1 / 27

Portfolio Theory

Portfolio Theory. Chapter 5 Risk aversion Utility function and indifference curves The capital allocation decision Portfolio of one risk-free asset and one risky asset Extension to the case of two risky assets. p = 0.6. W 1 = 150; Profit = 50. 1-p = 0.4. W 2 = 80; Profit = -20.

lainey
Download Presentation

Portfolio Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Portfolio Theory Chapter 5 • Risk aversion • Utility function and indifference curves • The capital allocation decision • Portfolio of one risk-free asset and one risky asset • Extension to the case of two risky assets

  2. p = 0.6 W1 = 150; Profit = 50 1-p = 0.4 W2 = 80; Profit = -20 Example of a Risky Investment W = 100 E(r) = pr1 + (1-p)r2 = ? 2 = p[r1 – E(r)]2 + (1-p) [r2 – E(r)]2  = ?

  3. p = 0.6 W1 = 150 Profit = 50 Risky Investment W2= 80 Profit = -20 1-p = .4 Risk Free T-bills Profit = 5 Expanding the choice set 100 Risk Premium = 22% - 5% = 17%

  4. Risk Preference/Attitude/Tolerance • Investor’s view of risk • Risk Averse • Risk Neutral • Risk Seeking / loving • Utility Function • Private value, in our case, of risk and return • Example: One popular utility function used in investments is U = E ( r ) – 0.5 A s2 • “A” measures the degree of risk aversion

  5. Risky Investment Example U = E ( r ) - 0.5 A s2 = 0.22 - 0.5 A (0.34) 2 Risk AversionAUtility High 5 - 0.07 3 0.05 Low 1 0.16 Compared to utility of T-bill = 0.05

  6. Preferences on a Diagram

  7. Indifference curves • U = E ( r ) - 0.5 A s2 Three variables on a 2-dimensional diagram Links portfolios that yield the same level of utility

  8. Indifference curve example • All four portfolios yield the same level of utility (U = 0.02) • Hence, they would lie on the same indifference curve in the diagram

  9. E(rp) Increasing Utility P Indifference Curves

  10. The capital allocation decision • Task: allocate capital (funds) between risky and risk-free assets • Examine risk/return tradeoff • If financial market is efficient, no free lunch plan • Demonstrate how different degrees of risk aversion will affect allocations between risky and risk-free assets

  11. Portfolio Mathematics • The rate of return of a portfolio is a weighted average of the return of each asset in the portfolio, with the portfolio proportions as weights • Consider a two-asset case: rp = w1r1 + w2r2 • Example of weights: (0.5, 0.5), (0.4, 0.6), (0.3, 0.7)

  12. Portfolio Mathematics • Mean return of a portfolio E(rp) = w1E(r1)+ w2E(r2) • Variance of portfolio (proxy for risk) • When two assets with variances, 12 and 22 , are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is: • What if one of the assets is risk-free?

  13. Portfolio of one risky asset and one risk-free asset • Risky portfolio P: E(rp) = 15% and p = 22% • Risk-free asset: rf= 7% and rf = 0% • And the proportions invested: y in P and (1-y) in F

  14. E(rc) = yE(rp) + (1 - y)rf = rf + y[E(rp) – rf] where rc = “combined” portfolio return If, for example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13% Expected return of the “combined” portfolio

  15. Standard Deviation of the combined portfolio • Recall that the variance is: • Apply to the current example: • This collapses to the following standard deviation:

  16. Standard deviation of combined portfolio • If y = 0.75, then c = 0.75(0.22) = 0.165 or 16.5% • If y = 1, then c = 1(0.22) = 0.22 or 22% • If y = 0, then c = 0(0.22) = 0%

  17. The Investment Opportunity Set with a Risky Asset and a Risk-Free Asset (CAL)

  18. Which portfolio to pick? (A = 4)

  19. The Portfolio that Maximizes Utility

  20. Algebraically.... (in order to be exact) • Maximize U = E(rc) – 0.5Ac2 with respect to y • Substitute in E(rc) and c2 • The solution is: • Numerical example:

  21. y* = 0.41 • Given this optimal weight, what is • the portfolio mean? E(rC) = 0.41 x 0.15 + (1 – 0.41) x 0.07 = 0.1028 or 10.28% • and the portfolio standard deviation? C= 0.41 x 0.22 = 0.0902 or 9.02%

  22. Diagrammatically

  23. Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4

  24. Leveraged portfolios • Extend beyond y = 1 • Borrow at the risk-free rate and invest in more than 100% of the portfolio in the risky asset • Can think of as short selling the risk-free asset • Using 50% leverage: E(rc) = (-0.5) (0.07) + (1.5) (0.15) = .19 c = (1.5) (0.22) = .33

  25. E(r) P Borrower 7% Lender  p= 22% CAL with Risk Preferences

  26. An Application of the CAL • Client likes the fund, but not like the volatility (of the returns) • In this case, manager may let the client choose the level of volatility • If client wants a higher level of volatility, use leverage • If client wants a lower level of volatility, put a portion in cash • Slide up or down the CAL

  27. The Opportunity Set with Differential Borrowing and Lending Rates

More Related