05-Risk Return and CAL_2014

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Risk, Return and the CapitalAllocation Line

Marriott School of Management

Fin 410

Fall 2014

Rob Schonlau

Last updated September 15, 2014

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Your friends temporarily entrust 2 million to

you to invest. They asked you to invest it in

the best possible m nner

Earlier lectures discussed the types of assets you could invest inas well as the return and risk measures used to think aboutfinancial performance.

But nothing has been said thus far about how to best combineassets into the optimal portfolio. This lecture builds on earlier

intuition about risk and return and considers a simplified situationwhere you can only choose between investing in a risk-free assetand a single risky asset and asks the question: how much of your

wealth should go into the risk-free asset? How much in therisky asset? Why?

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Lecture 5 outline

Introduce the asset allocation choice:

Assume there are 2 assets in which to invest. One is a riskyasset and the other is a risk free asset. The asset allocation

question is how much of your wealth should be invested inthe risky asset? How much in the risk free asset?

Introduce the Capital Allocation Line (CAL) and Sharpe ratio

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The asset allocation question restated for

intuition. . . How much do you punch the accelerator? How fast do you want

to go? How do you make this decision? On some level, you arebalancing the expected risk against the expected enjoyment youget from speed.

Greater speed can be thrilling but it can also get pretty ugly.

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Human preferences

All else equal, people prefer higher expected returns. Thusgiven a choice between two investments of the same risk they willchoose the one with higher expected return.

All else equal, people prefer lower risk investments. Thus given achoice between two investments with the same expected returnthey will choose the one with lower risk.

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Consider a simplified framework

You can choose between 2 assets:

A risk-free bond Arisky asset (this could be a portfolio)

You can go long or short in your positions

Shorting the risk-free bond is like borrowing to buy more of therisky asset.

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What is a risk-free asset?

If we define risk in terms of standard deviation then thestandard deviation of the risk-free assets returns should bezero.

Because a risk free assets returns are not uncertain they aretreated like a constant when using expectations. =

In practice we treat short-term US T-Bills as risk-free assets.

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What is a risky asset?

A risky asset has uncertain future returns. (I.e. the expected 0). Conceptually we can think of a portfolio of riskyassets as a single risky asset.

In practice we could consider a single stock, a corporate bond,or a local government bond as this asset. Similarly we couldcombine collections of these assets to form a new single riskyasset (portfolio).

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Borrowing as short-selling

You can think of borrowingas short-sellinga risk-free bond. Seethe cash inflow/outflow comparison below.

Borrowing vs short-selling:Short-sell 1 bond

Borrow $909.09 at 10% FV = 1000, Price = 909.09

Get $909.09 now Get $909.09 nowPay $1000 in future Pay $1000 in future

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The question again

What fraction of investment equity do you put in the risk-free bond?What fraction do you put in the risky asset?

The more wealth you put in the risky asset the higher yourexpected return.

The more wealth you put in the risky asset the higher the risk ofoutcomes far different from the expected return.

We will use Statistics Rules #1 and #2 to think about portfolioexpected return and risk.

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Lecture 5 outline

Introduce the asset allocation choice:

Assume there are 2 assets in which to invest. One is a riskyasset and the other is a risk free asset. The asset allocation

question is how much of your wealth should be invested inthe risky asset? How much in the risk free asset?

Introduce the Capital Allocation Line (CAL) and the Sharpe ratio

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Summarize the portfolios expected return and

standard deviation

Two equations describe the expected return and risk in your portfolio.Given human preferences, we want to maximize E(rp) and minimizeby choosing the weight (percent) of our wealth we invest in each

asset.

Expected Return:

Standard Deviation:

( ) ( ) 1p S fE r wE r w r

p S

w

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Plot the expected return and standard deviation

combinations of different portfolios created by

changing the percent of your wealth invested in the

risky asset.

Assume: Equations:

E[rs] = .08 E[rp] = wE[rS]+(1-w)rf

s = .12 p = ws

rf = .04

Risky (w) Risk-Free(1-w)

A: 0% 100%

B: 100% 0%

C: 50% 50%D: 150% -50% 4%

E[rp]

p

A

8%

.12

B

.06

6% C

.18

10%D

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Capital Allocation

Line (CAL)

What is the equation

for the CAL line on the

previous slide?

What is the intuition for the y-axis intercept?

What is the slope? Rise-over-run for any two points.

Point 1 (x,y): p = 0, E[rp] = rf Point 2 (x,y): p = s , E[rp] = E[rs]

s

fs rrE

Run

Rise

][

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Capital Allocation Line This is just the equation for a line!

( )

( ) S f

p f P

S

E r r

E r r

Y variable Intercept

Slope

X variable

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CAL Example

E[rs] = .08

s = .12

rf = .04

E[rp] = wE[rS]+(1-w)rf

p = ws

Risky Risk-Free

A: 0% 100%

B: 100% 0%C: 50% 50%

D: 150% -50%4%

E[rp]

p

A

8%

.12

B

.06

6% C

.18

10%D

[ ][ ]

s f

p f p

s

E r rE r r

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Sharpe Ratio


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Sharpe Ratio

Sharpe Ratio =

This ratio is also called the reward-to-volatility or reward-to-variability ratio. As you add more of the risky asset to yourportfolio the expected risk premium increases but so does thedenominator.

All else equal, given a choice between two Sharpe ratios youwould prefer the larger one because the expected financial returnwould be higher for each unit of risk.

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Capital Allocation Line

How much do you punch the accelerator? You make the call . . . .

Risk-averse investors will invest more in the risk-free asset.

Risk-tolerant investors will invest more in the risky asset.

Buteveryone should always prefer portfolios with higherSharpe ratios.

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Targeting E[r] with Vanguard

Assume:

The expected return on a Vanguard fund is 12% and thestandard deviation is 0.16.

The risk-free rate is 7%. Assume you can borrow and lend at this rate.

What portfolio weights would you use if you wanted to allocate yourwealth between the risky asset (the fund) and the risk-free asset in

such a way that your portfolio had an expected return of 17%?

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First write the portfolio return formula:

Then use statistics rule 1 to apply expectations:

Invest 200% of investment equity in risky portfolio by borrowingat risk-free rate.

w

w

ww

rwrwErE fsp

2

05.07.17.

07).1(12.017.0

)1(][][

fsp rwwrr )1(

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With a weight of 2.0 in the risky asset, the standard deviation ofthe portfolio would be 2.0*0.16 = 0.32.

What is the intercept and slope of the CAL?

Intercept = 0.07 Slope = (0.12-0.07)/0.16 = 0.3125

17% = 0.07+.3125*.32 [ ][ ] s f

p f p

s

E r rE r r

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E[r]

p

.07

.16

.12

.32

.17

Position Stdev100% in risky portfolio 0.16Desired position 0.32

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Targeting E[r] with Vanguard (continued)

Suppose you have $1000 to invest (investment equity) and theprice of the risky asset is $18/share

Then, according to our calculations, you should buy $2000 ofthe Vanguard Fund.

Buy 2000/18 =111 shares You have to borrow $1000 at 7% to help buy these shares

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Targeting with Vanguard

Suppose you want the standard deviation of your portfolio to be0.25. Remember that the standard deviation of the risky asset byitself is .16. How much should you invest in the risky asset?

Use statistics rule 2

Invest 156% of investment equity in risky portfolio by borrowing atrisk-free rate.

w

w

56.116.0/25.0

16.025.0

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With a weight of 1.56 in the risky asset,

We can also use the equation for the CAL line to find theexpected return:

14.8% = 0.07+.3125*.25

%8.1407.*)56.(12.0*56.1][ prE

[ ][ ]

s f

p f p

s

E r rE r r

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E[r]

p

.07

.16

.12

.25

.148

Position Stdev100% in risky portfolio 0.16Desired position 0.25

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Suppose you have $1000 to invest (investment equity) and theprice of the risky asset is $18/share.

Then, according to our calculations, you buy $1560 of theVanguard Fund.

Buy 1560/18 =87 shares You have to borrow $560 at 7% to buy these shares

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Passive Investing Approach

Select the passive portfolio with the highest Sharpe Ratiopossible

Benefits of passive investing:

No need to spend time researching stocks No need to pay someone else to do research

Indexing is one of the most common passive strategies.

The CAL drawn between the risk-free rate and a broad index ofcommon stocks is called the capital market line (CML).

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05-Risk Return and CAL_2014