investment principle (demo in eiu)

8/10/2019 Investment Principle (Demo in EIU)

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AGENDA

CAPITAL ALLOCATIONBETWEEN RISK AND RISK-

FREE ASSET

CAPITAL ALLOCATIONBETWEEN TWO RISKY

ASSETS


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ASSET ALLOCATION

PROCESS OF CAPITAL ALLOCATION

RISK-FREE ASSET PROPERTIES OF THE COMBINED PORTFOLIO

THE CAPITAL ALLOCATION LINE

OPTIMAL PORTFOLIO

CAPITAL ALLOCATION BETWEEN RISK

AND RISK-FREE ASSET


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ASSET ALLOCATION

Capital allocation is the choice of the proportion

of portfolio.

Example: You have $10,000 to invest. Following

your brokers recommendation you have therisky asset P as 60% stock and as 40% bonds(recall that even government bonds are risky,unless held to maturity). When you choose y

(risky assets proportion) = 0.6, the combinedportfolio C is:


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With y = 0.8, the combined portfolio C would be?

You make yourself to figure it out.

ASSET ALLOCATION


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Examine risk/return tradeoff

Demonstrate how different degrees of risk aversionwill affect allocations between risky and risk free

asset. Consider the optimal risky portfolio as given and

analyze the allocation decision between the riskyasset and the risk-free asset (T-bills)

Rate of return:

PROCESS OF CAPITAL ALLOCATION


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Technically, the risk-free asset is default-free andwithout inflation risk (a price-indexed default-freebond)

In practice, Treasury bills come closest, because:- Short term means little interest-rate or inflation risk

- Default risk is practically zero, since the governmentwould no default

RISK-FREE ASSET


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PROPERTIES OF THE COMBINED PORTFOLIO

Notation:

rf= rate of return on the risk-free asset

rp= rate of return on the risky asset

rc= rate of return on the complete portfolio(including both the risk-free asset and the risky

asset)

y = proportion of the investment budget to be placed

in the risky asset p= standard deviation of the return on the risky

asset

c= standard deviation of the return on the completeportfolio


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Expect rate of return:

Variance:

Standard Deviation:



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Solve for y:

Replace the equation for the expected rate ofreturn:

This defines a line in the mean-variance space the capital allocation line (CAL)



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The CAL gives the trade-off between risk andreturn or the CAL describes all risk-returncombinations available to investor.

It allows to see what expected return on thecombined portfolio (E[Rc]) is attainable for a givenlevel of risk (c).

is the Reward-to-Variability ratio of P

is the risk premium on P

CAPITAL ALLOCATION LINE


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Example: Rf = 7%

E (Rp) = 15%

p = 22% Risk premium = 15% - 7% = 8%

Rate of return of portfolio:

Standard Deviation for portfolio:



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We have shown how to develop the CAL, the graphof all feasible riskreturn combinations availablefrom different asset allocation choices. The investorconfronting the CAL now must choose one optimalportfolio, C, from the set of feasible choices.

Preview formula of Utility:

U = R(rc) 1/2A

where A is the coefficient of risk aversion and 1/2 is ascale factor.

OPTIMAL PORTFOLIO


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The investor attempts to maximize utility, U, bychoosing the best allocation to the risky asset, y.

We derive the formula:

the solution is given by the first-order constraint

solving for y gives the optimal choice of investment inthe risky portfolio

OPTIMAL PORTFOLIO


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Going back to our previous example the optimalssolution for an investor with a coefficient of riskaversion A = 4, 4 is:

n other words, this particular investor will invest41% of the investment budget in the risky asset and

59% in the risk-free asset which Utility ismaximized.

Notice that 3.28/9.02 = .36, which is the reward-to-variability ratio assumed for this problem.

OPTIMAL PORTFOLIO


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You manage a risky portfolio with an expected rate of return of 18% and astandard deviation of 28%. The T-bill rate is 8%.

1. Your client chooses to invest 70% of a portfolio in your fund and 30% in aT-bill money market fund. What is the expected value and standarddeviation of the rate of return on his portfolio?

2. Suppose that your risky portfolio includes the following investments in thegiven proportions:

Stock A: 25%

Stock B: 32%

Stock C: 43%

What are the investment proportions of your clients overall portfolio,including the position in T-bills?

3. Your clients degree of risk aversion is A = 3.5. What proportion, y, of the

total investment should be invested in your fund? What is the expectedvalue and standard deviation of the rate of return on your clients

optimized portfolio

PREVIEW EXERCISE


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A PORTFOLIO WITH TWO RISKY ASSETS

OPTIMAL PORTFOLIO

CAPITAL ALLOCATION OF TWO RISKY

ASSETS


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The Return of portfolio:

R1is the return and w is the weight of asset 1R2is the return and (1-w) is the weight of asset 2

The weights of the portfolio sum to 1.

The variance of the portfolio:



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Recalling that:

The variance of the return on a portfolio of tworisky assets can be expressed as a function of thecorrelation coefficient:


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There are 4 cases of correlation coefficient:

Perfect positive correlation



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Perfect negative correlation,

We make first order constraint, we conclude:



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No correlation,



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