5.5asset allocation across risky and risk free portfolios 5-1

5.5 Asset Allocation Across Risky and Risk

Free Portfolios

5-1

Allocating Capital Between Risky & Risk-Free Assets

Possible to split investment funds between safe and risky assets

Risk free asset rf : proxy; ________________________ Risky asset or portfolio rp: _______________________

Example. Your total wealth is $10,000. You put $2,500 in risk free T-Bills and $7,500 in a stock portfolio invested as follows:– Stock A you put ______– Stock B you put ______– Stock C you put ______

$2,500

$3,000

$2,000

T-bills or money market fund

risky portfolio

$7,500

5-2

Weights in rp

– WA =

– WB =

– WC =

The complete portfolio includes the riskless

investment and rp.

$2,500 / $7,500 = 33.33%

$3,000 / $7,500 = 40.00%

$2,000 / $7,500 = 26.67%

100.00%

Wrf = ; Wrp =

In the complete portfolio

WA = 0.75 x 33.33% = 25%; WB = 0.75 x 40.00% = 30%

WC = 0.75 x 26.67% = 20%;

25% 75%

Stock A $2,500Stock A $2,500

Stock B $3,000Stock B $3,000

Stock C $2,000Stock C $2,000

Wrf = 25%

Allocating Capital Between Risky & Risk-Free Assets

5-3

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%

y = % in rpy = % in rp (1-y) = % in rf(1-y) = % in rf

Example

5-4

E(rC) = E(rC) =

Expected Returns for Combinations

E(rC) =

For example, let y = ____

E(rC) =

E(rC) = .1175 or 11.75%C = yrp + (1-y)rf

C = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5%

c =

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


yE(rp) + (1 - y)rf

yrp + (1-y)rf

Return for complete or combined portfolio

0.75

(.75 x .14) + (.25 x .05)

5-5

Complete portfolio

Varying y results in E[rC] and C that are ______ ___________ of E[rp] and rf and rp and rf

respectively.

E(rc) = yE(rp) + (1 - y)rf

c = yrp + (1-y)rf

linearcombinations

This is NOT generally the case for the of combinations of two or more risky assets.

5-6

E(r)

E(rp) = 14%

rf = 5%

22%0

P

F

Possible Combinations

E(rp) = 11.75%

16.5%

y =.75

y = 1

y = 0

5-7

CALCAL(Capital(CapitalAllocationAllocationLine)Line)

Combinations Without Leverage

Since σrf = 0

σc= y σp

If y = .75, thenσc=

If y = 1σc=

If y = 0σc=

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


75(.22) = 16.5%

1(.22) = 22%

0(.22) = 0%

E(rc) = yE(rp) + (1 - y)rf

y = .75E(rc) =

y = 1E(rc) =

y = 0E(rc) =

(.75)(.14) + (.25)(.05) = 11.75%

(1)(.14) + (0)(.05) = 14.00%

(0)(.14) + (1)(.05) = 5.00%

5-8

Using Leverage with Capital Allocation Line

Borrow at the Risk-Free Rate and invest in stock

Using 50% Leverage

E(rc) =

c =

rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


rf = 5%rf = 5% rf = 0%rf = 0%

E(rp) = 14%E(rp) = 14% rp = 22%rp = 22%


(1.5) (.14) + (-.5) (.05) = 0.185 = 18.5%

(1.5) (.22) = 0.33 or 33%

E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5%= 5%

22%22%00

PP

FF

Possible CombinationsPossible Combinations

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

y =.75y =.75

y = 1y = 1

E(rE(rCC) =18.5%) =18.5%

33%33%

y = 1.5

y = 1.5

y = 0y = 0

5-9

Risk Premium & Risk Aversion

• The risk free rate is the rate of return that can be earned with certainty.

• The risk premium is the difference between the expected return of a risky asset and the risk-free rate.

Excess Return or Risk Premiumasset =

Risk aversion is an investor’s reluctance to accept risk.

How is the aversion to accept risk overcome?

By offering investors a higher risk premium.

E[rasset] – rf

5-10

Risk Aversion and Allocation Greater levels of risk aversion lead investors to

choose larger proportions of the risk free rate

Lower levels of risk aversion lead investors to choose larger proportions of the portfolio of risky assets

Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5%= 5%

22%22%00

PP

FF

Possible CombinationsPossible Combinations

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

E(rE(rpp) = 11.75%) = 11.75%

16.5%16.5%

y =.75y =.75

y = 1y = 1

E(rE(rCC) =18.5%) =18.5%

33%33%

E(rE(rCC) =18.5%) =18.5%

33%33%

y = 0y = 0

y = 1.5

5-11

E(r)E(r)E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5% = 5%

= 22%= 22%00

PP

FFFF

rprprprp

) Slope = 9/22) Slope = 9/22

E(rE(rpp) - ) - rrff = 9% = 9%


P or combinations of P or combinations of P & Rf offer a return P & Rf offer a return per unit of risk of per unit of risk of 9/22.9/22.

5-12

Quantifying Risk Aversion

25.0 pfp ArrE E(rp) = Expected return on portfolio p

rf = the risk free rate

0.5 = Scale factor

A x p2 = Proportional risk premium

The larger A is, the larger will be the _________________________________________ investor’s added return required to bear risk

5-13

Quantifying Risk AversionRearranging the equation and solving for A

Many studies have concluded that investors’ average risk aversion is between _______

σ

rrEA

p

fp

2.50

)(

2 and 4

5-14

Using A

What is the maximum A that an investor could have and still choose to invest in the risky portfolio P?

Maximum A =

E(r)E(r)

E(rE(rpp) = 14%) = 14%

rrff = 5%= 5%

= 22%= 22%00

PP

FF

rprp

) Slope = 9/22) Slope = 9/22

E(rE(rpp) ) -- rrff = 9%= 9%


σ

rrEA

p

fp

2.50

)(

0.220.5

0.050.14A

2

3.719

3.719

5-15

Sharpe Ratio

• Risk aversion implies that investors will accept a lower reward (portfolio expected return) in exchange for a sufficient reduction in risk (std dev of portfolio return)

• A statistic commonly used to rank portfolios in terms of the risk-return trade-off is the Sharpe measure (also reward-to-volatility measure)

• The higher the Sharpe ratio the better• Also the slope of the CAL

Sharpe ratio

p

fp rrES

return excess portfolio of dev std

premiumrisk portfolio

• Example: You manage an equity fund with an expected return of 16% and an expected std dev of 14%. The rate on treasury bills is 6%.

71.014

616

deviation Standard

premiumRisk

S

5.6 Passive Strategies and the Capital Market Line

5-18

A Passive Strategy

• Investing in a broad stock index and a risk free investment is an example of a passive strategy.

– The investor makes no attempt to actively find undervalued strategies nor actively switch their asset allocations.

– The CAL that employs the market (or an index that mimics overall market performance) is called the Capital Market Line or CML.

5-19

Active versus Passive Strategies• Active strategies entail more trading costs than

passive strategies.• Passive investor “free-rides” in a competitive

investment environment.• Passive involves investment in two passive

portfolios– Short-term T-bills– Fund of common stocks that mimics a broad

market index– Vary combinations according to investor’s risk

aversion.

5-20

5.5asset allocation across risky and risk free portfolios 5-1

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