Lecture 2: Risk AversionBKM: 6.1–6

BMAN20072 Investment Analysis

Hening Liu

Outline

I Mean-variance utility function and mean-variance criterion

I Indifference curve

I Combining risk-free and risky assets

I Capital allocation line (CAL)

I Risk tolerance and asset allocation

Risk-return trade-off

I GambleI risk taking for no purpose but enjoyment of risk itself

I SpeculationI undertaking of risk involved because one perceives a favorable

risk-return trade-off

I Risk averse investorsI only want risk-free investment opportunities or speculative

prospects with positive risk premiumI penalizes expected rate of return of risky investment by

accounting for risks involved

Risk-return trade-off

Utility score

U = E (r)− 1

2Aσ2

I Notation: U is utility value; E (r) is expected return; A isinvestor’s coefficient of risk aversion; σ2 is variance of returns.

I Investor’s risk attitude:

I risk averse (A > 0): variance of returns contributes negativelyto utility value

I risk neutral (A = 0): cares only about the level of expectedreturn

I risk lover (A < 0): adjust utility upward for the “fun” of risk

I We focus on risk averse investors (A > 0)

Risk-return trade-off

Risk-return trade-off

Mean-variance criterion

I Consider two portfolios A and B

I A dominates B ifE (rA) ≥ E (rB)

andσA ≤ σB

and at least one inequality is strict (rules out the equality).

Risk-return trade-off

Risk-return trade-off

Indifference curve

I links all points with same utility value on a diagram that plotsE (r) (on the vertical co-ordinate) against σ (on the horizontalco-ordinate)

I steeper for more risk averse investors — a small increase in σmust be accompanied by a large increase in E (r) to yield thesame utility value

I higher (in northwest direction) for greater utility level

Risk-return trade-off

Combining risk-free and risky assets

Expected return from combining the risk-free asset and risky assets

rC = yrp + (1− y) rf

σC = yσp

E (rC ) = yE (rp) + (1− y) rf

= rf + y [E (rp)− rf ]

= rf +σC

σp[E (rp)− rf ]

I rC : returns of the combined portfolioI rp: returns of the risky portfolio (a portfolio of many risky

assets)I rf : risk-free rateI y : weight on the risky portfolioI σC : standard deviation of returns of the combined portfolioI σp: standard deviation of returns of the risky portfolio

Combining risk-free and risky assets

Combining risk-free and risky assets

Capital allocation line (CAL)

E (rC ) = rf + σCE (rp)− rf

σp

I depicts all the risk-return combinations available for a risk freeasset and a risky portfolio P

I slope of CAL (E (rp)− rf

σp) equals

I increase in returns of the combined portfolio per unit ofadditional standard deviation

I incremental return per incremental riskI reward to variability ratioI Sharpe ratio

Combining risk-free and risky assets

CAL kinks when

I investors borrow to invest more in risky asset (y > 1) andI borrowing rate>lending rate

Risk tolerance and asset allocationFor a risk aversion A, as the weight (y) on the risky portfolioincreases

I utility increases up to a certain level but eventually declinesI this is because volatility catches up to offset gains in expected

returns

Risk tolerance and asset allocation

I Which combination of the risk-free asset and the riskyportfolio gives maximum utility?

I The optimal portfolio weight is

y∗ =E (rp)− rf

Aσ2p

I rp: returns of the risky portfolio (a portfolio of many riskyassets)

I rf : risk-free rateI y : weight on the risky portfolioI σp: standard deviation of returns of the risky portfolio

Risk tolerance and asset allocation

Risk tolerance and asset allocation

Optimal combined portfolio depends on risk aversion

Passive investment strategy

I Why passive strategy?I avoids any direct or indirect security analysisI achieves diversification. For example, a well-diversified

portfolio of common stocks could be S&P500, NYSE or CRSPvalue-weighted portfolio

I Capital market line (CML)I is a CAL where the risky portfolio P comprises of a

well-diversified market portfolioI depicts combinations available between the risk-free asset and

the market portfolio

Passive investment strategy

Passive investment strategy

What is an average value of investors’ risk aversioncoefficient?

I Assume S&P500 to be the market portfolio (portfolio P)

I Assume a risk premium of 8.4% and a standard deviation of20.5% (1926–2005)

I Assume that S&P500 carries 76% of all invested wealth

y∗ =E (rp)− rf

Aσ2p

0.76 =0.084

A× 0.2052

A = 2.6

Exercise Question I

Consider a portfolio that offers an expected rate of return of 12%and a standard deviation of 18%. T-bills offer a risk-free 7% rateof return. What is the maximum level of risk aversion for whichthe risky portfolio is still preferred to bills?

I The investor can only invest in either the risky portfolio or therisk-free T-bills.

I The utility value of the risky investment is

u = 0.12− 1

2A×

(0.182

)I The utility value of the risk-free investment is urf = 0.07

I We are looking for A such that u = urf . This gives usAmax = 3.086.

I If A > Amax , the risk-free investment is strictly preferred. IfA < Amax , the risky portfolio is strictly preferred.