## Stock Examples - Background |
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Every stock in the market has a certain return. We assume that this return has a normal | ||||||||

distribution. This means that we can completely describe this distribution by two terms. The | ||||||||

first term is the mean (expected return) and the second is the variance of returns. We can also | ||||||||

compute a covariance of returns between any pair of stocks -- those that move together will | ||||||||

have positive covariances, those that move in opposite directions will have negative | ||||||||

covariances. If we know the expected return and variance of several stocks, we can put | ||||||||

together a portfolio of these stocks that has a desired variance (risk) with a certain expected | ||||||||

return. The Solver is used to pick the portfolio with the smallest variance for a certain | ||||||||

expected return. (Or it could be used to find the highest expected return for a certain | ||||||||

variance.) | ||||||||

How do we calculate the expected return and variance of a portfolio? Harry Markowitz | ||||||||

developed a method that computes the portfolio variance as the sum of the individual stock | ||||||||

variances and covariances between pairs of stocks in the portfolio, weighted by the relative | ||||||||

proportion of each stock in the portfolio. From a mathematical point of view this is the right | ||||||||

thing to do. However, since all covariance terms between all stocks must be known, this | ||||||||

requires many calculations. A portfolio with 100 different stocks would require more than | ||||||||

5000 covariance terms, for instance. | ||||||||

William Sharpe devised another method of determining the expected return and variance of | ||||||||

a portfolio. This method assumes that the return of each stock is composed of two parts: | ||||||||

One part (denoted by beta) is dependent on the overall market's performance, and the other | ||||||||

(denoted by alpha) is independent of the market. So we can write the expected return as | ||||||||

Return = Beta * Market + Alpha + Residual | ||||||||

In this equation, alpha and beta are constants that are different for each stock; the residual | ||||||||

term is a random variable with an expected value of zero. When this formula is used, a | ||||||||

portfolio with 100 stocks would only require 302 terms to fully describe its distribution (100 | ||||||||

alphas, 100 betas, 100 residual variances, plus the market return and the market variance). | ||||||||

Compared to the more than 5000 terms of the Markowitz method, this is a big improvement. | ||||||||

However, although the Sharpe method requires less work, it is not as accurate as the | ||||||||

Markowitz method. | ||||||||

Bond Examples - Background | ||||||||

Bonds are generally considered less risky than stocks. If a bond is of high credit quality, its | ||||||||

price changes will depend almost entirely on interest rate changes. If interest rates go up, the | ||||||||

price of a bond goes down, while if interest rates go down, the price of a bond goes up. | ||||||||

However, most bonds pay interest in an annual (or semi-annual) coupon, and the interest can | ||||||||

be reinvested. If interest rates go up, the price of the bond does go down, but the coupon could | ||||||||

be reinvested at the higher interest rate. These two effects cancel each other out at the end of | ||||||||

the duration of a bond. The duration of a bond is the average time in which a bond is repaid. | ||||||||

For example, if a bond has a maturity of 3 years with face value $1,000 and an annual coupon | ||||||||

payment of $100 and a yield of 10%, the duration is 2.735 years. Thus, one way of protecting | ||||||||

against fluctuation of interest rates is to have a portfolio whose duration is equal to one's own | ||||||||

investment time horizon. A portfolio of bonds has a duration that is the weighted average of | ||||||||

the duration of the individual bonds. | ||||||||

In model Bond1 we assume we know the duration of the bonds available and want to maximize | ||||||||

the yield of the portfolio while keeping the duration equal to a given investment time horizon. | ||||||||

This technique is known as bond portfolio 'immunization.' Bond2 is another immunization | ||||||||

model, but this time the duration for each bond is calculated from the face value, maturity, | ||||||||

annual coupon and yield to maturity of each bond. | ||||||||

In model Bond3 we look at another common way to protect against interest rate fluctuations. | ||||||||

When we acquire a portfolio of bonds, we can exactly calculate the cash flow that arrives from | ||||||||

this portfolio. This consists of the coupons plus the face values of the bonds that mature. | ||||||||

This also works the other way around. If we know we need to have certain amounts of cash | ||||||||

available at certain periods, we can put together a portfolio that does exactly this. The Solver | ||||||||

chooses that portfolio that covers the required amounts in each year and costs the least. | ||||||||

Bond4 is another example of this technique called 'Exact Matching.' This time we allow | ||||||||

excess funds in a certain period to be reinvested to meet requirements in a future period. | ||||||||