Advertisement

38 terms

Steps of Valuing a Bond

+Estimate cash flows

+Determine appropriate discount rate

+Calculate the present value of the estimated cash flow

+Determine appropriate discount rate

+Calculate the present value of the estimated cash flow

Situations Where Estimating Cash Flows is Difficult

+Principal repayment stream is not known with certainty

+Coupon payments are not known with certainty

+Bond is convertible

+Coupon payments are not known with certainty

+Bond is convertible

Arbitrage Free Valuation

When a bond has each of its cash flows discounted using a discount rate that is specific to the maturity of each cash flow;

Spot rates used are required rate of returns on zero coupon bonds maturing at a given time;

The value of a bond based on spot rates must be equal to the value of its parts or there is an arbitrage opportunity

Spot rates used are required rate of returns on zero coupon bonds maturing at a given time;

The value of a bond based on spot rates must be equal to the value of its parts or there is an arbitrage opportunity

Steps of Arbitrage Free Valuation

*Value the security using spot values

*Compare the value to the market price

*Compare the value to the market price

Sources of Bond Return

+Coupon payments

+Recovery of principal at maturity

+Reinvestment income

+Recovery of principal at maturity

+Reinvestment income

Current Yield

The yield from the bond's annual coupon payments;

Offers little information;

Current Yield = (Annual Cash Coupon Payment)/(Bond Price)

Offers little information;

Current Yield = (Annual Cash Coupon Payment)/(Bond Price)

Yield to Maturity

The IRR of a bond's price and promised cash flows;

Stated as two times the semiannual coupon payments implied by the bond's price

Stated as two times the semiannual coupon payments implied by the bond's price

Bond Equivalent Yield =

[(1 + Annual YTM) ^ (1/2) - 1] * 2;

Referred to as the semiannual yield to maturity or semiannual-pay yield to maturity

Referred to as the semiannual yield to maturity or semiannual-pay yield to maturity

Yield to Call

The yield on callable bonds that are selling at a premium to par;

Can be less than the yield to maturity if the bond is trading at a premium;

Calculate the same way as yield to maturity but the call price is used instead of par and the time period only runs to the next call

Can be less than the yield to maturity if the bond is trading at a premium;

Calculate the same way as yield to maturity but the call price is used instead of par and the time period only runs to the next call

Yield to Worst

The worst yield outcome of any of the possible call provisions

Yield to Refunding

Used when a bond is callable and rates make sense for it to be called, but the bond covenants contain provisions giving protection from refunding until a future date;

Same calculation as yield to call but date used is the first date refunding is allowed

Same calculation as yield to call but date used is the first date refunding is allowed

Yield to Put

Used if a bond has a put option and is selling at a discount;

Calculated the same way as yield to maturity but with the put price as the price and put date as the date

Calculated the same way as yield to maturity but with the put price as the price and put date as the date

Cash Flow Yield

Used for mortgage-backed securities and other amortized asset-backed securities;

Includes assumptions on how prepayments are likely to occur;

Once monthly cash flow projections are made, can calculate a CFY as a monthly IRR based on the market price of the security;

Bond Equivalent Yield = [(1 + Monthly CFY) ^ 6 - 1] * 2

Includes assumptions on how prepayments are likely to occur;

Once monthly cash flow projections are made, can calculate a CFY as a monthly IRR based on the market price of the security;

Bond Equivalent Yield = [(1 + Monthly CFY) ^ 6 - 1] * 2

Limitation of Yield to Maturity

Doesn't tell the compounded rate of return that will be realized on a fixed income security;

Assumes reinvestment at the yield to maturity

Assumes reinvestment at the yield to maturity

When Bond at Par....

Coupon Rate = Current Yield = Yield to Maturity

When Bond at Discount...

Coupon Rate < Current Yield < Yield to Maturity

When Bond at Premium...

Coupon Rate > Current Yield > Yield to Maturity

Bootstrapping

Method of constructing a Treasury yield curve using the yield to maturities of different maturities

Steps of Bootstrapping

*Begin with 6-month spot rate

*Set value of the 1-year bond equal to present value of the cash flows with the 1-year spot rate divided by two as the only unknown

*Solve for 1-year spot rate

*Use 6-month and 1-year spot rates and equate the present value of the cash flows of the 1.5-year bond to its price, with 1.5-year bond as the only unknown

*Solve for 1.5-year bond

*Set value of the 1-year bond equal to present value of the cash flows with the 1-year spot rate divided by two as the only unknown

*Solve for 1-year spot rate

*Use 6-month and 1-year spot rates and equate the present value of the cash flows of the 1.5-year bond to its price, with 1.5-year bond as the only unknown

*Solve for 1.5-year bond

Nominal Spread

The difference between a bond's YTM and a similar Treasury's YTM;

Uses a single discount rate;

Ignores the shape of the yield curve and is technically only correct if yield curve is flat

Uses a single discount rate;

Ignores the shape of the yield curve and is technically only correct if yield curve is flat

Zero Volatility Spread

The equal amount that must be added to each rate on the Treasury spot yield curve in order to make the present value of the risky bond's cash flow equal to its market price;

Measures spread to Treasury spot rates necessary to produce a spot rate curve that correctly prices a risky bond;

For a risky bond, the value obtained from discounting expected cash flows at Treasury spot rates will be too high since Treasury spot rates are lower than they would be for a risky bond

Measures spread to Treasury spot rates necessary to produce a spot rate curve that correctly prices a risky bond;

For a risky bond, the value obtained from discounting expected cash flows at Treasury spot rates will be too high since Treasury spot rates are lower than they would be for a risky bond

Factors Influencing Difference Between Nominal and Zero-Vol Spreads

~The steeper the benchmark spot rate curve, the greater the difference between the two and an upward/downward sloping curve produces a Z spread greater/smaller than nominal spread

~The shorter the maturity, the greater the difference

~The shorter the maturity, the greater the difference

Option Adjusted Spread

The spread to the Treasury spot curve that the bond would have if it were option-free

Forward Rate

Borrowing/lending rate for a loan to be made at a future date;

Borrowing for three-years at a three year rate or for 1-year periods, three in succession, should cost the same

Borrowing for three-years at a three year rate or for 1-year periods, three in succession, should cost the same

Scenario Analysis

Measuring interest rate risk by plugging in different rates to the valuation model and looking at the outputs

Duration/Convexity Approach

Approximates the actual interest rate sensitivity of the bond

Duration Relationships

*HIgher/lower coupon means lower/higher duration

*Longer/shorter maturity means higher/lower duration

*Higher/lower market yield means lower/higher duration

*Longer/shorter maturity means higher/lower duration

*Higher/lower market yield means lower/higher duration

Convexity

Makes so a bond's rate of devaluation fall the more yields rise

Effective Duration =

(Bond Price When Yields Fall - Bond Price When Yields Rise)/(2 ** Initial Price ** Change in Yield in Decimal Form)

Macaulay Duration

An estimate of a bond's interest rate sensitivity based on years until promised cash flow will arrive;

Cannot be used for bonds with options

Cannot be used for bonds with options

Modified Duration

Similar to Macaulay but takes into account YTM;

= (Macaulay Duration)/(1 + Periodic Market Yield)

= (Macaulay Duration)/(1 + Periodic Market Yield)

Interpretations of Duration

+Duration is the slope of the price-yield curve at the bond's current YTM

+Duration is a weighted average of the time until each cash flow

+Duration is the approximate percentage change in price for a 1% change in yield

+Duration is a weighted average of the time until each cash flow

+Duration is the approximate percentage change in price for a 1% change in yield

Portfolio Duration

The weighted average of each bond's duration;

Best with a parallel curve shift since not all bonds will have the same yield change

Best with a parallel curve shift since not all bonds will have the same yield change

Convexity

The curvature of the price-yield curve;

The more convexity, the worse the duration estimate will differ from actual change

The more convexity, the worse the duration estimate will differ from actual change

Duration/Convexity Bond Pricing =

[(-Duration ** Change in Yield) + (Convexity ** Change in Yield ^ 2)] * 100

Effective Convexity

Takes into account changes in cash flows from embedded options

Difference Between Modified and Effective Convexity

Modified convexity does not take options into account and effective convexity does

Price Value of a Basis Point

The dollar change in the price/value of a bond or portfolio when the yield changes by one basis point;

= Duration** 0.0001 ** Bond Value

= Duration