Risk, return, and diversification A reading prepared by Pamela Peterson Drake O U T L I N E 1. Introduction 2. Diversification and risk 3. Modern portfolio theory 4. Asset pricing models 5. Summary 1. Introduction As managers, we rarely consider investing in only one project at one time. Small businesses and large corporations alike can be viewed as a collection of different investments, made at different points in time. We refer to a collection of investments as a portfolio. While we usually think of a portfolio as a collection of securities (stocks and bonds), we can also think of a business in much the same way -- a portfolios of assets such as buildings, inventories, trademarks, patents, et cetera. As managers, we are concerned about the overall risk of the company's portfolio of assets. Suppose you invested in two assets, Thing One and Thing Two, having the following returns over the next year: Asset Return Thing One 20% Thing Two 8% Suppose we invest equal amounts, say $10,000, in each asset for one year. At the end of the year we will have $10,000 (1 + 0.20) = $12,000 from Thing One and $10,000 (1 + 0.08) = $10,800 from Thing Two, or a total value of $22,800 from our original $20,000 investment. The return on our portfolio is therefore: ⎛⎞$22,800-20,000 Return = ⎜⎟= 14% ⎝⎠$20,000 If instead, we invested $5,000 in Thing One and $15,000 in Thing Two, the value of our investment at the end of the year would be: Value of investment =$5,000 (1 + 0.20) + 15,000 (1 + 0.08) = $6,000 + 16,200 = $22,200 and the return on our portfolio would be: ⎛⎞$22,200-20,000 Return = ⎜⎟= 11% ⎝⎠$20,000 which we can also write as: Risk, return, and diversification, a reading prepared by Pamela Peterson Drake 1 ⎡⎤⎛⎞$5,000 ⎡ ⎛⎞$15,000 ⎤ Return = ⎢⎥⎜⎟(0.2) +=⎢ ⎜⎟(0.08) ⎥ 11% ⎣⎦⎝⎠$20,000 ⎣ ⎝⎠$20,000 ⎦ As you can see more immediately by the second calculation, the return on our portfolio is the weighted average of the returns on the assets in the portfolio, where the weights are the proportion invested in each asset. We can generalize the formula for a portfolio return, rp, as the weighted average of the returns of all assets in the portfolio, letting: • i indicate the particular asset in the portfolio, • wi indicate the proportion invested in asset i, • ri indicate the return on asset i, and • S indicate the number of assets in the portfolio The return on the portfolio is: rp = w1r1 + w2r2 + ... + wSrS , which we can write more compactly as: S rwpi= ∑ ri i1= Example: The return on a portfolio Problem Consider a portfolio comprised of three assets, with expected returns and investments of: Asset Expected return Investment A 10% $20,000 B 5% 10,000 C 15% 20,000 What is the expected return on the portfolio? Solution rp = 40% (10%) + 20% (5%) + 40% (15%) rp = 0.04 + 0.01 + 0.06 rp = 0.11 or 11% 2. Diversification and risk "My ventures are not in one bottom trusted Nor to one place; nor is my whole estate Upon the fortune of this present year. Therefore my merchandise makes me not sad" - William Shakespeare, Merchant of Venice. In any portfolio, one investment may do well while another does poorly. The projects' cash flows may be "out of synch" with one another. Let's see how this might happen. Let's look at the idea of "out-of-synchness" in terms of expected returns, since this is what we face when we make financial decisions. Consider Investment One and Investment Two and their probability distributions: Risk, return, and diversification, a reading prepared by Pamela Peterson Drake 2 Scenario Probability of Return on Return on Scenario Investment One Investment Two Boom 30% 20% -10% Normal 50% 0% 0% Recession 20% -20% 45% We see that when Investment One does well, in the boom scenario, Investment Two does poorly. Also, when Investment One does poorly, as in the recession scenario, Investment Two does well. In other words, these investments are out of synch with one another. Now let's look at how their "out-of-synchness" affects the risk of the portfolio of One and Two. If we invest an equal amount in One and Two, the portfolio's return under each scenario is the weighted average of One and Two's returns, where the weights are 50 percent: Scenario Probability Weighted average return Boom 0.30 [0.5 ( 0.20)] + [0.5 (-0.10)] = 0.0500 or 5% Normal 0.50 [0.5 ( 0.00)] + [0.5 ( 0.00)] = 0.0000 or 10% Recession 0.20 [0.5 (-0.20)] + [0.5 ( 0.45)] = 0.1250 or 12.5% The calculation of the expected return and standard deviation for Investment One, Investment Two, and the portfolio consisting of One and Two results in the following the statistics, Return on Return on Return on a Probability of Investment Investment portfolio comprised Scenario scenario One Two of One and Two Boom 30% 20% -10% 5% Normal 50% 0% 0% 0% Recession 20% -20% 45% 12.5% Expected return 2% 6% 4% Standard deviation 14.00% 19.97% 4.77% The expected return on Investment One is 2 percent and the expected return on Investment Two is 6 percent. The return on a portfolio comprised of equal investments of One and Two is expected to be 4 percent. The standard deviation of Investment One 's return is 14 percent and of Investment Two 's return is 19.97 percent, but the portfolio's standard deviation, calculated using the weighted average of the returns on investments One and Two in each scenario, is 4.77 percent. This is less than the standard deviations of each of the individual investments because the returns of the two investments do not move in the same direction at the same time, but rather tend to move in opposite directions. A. The role of covariance and correlation The portfolio comprised of Investments One and Two has less risk than the individual investments because each moves in different directions with respect to the other. A statistical measure of how two variables -- in this case, the returns on two different investments -- move together is the covariance. Covariance is a statistical measure of how one variable changes in relation to changes in another variable. Covariance in this example is calculated in four steps:1 Step 1: For each scenario and investment, subtract the investment's expected value from its possible outcome; 1 You should notice a similarity between the calculation of the covariance and the variance (that you learned in the Measuring Risk reading). In the case of the variance, we took the deviation from the expected value and squared it before weighting it by the probability. In the case of the covariance, we take the deviations for each asset, multiply them, and then weight by the probability. Risk, return, and diversification, a reading prepared by Pamela Peterson Drake 3 Step 2: For each scenario, multiply the deviations for the two investments; Step 3: Weight this product by the scenario's probability; and Step 4: Sum these weighted products to arrive at the covariance. Deviation of Deviation of Investment One's Investment Two's Product return from its return from its of the Weight the product by Scenario Probability expected return expected return deviations the probability Boom 0.30 0.1800 -0.1600 -0.0288 -0.00864 Normal 0.50 -0.0200 -0.0600 0.0012 0.00060 Recession 0.20 -0.2200 0.3900 -0.0858 -0.01716 covariance =-0.02520 As you can see in these calculations, in a boom economic environment, when Investment One is above its expected return (deviation is positive), Investment Two is below its expected return (deviation is negative). In a recession, Investment One's return is below its expected value and Investment Two's return is above its expected value. The tendency is for the returns on these portfolios to co-vary in opposite directions -- producing a negative covariance of -0.0252. We can represent this calculation in a formula, using pi to represent the probability, r to represent the possible return, the ε to represent the expected return: N Covariance p r -E r -E One,Two= ∑ i()( One,i One Two,i Two ) i=1 Let's see the effect of this negative covariance on the risk of the portfolio. The portfolio's variance depends on: 1. the weight of each asset in the portfolio; 2. the standard deviation of each asset in the portfolio; and 3. the covariance of the assets' returns. Let cov1,2 represent the covariance of two assets' returns. We can write the portfolio variance for a two- security portfolio as: 2 2 2 2 Portfolio variance, 2-security portfolio = w1 σ1 + w2 σ2 + 2 w1 w2 cov1,2. The portfolio standard deviation is the square root of the variance, or: Portfolio standard deviation, 2-security portfolio = w12σσ 12 + w 22 22 + 2w 1 w 2 cov 1,2 . More generally, the formula is: NNN 22 Portfolio standard deviation = ∑∑∑wiiσσ + wijijij wσ r i=1 i==≠ 1 j 1 j i Recognizing that the covariance is the product of the correlation and the respective standard deviations, we can also write the formula as: NNN 22 . Portfolio standard deviation = ∑∑∑wiiσ + wij w cov ij i=1 i==≠ 1 j 1 j i We can apply this general formula to our example, with Investment One's characteristics indicated with a 1 and Investment Two's with a 2, Risk, return, and diversification, a reading prepared by Pamela Peterson Drake 4 w1 = 0.50 or 50 percent w2 = 0.50 or 50 percent s1 = 0.1400 or 14.00 percent s2 = 0.1997 or 19.97 percent cov1,2 = -0.0252.