exhibits large skewness or kurtosis.

important bc fatness of tails shows a riskier investment- higher probability for a negative return. this is bad to investors.

kurtosis is actually a measure of excess kurtosis.

In finance, fat tails are considered undesirable because of the additional risk they imply. For example, an investment strategy may have an expected return, after one year, that is five times its standard deviation. Assuming a normal distribution, the likelihood of its failure (negative return) is less than one in a million; in practice, it may be higher. Normal distributions that emerge in finance generally do so because the factors influencing an asset's value or price are mathematically "well-behaved", and the central limit theorem provides for such a distribution. However, traumatic "real-world" events (such as an oil shock, a large corporate bankruptcy, or an abrupt change in a political situation) are usually not mathematically well-behaved. investors are price takers: individual trades don't affect prices.

single period investment horizon.

investments are limited to traded assets.

no taxes or transaction costs.

all investors are rational mean0variance optimizers.

info is costless and available to all.

investors have homogenous expectations.

All investors choose to hold the same portfolio: the market portfolio

• The proportion of each stock in the market portfolio is the market value of the stock expressed as a percentage of total market value

• The market portfolio is the tangency portfolio: the Capital Market Line (CML) is the best attainable CAL

The market is composed of many small investors, who are price-takers; i. e., perfect

competition. In reality this assumption was fairly realistic until recent years when institutional investors increasingly began to influence the market with their large transactions.

(b) All investors have the same holding period. Obviously, different investors have different goals, and thus have different holding periods.

(c) Investments are limited to those that are publicly traded. In addition, it is assumed that investors may borrow or lend any amount at a fixed, risk-free rate. Obviously, investors may purchase assets that are not publicly traded; however, the dollar volume of publicly traded assets is considerable. The assumption that investors can borrow or lend any

amount at a fixed, risk-free rate obviously is false. However, the model can be modified

to incorporate different borrowing and lending rates.

(d) Investors pay no taxes on returns and incur no transaction costs. Obviously, investors do

pay taxes and do incur transaction costs.

(e) All investors are mean-variance efficient. This assumption implies that all investors make

decisions based on maximizing returns available at an acceptable risk level; most investors probably make decisions in this manner. However, some investors are pure wealth maximizers (regardless of the risk level); and other investors are so risk averse that avoiding risk is their only goal.

(f) All investors have homogeneous expectations, meaning that given the same data all investors would process the data in the same manner, resulting in the same risk/return assessments for all investment alternatives. APT applies to well diversified portfolios and not

necessarily to individual stocks

• With APT it is possible for some individual stocks to be

mispriced - not lie on the SML

• APT is more general in that it gets to an expected return

and beta relationship without the assumption of the

market portfolio

• APT can be extended to multi factor models

CAPM depends on risk-return dominance. it assumes many small changes are required to bring the market back to eliquibrium. when an equilibrium price is violated many investors will make small portfolio changes, depending on their risk tolerance, until equilibrium is restored.

APT depends on a no arbitrage condition. it assumes a few large changes are required to bring the market back to equilibrium. under no-arbitrage argument of APT, each investor will take as large a position as possible so only a few investors must act to restore equilibrium.

Both the CAPM and the APT are market equilibrium models, which examine the factors that affect securities' prices. In equilibrium, there are no overpriced or underpriced securities. In both models, mispriced securities can be identified and purchased or sold as appropriate to earn excess profits.

The CAPM is based on the idea that there are large numbers of investors who are focused on risk-return dominance. Under the CAPM, when a mispricing occurs, many individual investors make small changes in their portfolios, guided by their degrees of risk aversion. The aggregate effect of their actions brings the market back into equilibrium. Under the APT, each investor wants an infinite arbitrage position in the mispriced asset. Therefore, it would not take many investors to identify the arbitrage opportunity and act to bring the market back to equilibrium.

implications for prices derived from APT are much stronger than those prices derived from CAPM. Duration is a measure of the time it takes to recoup one's investment in a bond, assuming that one

purchased the bond for, say, 1000 euro. Duration is shorter than term to maturity on coupon bonds as

cash flows are received prior to maturity. Duration equals term to maturity for zero-coupon bonds, as

no cash flows are received prior to maturity. Duration measures the price sensitivity of a bond with

respect to interest rate changes. The longer the maturity of the bond, the lower the coupon rate of the

bond, and the lower the yield to maturity of the bond, the greater the duration. Interest-rate risk

consists of two components: price risk and reinvestment risk. These two risk components move in

opposite direction; if duration equals horizon date, the two types of risk exactly offset each other,

resulting in zero net interest-rate risk. This portfolio management strategy is immunization. Some of the

problems associated with this strategy are: the portfolio is protected against one interest rate change

only; thus, once interest rates change, the portfolio must be rebalanced to maintain immunization;

duration assumes a horizontal yield curve (not the shape most commonly observed); duration also

assumes that any shifts in the yield curve are parallel (resulting in a continued horizontal yield curve); in

addition, the portfolio manager may have trouble locating acceptable bonds that produce immunized

portfolios; finally, both duration and horizon dates change with the mere passage of time, but not in a

lockstep fashion, thus rebalancing is required. Although immunization is considered a passive bond

portfolio management strategy, considerable rebalancing must occur, as indicated above. The portfolio

manager must consider the tradeoffs between the transaction costs and not being perfectly immunized

at all times. o Testing CAPM

• Capm is relationship about expected returns

• First, select a sample from 1 to T

• We want to get the returns of n stocks (a large number of stocks)

• CAPM must hold for all stocks, not just one stock

• We need to find out systematic source of risk and risk-free rate in the economy

• We want to test if the relationship holds for all stocks. In a good model, the relationship will hold for all stocks

• Run some regressions- estimate Betas and Alphas. For every stock, run the regression. If there are 500 assets, you must run 500 regressions

• The risk can differ amongst stocks

• So once you get your data, estimate for every stock at every point in time a regressions

• We get Bi estimates

• Now calculate average excess return for every stock

• Y= "gamma"

• Ri-Rj=y0+y1b1

• We want to have a gamma that is 0, on average for all stocks

• But we also want gamma1 y1 to equal excess reutrns of the market

• So even if gamme is 0, gamma 1 might not equal average return of markets

o Test outcomes for the single factor model

• Predicted SML is different than actual SML linke, as shown on graph

o Example: 500 largest European stocks, 2005-2014, market return-avg of 500 stocks

• Monthly market return chart shows there is a high volatility period

• First-stage regression

• Alphas are 0 or slightly positive, positively skewed

• What does this show us? That the CAPM doesn't hold

• It is normal that if you estimate some alphas, even if the CAPM holds, there will be some different alphas

• Then a regression for the betas

o Distribution is centered around 1. Is this a surprise?

o Where should the mean be? Around 1. Average return for 500 regressions should be 1, bc youre using the overall return of the market

o Second-stage regression

• Rwgress the average return of every stock ...

• 500 stocks to do this to, giving you 500 average returns

• results from regression:

• .0009,.0067

• .0007,.0008

• these are always different than 0, bc there will always be some risk premium

• is this risk premium equal to the risk premiu of the market?

• Compare .0067 to average return on market (Rm=0.0077)

• Accept or reject CAPM? Accept the CAPM bc the alphas are 0, and the market risk premium is not different from that of the market.

• We don't have enough confidence to say .67 and .77 are different from each other, bc the 77 is volatile enough to say theyre difft from each other coupon rate > current yield > YTM.

current yield exceeds YTM bc YTm accounts for the built in capital loss on the bond. the bond bought at premium of 1276 will eventually fall in value to 1000 at maturity.

current yield: investments annual income (interest or dividends) divided by current price of security. this measure looks at current price of bond instead of its face value.

current yiel= annual coupon pmt / current market value of bond. based on purchase price

yield to maturity: total return anticipated on a bond if the bond is held until the end of its lifetime. YTM is expressed as an annual rate. its the IRR of an investment bond if the investor holds the bond until maturity. based on par value, purchase price, duration, and coupon rate.

coupon rates remain stable and don't change.

• If FV is 1000 and bond is selling at a premium, then the price will be high. So the current yield will be relatively lower.

• For discount bonds, reverse this.

• Current yield is only what you get over next year

o Ex: 2 year bond with FV 1000. At time 1 year we receive 100 in coupon payment. In year 2 we receive 1,100 (FV + final coupon payment)

• Coupon = 10% and yield = 10%

• YTM=realized rate of return here

• If reinvestment rate drops to 8%? We still get our 100 at end of year 1, and reinvest that 100 at 8%/ so we receive total of 108 from this coupon payment at end of year 2, plus 1100 from year 2. So here, the r=9.91%, lower than the YTM.

• The YTM may not be the actual return that we get. substitution swap: long in some undervalued asset and short in overvalued asset with similar risk profile. An exchange that is carried out by trading a fixed-income security for a higher yielding bond with similar features. A substitution swap involves the swapping of one bond for another bond that has a higher yield, but has a similar coupon rate, maturity date, call feature, credit quality, etc. A substitution swap allows the investor (such as a firm) to increase returns without altering the terms or risk level of the security. Investors will participate in substitution swaps when they believe there is a temporary discrepancy in bond prices due to market disequilibrium.

intermarket swap: sub across sectors in overall market in terms of riskiness, with different yields on the market. A swap transaction meant to capitalize on a yield discrepancy between bond market sectors. Intermarket spread swaps are based upon expectations of yield spreads between different bond sectors or spots on the yield curve. By entering a swap, parties are able to gain exposure to the underlying bonds, without having to directly hold the securities.

rate anticipation swap: suppose you expect in future IR to go down. optimal thing to do? which bonds to start buying? long term bonds. A type of swap in which bonds are exchanged according to their current duration and predicted interest rate movements. A rate anticipation swap is often made in order to take advantage of more profitable bond opportunities. Rate anticipation swaps are speculative in nature, since they depend on the outcome of the expected interest rate change. Various bond types respond differently to rising or falling interest rates and those who participate in rate anticipation swaps generally choose bonds based on performance.

pure yield pickup: if you observe that the yield curve is upwards sloping, you simply start buying long term bonds to benefit from IR increase. A transaction in which bonds with lower returns are swapped for bonds with higher returns. With a pure yield pickup swap the sole purpose of the transaction is to increase yield, the new bonds will have a similar maturity and risk rating as the old bonds; only the coupon will differ. The greater the volatility of the underlying stock, the greater the option premium; the more

volatile the stock, the more likely it is that the option will become more valuable (e. g., move

from an out of the money to an in the money option, or become more in the money). For call

options, the lower the exercise price, the more valuable the option, as the option owner can

buy the stock at a lower price. For a put option, the lower the exercise price, the less valuable the option, as the owner of the option may be required to sell the stock at a lower than market

price