Business 35150 John H. Coc hraneProblem Set 111. Let’s actually solve a few very simple classic portfolio problems. Our investor starts with wealthand considers investing in a single stock (the index) with return +1and a bond with return.Denote+1= +1− . The investor consumes tomorrow only, so his objective ismax{} [(+1)] = £(+1)¤= nh(+1+ ))io(a) Find the first-order condition for this maximization (take the derivative with respect to )and express it in the form £0(·)+1¤=0. Notice that this is our friend  = (),andweare now determining +1= +1= +1given the properties of returns, not determining¡+1¢give n the properties of consumption.(b) Suppose utility is quadratic (+1)=−12(∗− +1)2where ∗is a number. Show that theoptimal portfolio weight depends only on the mean and variance of the excess return. (Youwillgetaformulathatdependson() and (2)=()2+ 2().)(c) Suppose utility is exponential()=−−; 0()=− is the “coefficient of absolute risk aversion.” Find the optimal portfolio weight  and showthat  depends only on mean and variance (+1) and 2(+1).To solve this case first take the expectation of utility, assuming that +1is normally distrib-uted, and using the fact that if  is normally distributed ()=()+122(). Then takethe derivative with respect to  and set it to zero. Sometimes it’s easier to do it this wayrather than find the first-order condition 0=£0(+ )¤then take expectations,and then take derivative with respect to .2. This is a very simple version of the power-lognormal “real” portfolio problem that we are solving.The investor wan ts to maximize  [()] = ³1−´, by investi ng from time zero to time starting with initial wealth 0. The investor can put money in a stock whose log return isnormally distributed with mean arithmetic return  and standard deviation , and a bond thatearns =0. We will use the following fact: If the investor holds a constantly rebalanced portfoliowith weight  in the stock, wealth at time  is also lognormally distributed,0= [−1222] +√; ˜(0 1) (1)(I derive this below FYI)(a) Use  []=()+122()to find h1−i.(b) Now, find the optimal portfolio. Maximize the last expression by setting the derivative ofh1−iwith respect to  to zero, and solve for .(c) What is the effect of  in your optimal formula? Should a long-horizon investor hold morestocks because stocks are safer in the long run (with these assumptions!)2(d) Even if returns are independent over time, a friend argues, stocks are safer for long-runinvestors. True, the one year return has a 20% standard deviation, but 25-year averagereturns1(1+ 2+  + ) have a 20%/√25 = 4% standard deviation, while the averagereturn is the same for any horizon. He also points out that the Sharpe ratio(1+2++)(1+2++)rises with the square root of horizon. Is the implication that stocks are better for long-runinvestors in this situation right?(e) What considerations are left out of this problem that might tilt the optimum towards stocksfor longer horizon investors?(Note, not necessary to do the problem. Where did (1) come from? To do this you need a little and . is stock price, and the stock’s instantaneous arithmetic returns follow=  + A bond has risk free rate .(is a normally distributed random variable with mean 0 andvariance  It is the continuous time version of the we have been using to describe time series.This just says +1=+1−=  + +1.)Now, if you put weight  into the stock, wealth  evolve s as= +(1− )=  ( + )+(1−)Using Ito’s lemma, (second order expansion and 2= ) log  =∙ +(1−)−1222¸ + log − log 0=∙¡ − ¢+ −1222¸ + Z=0and withR=0=√˜(0 1),0= [+(−)−1222] +√I specialized to =0.)3. Note: This is an especially good problem. You are considering investing in two managers, andof course the mark et index. You have a mean-variance objective with risk aversion  =2.Yourassessmen t of the market portfolio is a mean ()=8% volatility () = 20%.YourunCAPM regressions for the two managers= + + with result 1=22=2;() = 10% for both managers, and the residuals  have correlation−05. Your believe 1= −03%, 2=12%.(a) Find the optimal allocation to the market index and to the t wo managers. (Hint: Be carefulabout units, i.e. should you express 10% as 10, 1.10, 0.10?) Express the answer in terms ofaweighton the excess market return and weights on “portable alpha” or beta-hedged portfolios for each of the two managers — (−= +, the manager’s returnless his beta times the market return) — i.e. write the optimal portfolio return in the form,= + 1¡1− 1¢+ 2¡2− 2¢and find the weights 12.3(b) Find the weights in terms of the actual zero-cost investments, i.e. and 12.i.e.=ˆ+ˆ11+ˆ22. (I used hats because these may be different from part a)(c) Find the weights in terms of actual investments, i.e. = ∗+ ∗+ ∗11+ ∗22(again, ∗ because these might be different from the weights in a and b.(d) Compare the three sets of weights. Which are the same, which are different?(e) In this example, you end up investing a positiv e amount with a manager who has negativealpha. How is this