Stock and Bond Examples - Background

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.

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.