Stock 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
|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
|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.