Portfolio Theory

# Portfolio Theory - PortfolioTheory LearningOutcomes Capital...

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Portfolio Theory Professor David McLean Alberta School of Business

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Learning Outcomes Capital allocation decisions How much to invest in the risky vs. risk free assets? Diversification and portfolio risk Why many assets are better than just a few The effect of correlation on risk and 2
One Risky Portfolio and a Risk-Free  Asset Consider a risky portfolio (P) E(rp) = 15% and σp = 22% Consider the risk free asset (rf) rf = 7% and σ rf = 0% We invest proportions “y” and “1-y” in each to create a combined portfolio “C” y in P and (1- y ) in rf Note that y + (1- y ) = 1 3

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Portfolio C’s Expected Return What is portfolio C’s expected return? E(rc) = yE(rp) + (1-y)rf We can rewrite this as: E(rc) = rf + y[E(rp) – rf] = risk free + y * risk premium Assume y = 75% 4
Portfolio C’s Risk Recall that σp = 22% By definition, the risk free asset has zero risk and thus zero covariance with all other assets σc = y * 22% = 0.75 * 22% = 16.5% 5

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The Risk and Return of Various  Combinations of P and rf 6 E(r) E(rp) = 15% rf = 7% 22% 0 P F 16.5% E(rc) = 13% C
Interpretation of the Line If an investor chooses to invest only in the risky asset Then y = 1 and σC = 22% If the investor chooses to invest only in the risk free asset Then y = o and σC = 0 Midrange portfolios like C have y between 0 and 1 If y = .75 then σC = .75(.22) = .165 or 16.5% Questions: What is the slope of the line? 7

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Capital Allocation Line (CAL) 8 E(r) E(rp)=15% rf = 7% p = % 22 P F S = 8/22 E(rp) - rf = 8%
Capital Allocation Line (CAL) The CAL depicts all possible combinations of P and rf The slope of the CAL, S, is a return-to-risk ratio Hence, the CAL reflects how much reward (return) you get in exchange for incremental unit of risk 9 p f p σ r ) E(r S =

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The CAL and Leverage Leverage refers to using borrowed funds to invest in the risky asset Put differently, in a leveraged portfolio y > 1 Let’s assume that y = 150% rc = (-.5) (.07) + (1.5) (.15) = 0.19 σc = (1.5) (.22) = 0.33 (higher than an 10
Risk Aversion and the CAL? Where is the best place to be on the CAL? It depends on your risk aversion The CAL reflects a risk return tradeoff ; where you want to be on the CAL depends on how you feel about risk Recall the utility function: U = E ( r ) – .5 A σ 2 We can solve for y*, the portfolio weight that maximizes utility 11

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y* and The Optimal Portfolio 12 U = E(r c ) - 1 2 A 2 c E(r c
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