1. Asset Allocation Week 4

2. Asset Allocation: The Fundamental Question How do you allocate your assets amongst different assets ? –There are literally thousands of assets available to you for investment purposes. Which ones will you invest in, and how much will you invest in each of these? –In this class, we will limit our discussion only to the universe of stocks and one risk-free asset (the Treasury). However, the same ideas can be extended to any set of risky assets, like real estate, hedge fund investments, etc. –We can divide this question into two separate decisions.

3. The Two Decisions How do you allocate your assets amongst different assets? There are two decisions that you have to make: A. How will you allocate between the risk-free asset and the portfolio of risky assets (stocks)? –To figure this out, ask yourself this question: Of all the money you have available to you for investment, how much do you want to keep in “cash”? –We shall see that there is no “best” way to allocate – your allocation will depend on your preferences and risk tolerance. Thus, your decision will depend on criteria like your current age, total wealth, current financial commitments, etc. B. How will you allocate between different risky assets within the portfolio of risky assets. –We shall see that there is an optimal allocation between risky assets – one “best” way to divide all your cash between the risky stocks. –This is the primary question we shall deal with over here.

4. Assumptions Traditionally, when we decide asset allocation, we will assume that all the assets are fairly priced. –If, instead, one asset is not fairly priced (it is under- or over-valued), it may be optimal for you to simply allocate all your money into that one asset! Moreover, we will assume that we know, or can estimate from past history, all that we need to know about the expected returns, volatilities, and correlations of our stocks.

5. The Objective of Allocation What should be our objective when we decide to allocate between different assets? – For example, why should we not invest in only one asset? We may not wish to invest in one asset as one of our goals is to diversify (and, thus, reduce) risk. Specifically, we will set our objective as: Earn the highest return per unit of risk. –We will measure our return as “excess returns”: R – Rf –We will measure our risk by the volatility of the return. The volatility is the standard deviation of the return.

6. The Sharpe Ratio Given our objective of maximizing the return per unit of risk, we will use a metric based on the expected return and volatility of the asset that is commonly known as the “Sharpe Ratio”. The Sharpe ratio measures the tradeoff between risk and return for each portfolio. –Sharpe Ratio = (R-Rf)/(Vol). We will use the Sharpe ratio as our criteria for choosing between different allocations.

7. Maximizing the Sharpe Ratio It is important to note that maximizing the Sharpe ratio, i.e., maximizing the excess return per unit of risk is not equivalent to either (a) maximizing the return, or (b) minimizing the risk. Example: –Between 1/1994 and 9/2004, the average return earned on a stock of KO was 11.85%/year. Over the same period, the return earned on PEP was 14.69%/year. The volatility of KO’s return was 25.35%/year, and the volatility of PEP’s return was 24.28%/year. Thus, PEP earned a higher return with a lower risk than KO over this period. –Qt: Suppose you expect KO and PEP to perform the same over the next 10 years. Does this mean that you should invest all your money in Pepsi, and nothing in Coke? Answer: No.

8. Notations and Useful Formulae Let there be two assets, Asset 1 and Asset 2. R 1, R 2 = expected returns on Asset 1 and Asset 2, respectively. Vol 1, Vol 2 = volatilities of the returns on Asset 1 and Asset 2, respectively. –The volatility is the standard deviation of the returns. Rho 12 = correlation between returns on Asset 1 and Asset 2 W 1 = proportion in Asset 1. W 2 = proportion in Asset 2. Rp= expected return on portfolio of the two assets = w 1 R 1 + w 2 R 2 Vol p = Volatility of portfolio of the two assets = (w 1 )^2 (Vol 1 )^2 + (w 2 )^2 (Vol 2 )^2 + 2 x Rho 12 x w 1 x w 2 x Vol 1 x Vol 2

9. Asset Allocation A. Risky vs. Riskless Asset First, consider the allocation between the risky and riskless asset. Rf = expected return on riskfree asset. Rp= expected return on risky portfolio. Volatility of riskfree asset = 0. W 1 = proportion in riskfree asset. W 2 = proportion in risky asset. Is there an optimal w 1, w 2 ? We shall show that the choice of w 1, w 2 is individual- specific. Thus, there is no one best portfolio allocation.

10. Portfolio of Risky + Riskless Asset To calculate the portfolio return and portfolio variance when we combine the risky asset and riskless asset, we can use the usual formulas, noting that the volatility of the riskfree rate is zero. Portfolio Return = w 1 Rf + w 2 Rp. Portfolio Variance = (w 1 )^2 (0) + (w 2 )^2 (vol of risky asset)^2 + 2 (correlation) (w 1 )(w 2 ) (0)(vol of risky asset). Portfolio Volatility = w 2 * (vol of risky asset). This simplification in the formula for the portfolio volatility occurs because the vol of the riskfree asset is zero. To understand the tradeoff between risk and return, we can graph the portfolio return vs the the portfolio volatility. The following graph shows this graph for the case when the mean return for the riskfree asset is 5%, the mean return for the risky asset is 12%, and the volatility of the risky asset is 15%.

11. Riskfree Return=5%, Risky Return=12%, Vol of Risky Asset=0.15

12. Portfolio Return vs. Portfolio Volatility

13. How to allocate between the riskfree asset and the risky stock portfolio. The conclusion we draw from the straight-line graph is that: when we combine a riskfree asset with the risky stock portfolio, all portfolios have the same Sharpe ratio. Therefore, it is not possible to make a decision on allocation between the riskfree asset and the risky stock portfolio based solely on the Sharpe ratio. Instead, we will have to take into account individual- specific considerations. There is no single allocation here that is best for all investors. Your decision to allocate between the risky asset and the riskfree asset will be determined by your level of risk aversion and your objectives, depending on factors like your age, wealth, horizon, etc. The more risk averse you are, the less you will invest in the risky asset. Although different investors may differ in the level of risk they take, they are also alike in that each investor faces exactly the same risk- return tradeoff.

14. B. Portfolio of Risky Assets We discussed the allocation between the risky (stock) portfolio and the riskless (cash) portfolio. Now we will consider the other decision that an investor must make: how should the investor allocate between two or more risky stocks? Once again we will assume that investors want to maximize the Sharpe ratio (so that investors want the best tradeoff between return and volatility).

15. Determining the Optimal Portfolio If we can plot the portfolio return vs. portfolio volatility for all possible allocations (weights), then we can easily locate the optimal portfolio with the highest Sharpe ratio of (Rp - Rf)/(Vol of portfolio). When we only have two risky assets, it is easy to construct this graph by simply calculating the portfolio returns for all possible weights. When we have more than 2 assets, it becomes more difficult to represent all possible portfolios, and instead we will only graph only a subset of portfolios. Here, we will choose only those portfolios that have the minimum volatility for a given return. We will call this graph the minimum variance frontier. Once we solve for this minimum variance frontier, we will show that there exists one portfolio on this frontier that has the highest Sharpe ratio, and thus is the optimal stock portfolio. Because there exists one specific portfolio with the highest Sharpe ratio, all investors will want to invest in that portfolio. Thus, the weights that make up this portfolio determines the optimal allocation between the risky assets for all investors.

16. Frontier with KO and PEP As an example, consider a portfolio of KO and PEP. What should be the optimal combination of KO and PEP? –Refer to excel file. As we only have two assets here, we can easily tabulate the Sharpe ratio for a range of portfolio weights, and check which portfolio has the highest Sharpe ratio. The next slide shows the results. In the calculation of the Sharpe ratio, it is assumed that the riskfree rate is constant (which is not strictly true). The portfolio mean and portfolio return are calculated over the 10-year sample period 1994-2004, with monthly data. As can be seen, the optimal weight for a portfolio (to get the maximum Sharpe ratio) is about 28% for KO. –If the exact answer is required, we can easily solve for it using the solver in Excel..

17. Volatility-Return Frontier Consider the graph of the portfolio return vs. Portfolio volatility. Graphically, the optimal portfolio (with the highest Sharpe ratio) is the portfolio that lies on a tangent to the graph, drawn such that it has the risk-free rate as its intercept. This is because the slope of the line that passes connects the risk-free asset and the risky portfolio is equal to the Sharpe ratio. Thus, the steeper the line, the higher the Sharpe ratio. The tangent to the graph has the steepest slope, and thus the portfolio that lies on this tangent is the optimal portfolio (having the highest Sharpe ratio). This tangent is also called the “capital allocation line”. All investments represented on this line are optimal (and will comprise of combination of the riskfree asset and risky stock portfolio).

18. Portfolio Return-Volatility Frontier KO + PEP, 1994-2004