Suppose that the S&P 500, with a beta of 1.0, has an expected return
of 13 percent and T-bills provide a risk-free return of 5 percent.

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. Based on 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?

Hi!!


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?

Since T-Bills offer a risk free rate of return, their beta is zero.

Portfolio (i):
                0% of S&P500
              100% of T-Bills

Beta(i) = weighted sum of the betas of each asset =
                  = 0 * 1 + 1 * 0 =
                  = 0
r(i) = weighted sum of the returns of each asset =
     = 0 * 13% + 1 * 5% =
     = 5%



Portfolio (ii):
               25% of S&P500
               75% of T-Bills

Beta(ii) = 0.25 * 1 + 0.75 * 0 = 0.25

r(ii) = 0.25 * 13% + 0.75 * 5% = 7%



Portfolio (iii):
                50% of S&P500
                50% of T_Bills

Beta(iii) = 0.5 * 1 + 0.5 * 0 = 0.5

r(iii) = 0.5 * 13% + 0.5 * 5% = 9%



Portfolio (iv):
               75% of S&P500  
               25% of T-Bills

Beta(iv) = 0.75 * 1 + 0.25 * 0 = 0.75

r(iv) = 0.75 * 13% + 0.25 * 5% = 11%



Portfolio (v):
              100% of S&P500
                0% of T-Bills

Beta(v) = 1 * 1 + 0 * 0 = 1

r(v) = 1 * 13% + 0 * 5% = 13%

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

In this case the risk/return trade-off statement is valid:
When beta goes down (i.e. the risk decrease) the expected return
increases, that is, more risk offer a higher expected return.

Note that in this particular case the rate of each portfolio is:
rP = Beta_P * S&P500 rate + (1-Beta_P)*T-Bills rate =
   = Beta_P*(S&P500 rate - T-Bills rate) + T-Bills rate 

There is a linear positive relationship between rate and risk, more
risks implies more expected returns.

            ---------------------

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

The security market line is the line (graph) representing the
relationship between expected return and market risk.
S&P500 index is widely used as a benchmark for the overall market
performance because it is considered the better representation of the
U.S. market (in fact, to many it is the definition of the market).
Because of that the S&P500 expected return is considered the market
expected return.
T-Bills rate is used as the Risk free rate for the Financial Analysis.
The above paragraph lead us to rewrite the equation:
rP = Beta_P*(S&P500 rate - T-Bills rate) + T-Bills rate 
as
rP =Beta_P*(rM - rF) + rF
where:
rP = rate of return of the portfolio
Beta_P = beta of the portfolio
rM = Expected market return
rF = risk free rate

but this is the CAPM formula!! and also the equation used to represent
the security market line.

So answering part b) we have found the security market line relationship.

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I hope that this helps you. Feel free to request for a clarification
if you need it.

Regards.
livioflores-ga

Thank you very much!