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.