Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows: Stock Expected Return Standard Deviation A 12 % 3 % B 19 9 Correlation = –1 Suppose that it is possible to borrow at the risk-free rate, rf. What must be the value of the risk-free rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.) (Do not round intermediate calculations. Round your answer to 3 decimal places.)

0.135 or 13.5%

Explanation:

Given in the question are the following:

ERA = Expected return of Stock A = 12% = 0.12

ERB = Expected return of Stock B = 19% = 0.19

SDA = Standard deviation of Stock A = 3% = 0.03

SDB = Standard deviation of Stock B = 9% = 0.09

CAB = Correlation between A and B = -1

The correlation of -1 between Stock A and Stock B indicates that there a perfect negative correlation between the two stocks. Therefore, we can create a risk-free portfolio which its rate of return will be the risk-free rate in equilibrium.  

If we let wA denotes the proportion of investment in Stock A, and let wB denotes the proportion of investment in Stock B, the proportion of this portfolio can be obtained by setting its standard deviation equal to zero. Since there is a perfect negative correlation, the standard deviation of this portfolio (SDP) can be given as follows:

Absolute value [(wA × SDA) – (wB × SDB)] = SDP …………………………………….. (1)

Note that wB = (1 – wA) since the sum of the weight must be equal to 1.

Substituting all the relevant values into equation and set SDP = 0, we have  

[(0.03 × wA) − (0.11 × (1 - wA))] = 0

0.03wA – 0.11 + 0.11wA = 0

0.03wA + 0.11wA = 0.11

0.14wA = 0.11

wA = 0.11 ÷ 0.14 = 0.785714285714286

Since wB = 1 –wA, therefore:

wB = 1 - 0.785714285714286 = 0.214285714285714

The expected rate of return of the portfolio (ERP) can be estimated as follows:

ERP = (wA × ERA) + (wB × ERB)   (2)

Substituting all the relevant values into equation (2), we have:

ERP = (0.785714285714286 × 0.12) + (0.214285714285714 × 0.19)  

       = 0.0942857142857143 + 0.0407142857142857

ERP = 0.135 or 13.5%

Therefore, the value of the risk-free rate must be 13.5%.

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