective duration.

###
Price Value of a basis point (PVBP) =

(given: duration)

PVBP is an estimate of the ∆ in the value of a bond or a bond portfolio...for a 1 basis point ∆ in yield.

1 basis point = 0.0001

PVBP = duration x 0.0001 x bond (or portfolio) value

### The price value of a basis point (PVBP) for a 7-year, 10% semiannual pay bond with a par value of $1,000 and yield of 6% is closest to:

PVBP = initial price - price if yield changed by 1 bps.

Initial Price → Price with change:

FV = 1000 → 1000

PMT = (0.05 x $1000) = $50 → $50

N = 7 x 2 = 14 → 14

i = 6%/2 = 3% → 3.005

CPT PV = 1225.92 → CPT PV = 1225.28

PVBP = 1,225.92 - 1,225.28 = 0.64

PVBP is always the absolute value

### What is the relationship of PVBP and a bond's price?

The larger the PVBP, the more volatile the bond's price

PVBP = price value of a basis point

### Impact of yield volatility on Interest rate risk of a bond

Yield volatility = std deviation of the changes in the yield of a bond.

Uncertainty about a bond's future price due to ∆yield results from both:

a) a bond's price sensitivity to ∆y AND

b) from volatility of it's yield in the market

###
In addition to effective duration, analysts often use measures such as Value-at-Risk (VaR) to estimate the price sensitivity of bonds to changes in interest rates because these measures also incorporate the effects of:

A) time to maturity.

B) embedded options.

C) yield volatility.

(C)

Measures of price risk such as VaR account for yield volatility. Effective duration includes the effects of time to maturity and embedded options.

- The volatility of a bond's yield should be considered along with the bond's effective duration when estimating its price sensitivity to interest rates.