Lesson 5. Risky assets

Risky assets

Prof. Beatriz de Blas

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

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.

% W₁ = $150;000

W = $100; 000 p = 0:6

1 p = 0:4

& W₂ = $80;000

Mean or expected end-of-year wealth

E(W) = pW₁+ (1 p)W₂ = (0:6 150;000) + (0:4 80;000) = 122;000

Variance

2 _{= p [W}

1 E(W)])2 + (1 p) [W2 E(W)]2 =

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.

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 re-turns.

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

6-8

6-8

Indifference Curves

Expected Return

Standard Deviation

Increasing Utility

Expected return is a good.

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.

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 expo-sure to the risk of any particular security is limited.

Review of portfolio mathematics (handout)

Expected returns

Variance and standard deviation

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)

If we plot the portfolio characteristics in the expected return-standard deviation plane we have the investment opportunity set

20 15 10 5 0 15 12.5 10 7.5 5 2.5 0 sigma E(r) sigma E(r)

Capital allocation line (CAL) or Capital market line (CML): all the risk-return 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

4. 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

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

E(r_P) = w_DE(r_D) + w_EE(r_E) = E(r_E) + w_D [E(r_D) E(r_E)]

2

P = wD2 2D + w2E E2 + 2wDwECov(rD; rE)

P =

h

w_D2 2_D + w2_E 2_E + 2w_Dw_ECov(r_D; r_E)i1=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.

Remarks:

1. Diversi…cation e¤ect occurs whenever the correlation between the two se-curities is below 1.

2. 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

4. 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.

5. 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

The capital market line is 0.2 0.15 0.1 0.05 0 0.15 0.125 0.1 0.075 0.05 0.025 0 sigma_P E[r_P] sigma_P E[r_P]

E(r_P) = 0:10 + 0:04 0:20 P

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

5.1 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(rP) rF

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.

The separation property states that the portfolio choice problem may be sepa-rated 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.