3. Suppose that the S&P 500, with a beta of 1.0, has an expected return of 13% a

Question

3. Suppose that the S&P 500, with a beta of 1.0, has an expected return of 13% and T-bills provide a risk-free return of 4%.

a). What would be the expected return and beta of portfolios constructed from these two assets with weights in the S&P 500 of (i) 0; (ii) .25; (iii) .5; (iv) .75; (v) 1.0?

b). On the basis of your answer to (a), what is the trade-off between risk and return, that is, how does expected return vary with beta?

c). What does your answer to (b) have to do with the security market line relationship?

Explanation / Answer

In this problem an investor has two asset to invest. First one is risk free T bill. It is a bond issued by federal government. It is risk free in the sense that it has no liquidity / default risk. Sometimes a firm may fail to pay debt when it bacomes due. It can be due to cash crunch. Government does not encounter this risk. If it appears the government can print notes and circulate in the economy. Thus government issued bonds are considered as risk free investment. Due to this nature its rsk indicator beta has a zero value.

Second investment asset is market portfolio S&P 500. It is a portfolio constructed from listed asset. Due to its diverse nature the portfolio has no company specific unsystematic risk. Only systematic risk is there. It is indicated by beta value 1.

Investor can construct a portfolio by mixing these two assets in different ratios. The return of this portfolio is the weighted average return of two assets. Also portfolio beta is the weighted average of beta value of these two assets. Thus portfolio rerurn and portfolio beta under different weights of S&P 500 are shown below:

i. S&P 500 weight is 0:

Consider S&P expected return E(S&P 500)= 13% and T bill return E( T bill) =4%.

Portfolio return = 0 x 13% + 1 x 4% = 4%

Beta of portfolio = 0 x 1 + 1 x 0 = 0

ii. S&P 500 weight is 0.25:

Portfolio return = 0.25 x 13% +0.75 x 4% = 3.25% +3% = 6.25%

Beta of porfolio = 0.25 x 1 +0.75 x 0 = 0.25

iii. S&P 500 weight of 0.50:

Portfolio return = 0.50 x 13% +0.50 x 4%^ = 6.5% +2% =8%

Portfolio beta = 0.50 x 1 +0.50 x 1 = 0.50

iv. S&P 500 weight of 0.75:

Portfolio return = 0.75 x 13% +0.25 x 4% = 9.75% + 1% = 10.75%

Portfolio beta = 0.75 x 1 +0.25 x 0 = 0.75

v. S&P 500 weight of 1:

Portfolio return = 1 x 13% + 0 x 4% = 13%

Portfolio beta = 1 x 1 + 0 x 0 = 1

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Answer of (b) Risk return trade off is a comparison of risk and return under different situation. It indicates portfolio return per unit of risk. In this example portfolio return and porfoli beta value indicating portfolio risk will be compared for five different portfolio. Porfolio with highest return per unit risk is selected. Conside the result above. It is summarized in the table below:

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Answer (c):

Above table compares return and risk of different portfolios. If you look at these figures then you will observe a linear diagram. In this diagram, horizontal axis measures beta value and vertical axis will indicate portfolio return. This linear diagram will start from a positive intercept of 4%. Its slope will be the difference between S&P 500 expected return and T bill expected return. Thus linear equation is:

Portfolio return = T-bill risk free return + portfolio beta ( S$P 500 expected return - T-bill treturn)

This equation is nothing but the equaton of security market line (SML). It indicates combination of risk fre investment with risky portfolio.

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