Lesson 5. Risky assets

Transcription

1 Lesson 5. Risky assets Prof. Beatriz de Blas May 2006

2 5. Risky assets 2 Introduction How stock markets serve to allocate risk. Plan of the lesson: 8 >< >: 1. Risk and risk aversion 2. Portfolio risk 3. Portfolio of one risky asset and one risk-free asset 4. Portfolio of two risky assets 5. Riskless borrowing and lending

3 5. Risky assets 3 1. Risk and risk aversion Simple prospect: investment opportunity in which a certain initial wealth is placed at risk, and thre are only two possible outcomes. W = $100; 000 Mean or expected end-of-year wealth % W 1 = $150; 000 p = 0:6 1 p = 0:4 & W 2 = $80; 000 E(W ) = pw 1 + (1 p)w 2 = (0:6 150; 000) + (0:4 80; 000) = 122; 000 Variance 2 = p [W 1 E(W )]) 2 + (1 p) [W 2 E(W )] 2 = = 0:6(150; ; 000) 2 +0:4(80; ; 000) 2 = 1; 176; 000; 000

4 5. Risky assets 4 Fair game: a prospect with zero risk premium. Risk averse investors are willing to consider only risk-free or speculative prospects with positive risk premia. Utility allows us to formalize the notion of risk aversion. Certainty equivalent rate of a portfolio is the rate that risk-free investments would need to o er with certainty to be considered equally attractive as the risky portfolio. A portfolio is desirable only if its certainty equivalent return exceeds that of the risk-free alternative.

5 5. Risky assets 5 Mean-variance criterion: A dominates B if E(r A ) E(r B ) and A B and at least one inequality is strict (rules out the equality). Investors will be equally attracted to portfolios with high risk and high expected returns compared with other portfolios with lower risk but lower expected returns. These equally preferred portfolios will lie in the mean-standard deviation plane on a curve that ocnnects all portfolio points with the same utility value, the indi erence curve.

6 5. Risky assets 6 Indifference Curves Expected Return Increasing Utility Standard Deviation 6 8 Expected return is a good. Standard deviation is a bad.

7 5. Risky assets 7 2. Portfolio risk Asset risk versus portfolio risk When deciding the best combination or portfolio of securities to hold, investors need to consider 1. the relationship between the expected return on individual securities and the expected return on a portfolio made up of these securities. 2. the relationship between the standard deviation of individual securities, the correlations between these securities, and the standard deviation of a portfolio made up of these securities.

8 5. Risky assets 8 Hedging: investing in an asset with a payo pattern that o sets exposure to a particular source of risk (insurance). Diversi cation: investments are made in a wide variety of assets so that exposure to the risk of any particular security is limited. Review of portfolio mathematics (handout) Expected returns Variance and standard deviation Covariance and correlation coe cient

9 5. Risky assets 9 3. Portfolio of one risky asset and one risk-free asset Suppose an investor has decided on the composition of the risky portfolio: a proportion y allocated to the risky portfolio, P ; the remaining proportion, 1 y; to the risk-free asset, F: r P is the risky rate of return on P; assume E(r P ) = 15%; P = 22% r F is the risk-free rate of return on F; assume r F = 7% risk premium on the risky asset is E(r P ) r F = 15% 7% = 8% Rate of return on the complete portfolio r C = yr P + (1 y)r F E(r C ) = ye(r P ) + (1 y)r F = r F + y [E(r P ) r F ] = 7 + y(15 7) C = y P = 22y

10 5. Risky assets 10 If we plot the portfolio characteristics in the expected return-standard deviation plane we have the investment opportunity set E(r) sigma

11 5. Risky assets 11 Capital allocation line (CAL) or Capital market line (CML): all the riskreturn combinations available to investors, E(r C ) = ye(r P ) + (1 y)r F = r F + y [E(r P ) r F ] using C = y P! y = C P E(r C ) = r F + C P [E(r P ) r F ] Reward-to-variability ratio, aka the price of risk, is the slope of the CAL, that is, de(r C ) d C = E(r P ) r F P

12 5. Risky assets Portfolio of two risky assets Objective: construct risky portfolios to provide the lowest possible risk for any given level of expected return! e cient diversi cation. The relationship between risk and expected return depends on the correlation coe cient: xy = Cov(x;y) x y Correlation coe cient Risk reduction xy = +1! No risk reduction is possible xy = +0:5! Moderate risk reduction is possible xy = 0! Considerable risk reduction is possible xy = 0:5! Most risk can be eliminated xy = 1! All risk can be eliminated

13 5. Risky assets 13 Consider a portfolio of 2 mutual funds: long-term debt securities (D), and stock fund in equity (E) Debt Equity E(r) 8% 13% 12% 20% Cov(r D ; r E ) 72 DE 0:30 weights w D w E = 1 w D r P = w D r D + w E r E = r E + w D (r D r E )

14 5. Risky assets 14 E(r P ) = w D E(r D ) + w E E(r E ) = E(r E ) + w D [E(r D ) E(r E )] 2 P = w2 D 2 D + w2 E 2 E + 2w Dw E Cov(r D ; r E ) P = h w 2 D 2 D + w2 E 2 E + 2w Dw E Cov(r D ; r E ) i 1=2 DE = Cov(r D; r E ) D + E The relationship depends on correlation coe cient 1 DE +1: The smaller the correlation, the greater the risk reduction potential. [See Minimum variance handout]

15 5. Risky assets 15 Remarks: 1. Diversi cation e ect occurs whenever the correlation between the two securities is below Individuals face an opportunity set or feasible set: the possible expected return-standard deviation pairs of all portfolios that can be constructed from a set of asset.s 3. Minimum variance portfolio: it is the portfolio with the lowest possible variance in the opportunity set.

16 5. Risky assets E cient set or e cient frontier: the section of the opportunity set above the minimum variance portfolio. It describes the optimal trade-o. No investor would want to hold a portfolio with an expected return below that of the minimum variance portfolio.

17 5. Risky assets Riskless borrowing and lending Assume investors can allocate their money across the T-bills and common stock of a rm: Merville Enterprise. Stock Risk f ree asset E(r) 14% 10% 0:20 0 Invest $1; 000 : $350 in stocks (i.e. 35%) and $650 in T-bills (i.e. 65%). Expected return on portfolio is E(r P ) = (0:35 0:14) + (0:65 0:10) = 11:4% and the standard deviation is P = 0:35 0:20 = 0:07

18 5. Risky assets 18 The capital market line is E[r_P] sigma_p E(r P ) = 0:10 + 0:04 0:20 P

19 5. Risky assets 19 Consider investment strategy I: borrow $200 at the risk-free rate and invest all money $1; 200 in stocks (i.e. 120%). Expected return E(r I ) = (1:2 0:14) + ( 0:20 0:10) = 14:8% which is higher than 14%: The standard deviation is I = 1:2 0:20 = 0:24 higher than 0:20: This situation is called leverage.

20 5. Risky assets The optimal portfolio With a risk-free asset available and the e cient frontier identi ed, we choose the capital allocation line with the steepest slope. Objective: nd the weights (i.e. combinations of risky and risk-free assets) that result in the highest slope of the CAL, that is, S P = E(r P ) r F P Then point in which S P is tangent to the e cient frontier of risky assets: optimal portfolio. The separation property: A portfolio manager will o er the SAME risky portfolio P to all clients regardless of their degree of risk aversion. They can separate their risk aversion from their choice of the market portfolio. Risk aversion only plays a role if selecting on the CAL: the more risk averse client will invest more in the risk-free asset and less in the optimal risky portfolio.

21 5. Risky assets 21 The separation property states that the portfolio choice problem may be separated in two independent tasks: 1. Determination of the optimal risky portfolio: (a) estimate expected returns and variances, (b) calculate covariances among risky assets, (c) calculate e cient set of risky assets, (d) determine tangency with risk-free return and e cient set; 2. Allocation of the complete portfolio to T bills versus the risky portfolio, depending on personal preferences.