213 Portfolio Models-Introduction Were the two portfolios on the envelope, then all of the individual stocks would fall on or inside the curved line in the graph. In our case, two of the stock returns (stock 1 and stock 4) fall outside the frontier created by combina- tions of portfolios x and y. Thus x and y cannot be efficient portfolios. In Chapter 9 you will learn to compute efficient and envelope portfolios, and as you will see there, this requires considerably more computation. 8.6 Summary In this chapter we have reviewed the basic concepts and mathematics of portfolios. In succeeding chapters we shall describe how to compute the variance-covariance matrix from asset returns and how to calculate efficient portfolios. Exercises 1. The exercise disk for this chapter contains monthly data for stock prices of Kellogg and IBM. Compute the return statistics and graph a frontier of combinations of the two stocks. 2 . Consider the two stocks below. Graph the frontier of combinations of the two stocks. Show the effect on the frontier of varying the correlation from -1 to +1. A B C D E F TWO STOCKS 1 Varying the correlation coefficient 2 Stock A Stock B 3 Mean 3.00% 8.00% 4 Sigma 15.00% 22.00% 5 Correlation 0.3000 3. The disk for these exercises gives 5 years of monthly prices for two Vanguard funds-the Vanguard Index 500 fund (symbol VFINX) and the Vanguard High-Yield Corporate Bond fund (VWEHX). The first of these funds tracks the Standard and Poor's 500, and VWEHX is a junk-bond fund. Compute the monthly returns and the frontier of combinations of these two funds. 4. Consider the two random variables X and Y whose values are given below. Note that X and Y are perfectly correlated, though perhaps not linearly correlated. Compute their cor- relation coefficient.210 Chapter 8 8.5 Envelope Portfolios An envelope portfolio is the portfolio of risky assets that gives the lowest vari- ance of return of all portfolios having the same expected return. An efficient portfolio is the portfolio that gives the highest expected return of all portfolios having the same variance. Mathematically, we may define an envelope port- folio as follows: For a given return u = E(rp), an efficient portfolio p = [x1, X2, ... , XN] is one that solves min _Exx,01 = Var(rp) subject to Exin = u= E(Ip) Exi = 1 The envelope is the set of all envelope portfolios, and the efficient frontier is the set of all efficient portfolios. As shown by Black (1972), the envelope is the set of all convex combinations of any two envelope portfolios." This means that if x = [X1, X2, ... , XN] and y = [y1, y2, ... , ya] are envelope portfolios and if a is a constant, then the portfolio Z defined by ax + (1-a) y1 z = ax+(1-a)y= ax2 + (1-a) yz LaxN + (1-a)yN_ is also an envelope portfolio. Thus, we can compute the whole envelope fron- tier if we can find any two envelope portfolios. 8. Chapter 9 discusses the difference between envelope and efficient portfolios. In a word: The efficient frontier is a subset of the envelope containing only the optimal portfolios. 9. We discuss Black's theorem more extensively in the next chapter.211 Portfolio Models-Introduction By this theorem, once we have found two efficient portfolios x and y, we know that any other efficient portfolio is a convex combination of x and y. If we denote the mean and variance of x and y by {E(rx), of } and {E(ry), o} }, and if z = ax + (1 - a)y, then E(rz ) = aE(rx ) + (1-a)E(ry ) 62 = a303 + (1-a)63 + 2a(1-a) Cov(x, y) = a of + (1-a)203 + 2a(1-a)xSy Further details of the calculation of efficient portfolios are discussed in Chapter 9. Portfolios x and y Are Not on the Envelope To show that the concepts of envelope and efficient portfolio is non-trivial, we show that the two portfolios whose combinations are graphed in the previous example are not either envelope or efficient. This is easy to see if we extend the data table to include numbers for the individual stocks