RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS

5.1 Risk and Risk Aversion 5.2 Capital allocation across

Risky and Risk-free Portfolios 5.3 Portfolios of one Risky

Asset and a Risk-free Asset 5.4 Risk Tolerance and Asset

Allocation 5.5 Passive Strategy: the

Capital Market Line

5.1 Risk and Risk Aversion

Speculation◦ Considerable risk Sufficient to affect the decision

◦ Commensurate gain Gamble

◦ Bet or wager on an uncertain outcome

Risk averse investors

◦ reject investment portfolios that are fair games or worse

◦ willing to consider only risk-free or speculative prospects with positive risk premiums

Intuitively one would rank those portfolios as more attractive with higher expected returns

◦ But if risk increases along with return, how to quantify the rate at which they are willing to trade off return against risk?

1 、你购买一项投资在一个月后跌去了 15%的总价值。假设该投资的其他任何基本面要素没有改变，你会？（ a ）坐等投资回到原有价值。（ b）卖掉它，以免日后如果它不断跌价，让你寝食难安，夜不成寐。（ c ）买入更多，因为如果以当初价格购买时认为是个好决定，现在应该看上去机会更好。

2 、你购买一项投资，在一个月后暴涨了 40%。假设你找不出更多的相关信息，你会？（ a ）卖掉它；b）继续持有它，期待未来可能更多的收益（c）买入更多 - 也许它还会涨的更高

3.你在某个电视竞赛中有下列选择。你会选 :（ a） 1000元现钞；b） 50%的机会获得 4000元（c） 20%的机会获得10,000元（ d） 5%的机会获得 100,000元

4.你在一项博彩游戏中，已经输了 500元。为了赢回 500元，你准备的翻本钱是：（ a ）不来了，你现在就放弃b） 100 元 c） 250元（d） 500元（e ） 超过500元 

Utility score: to compete portfolios based on the expected return and risk

Where U = utility E ( r ) = expected return on the asset or

portfolio A = coefficient of risk aversion = variance of returns

21( )

2U E r A

Certainty Equivalent Rate◦ The rate that risk-free investments would need to

offer with certainty to be considered equally attractive as the risky portfolio

◦ To choose between a risky portfolio and a safe one, we may interpret a portfolio’s utility value as its ”certainty equivalent” rate of return

◦ A portfolio is desirable only if its certainty equivalent return exceeds that of the risk-free alternative

A portfolio, E(R)=20%, SD=30%, T-Bills offer risk free rate=7%, if A=4 , or A=2, which one is better?

Risk Neutral (A=0)◦ judge risky prospects solely by their

expected rate of return◦The certainty equivalent rate is the E(R)

Risk lover (A<0)◦Adjust the E(R) upward to take into

account the fun of confronting the prospect’s risk, willing to engage in fair games and gambles

Mean-variance (M-V) criterion◦ Portfolio A dominates B if

◦ And

◦ And at least one inequality is strict

Northwest is preferred direction, simultaneously increase E(R) and decrease variance

( ) ( )A BE r E r

A B

How about quadrants II and III ?◦ depends on the exact nature of the investor’s risk

aversion◦ Identify all portfolios that are equally attractive as

portfolio P◦ Increase in SD lowers utility, must be

compensated for by an increase in E(R) Indifference curve

◦ Equally preferred portfolios will lie in the mean-standard deviation plane on a curve

Less risk averse

More risk averse

p

pE r

Observe individuals’ decisions when confronted with risk

Observe how much people are willing to pay to avoid risk

◦Insurance against large losses

An investor

◦ Risk aversion, A, with his wealth

◦ Probability p, 100% loss

◦ Probability 1-p, no loss

◦ Calculate utility score

E(R)= Variance= U=-p-0.5Ap(1-p)

How much will the investor pay for insurance against the potential loss? v=?

U=-p-0.5Ap(1-p)=-v

v=p+0.5Ap(1-p)

Risky asset investment entail accepting some risk in return for the prospect of earning more than the safe T-bill rate

Investors price risky assets so that the risk premium will be commensurate with the risk of the expected excess return

Trade-off between reward and risk, measure risk by the SD of excess return

Sharpe Ratio for Portfolios =

Risk PremiumSD of Excess

Return

Forecasting interest rate is important Interest rate

◦ The promised rate of return denominated in some unit of account over some time period

Nominal rate of interest ,R◦ Growth rate of your money

Real rate of interest, r◦ Growth rate of your purchasing power

Inflation rate, i 1+R=(1+r)(1+i) r≈R-i

5.2 Capital allocation across Risky and Risk-free Portfolios

Geom. Arith. Stan. Series Mean% Mean% Dev.%World Stk 9.80 11.32 18.05US Lg Stk 10.23 12.19 20.14US Sm Stk 12.43 18.14 36.93Wor Bonds 5.80 6.17 9.05LT Treas. 5.35 5.64 8.06T-Bills 3.72 3.77 3.11Inflation 3.04 3.13 4.27

Risk Stan. Sharpe

Series Prem. Dev.% Measure

World Stk 7.56 18.37 0.41US Lg Stk 8.42 20.42 0.41US Sm Stk 14.37 37.53 0.38Wor Bonds 2.40 8.92 0.27LT Treas 1.88 7.87 0.24

Risk: Long-term bonds /stocks/Treasury bills ◦ Riskier investments offer higher average returns

Construct the portfolio◦ Using securities from all asset classes

To control the risk of the portfolio◦ The fraction of the T-bills versus risky assets

Capital allocation◦ An example of asset allocation choice

among broad investment classes◦ How much to risk-free, how much to risky

A portfolio of risky assets as P and the risk-free assets as F (a completer portfolio)◦ P: a given portfolio comprising two mutual funds,

stock (E) and long-term bonds (B) Total portfolio value =$300,000

◦ F: Risk-free value =$ 90,000◦ P: Risky (E and B) =$210,000

E= 113,400/210,000=54% B=96,600/210,000=46% Weight of E and B in portfolio P is unchanged in

capital allocation

210,0000.7(risky assets, portfolio )

300,000

90,0001 0.3(risk-free assets)

300,000

y P

y

•The risky Portfolio P y=210,000/300,000 = 0.700

• E 113,400/300,000 = 0.378

• B 96,600/300,000 = 0.322

•Risk-Free Assets F 1-y= 90,000/300,000 = 0.300

•Portfolio C 300,000/300,000 = 1.000

•Reduce risk by decreasing the proportion y

•How to construct the complete portfolio if y=0.56 ?

Only government issue default-free bonds because of tax and control of money supply◦Guaranteed real rate only if the duration of

the bond is identical to the investor’s desire holding period

Treasury bills as risk-free asset◦ Short-term

insensitive to interest rate fluctuations Inflation uncertainty negligible

5.3 Portfolios of one Risky Asset and a Risk-free Asset

Examine risk-return combinations available to investors

Concern: y (allocated to risky portfolio P) 1-y (allocated to risk-free asset F) Expectation of the complete portfolio’s rate of

return

SD of the completer portfolio

1c P f

f P f

E r yE r y r

r y E r r

c py

Expectation of the complete portfolio’s rate of return◦ Base rate is the risk-free rate◦ Expected to earn a risk premium depends on

Risk premium of P Position in P

c f P fE r r y E r r

Base rate of the

portfolio

The risk premiu

m

rf = 7%rf = 7% rf = 0%rf = 0%

E(rp) = 15%E(rp) = 15% p = 22%p = 22%

y = % in py = % in p (1-y) = % in rf(1-y) = % in rf

0.07 0.15 0.07

0.22c

c p

E r y

y y

Plot the portfolio characteristics in expected return-SD plane (Discuss: y=0, y=1, 0<y<1)

c f P fE r r y E r r c py

Capital Allocation Line (CAL) -Investment Opportunity Set:◦The set of feasible expected return and

standard deviation pairs of all portfolios resulting from different values of y.

◦All the risk-return combination available

Capital Allocation Line (CAL)

Reward-to-Variability ratio (Sharpe ratio):

◦ Slope of CAL

◦ Incremental return per incremental risk

c f P fE r r y E r r

cc f P f

p

E r r E r r

P f

p

E r rS

Leverage (buy on margin)◦ The portfolio plotted on the CAL to the right of P

Left to P, lending at risk-free rate (y<1,1-y>0, buy risk-free asset)

Right to P, borrow at risk-free rate (short position in risk-free asset, y>1,1-y<0)

Example:◦ Budget $300,000, and borrow additional $120,000 at

7%, investing the total funds in the risky asset120,000 300,000

1.4300,000

y

7% 1.4*8% 18.2%

18.2 71.4*22% 30.8%, 0.36

30.8

c

c

E r

S

Leverage◦ Non-government investors are usually

demanded at higher interest rates on loans◦ Suppose borrowing rate is ◦ The slope of CAL:

◦ Left to P, lending at rf=7%, slope=0.36◦ Right to P, borrowing at 9%, slope=0.27

Rf= 9%B

15% 9%0.27

22%

BP f

p

E r rS

Rf= 7%

Rf= 9%B

RP= 15%

σP=22%

PY>1

Y<1

E(r)

σ

5.4 Risk Tolerance and

Asset Allocation

CAL: graph of all feasible risk-return combinations available from different asset allocation choices

Choose one optimal portfolio C from CAL, Trade-off between risk and return

Different Risk aversion-----Different positions in risky assets

Maximize utility by choosing the best allocation to the risky asset, y

Expected return of the complete portfolio is given by:

Variance is:

Utility is

c f P fE r r y E r r

21( )

2U E r A

c py

Highest U

Utility increases as y increases, but eventually it declines

2

2 2

1ax ( )

21

2

0

c cy

f P f p

M U E r A

r y E r r Ay

U

y

2

P f

p

E r ry

A

Inversely proportional to A and risk level

Directly proportiona

l to risk premium

Indifference curve: a graph in the expected return-standard deviation plane of all points that result in a given level of utility

Example:◦ A=4, start from all in risk free assets (Rf=5%),

U=0.05, (0,0.05)◦ Then add a risky portfolio, σ=1%, maintain the same

utility, find E(r) U=E(r)-0.5*A*0.01*0.01=0.05 , then E(r)=5.02%

(0.01,0.0502)◦ Repeat for many other levels of σ to find E(r) to

maintain U=0.05, get the combinations of (E(r) , σ)

intercept

Higher indifference

curves offer a higher

expected return for any given level of

risk;More risk-

averse investors have

steeper indifference

curves, require greater

increase in E(r) to

compensate for increase in risk

A=4, calculate E(rc) for U=0.07,U=0.078,U=0.08653,U=0.094

Calculate E(rc) on CAL for σ in column 1

Calculate E(rc) on indifference curve

Tangency point corresponds to SD and E(r) of the optimal complete portfolio

5.5 Passive Strategy: the Capital Market Line

CAL: risk-free asset and risky portfolio P How to construct the risky portfolio P:

◦ Passive strategy and active strategy Passive strategy involves a decision that avoids

any direct or indirect security analysis◦ A natural candidate for a passively held risky asset

would be a well-diversified portfolio of common stocks

Because a passive strategy requires devoting no resources to acquiring information on any individual stock or group we must follow a “neutral” diversification strategy

To select a diversified portfolio of stocks that mirrors the value of the broad market

CML: the capital allocation line provided by 1-month T-bills and a broad index of common stocks

A passive strategy generates an investment opportunity set that is represented by the CML

Involves investment in two passive portfolios◦ Short-term T-bills◦ Fund of common stocks that mimics a broad

market index Reasons for passive investing

◦ Active strategy entails costs◦ Free-rider benefit, well-diversified portfolio of

common stock will be a reasonably fair buy