Question

Suppose you observe the following situation:

Security	Beta	Expected Return
Cooley, Inc.	1.6	19%
Moyer Co.	1.2	16%

If the risk-free rate is 8 %, are the securities correctly priced? What would the risk-free rate have to be if they are correctly priced?

Answer 1

Part 1:

First we will check whether the securities are correctly priced or not.

We will use the capital asset pricing model formula, which states that:
Expected return = Risk free rate + Beta*(Market return -Risk free rate)
=>(Expected return-Risk free rate)/Beta=(Market return -Risk free rate)
Market return -Risk free rate=Market risk premium
=>Market risk premium=(Expected return-Risk free rate)/Beta

If the securities are correctly priced, then the market risk premium for both the securities should be equal.

Market risk premium for Cooley, Inc. is:
=(19% - 8%)/1.6
=0.11/1.6
=0.06875 or 6.88% (Rounded to 2 decimal places)

Market risk premium for Moyer Co. is:
=(16% - 8%)/1.2
=0.08/1.2
=0.066666667 or 6.67% (Rounded to 2 decimal places)

As the values of market risk premium do not match, these securities are not correctly priced.

Part 2:
Calculation of risk free rates:
The formula we will use to calculate the risk free rates is:
Expected return = Risk free rate + Beta*Market risk premium
Lets, denote risk free rate as RF and market risk premium as MRP

For Cooley Inc.
19% = RF + 1.6*MRP
=>RF=19%-1.6*MRP

For Moyer Co.
16% = RF + 1.2*MRP
=>16% -1.2*MRP = RF
RF=16% -1.2*MRP   

Equating both the equations, we get;
19%-1.6*MRP = 16% -1.2*MRP
=>19%-16% = 1.6*MRP -1.2*MRP
=>3% = MRP*(1.6 - 1.2)
=>0.03 = MRP*0.4
=>MRP=0.03/0.4=0.075

Now,
RF=16% -1.2*MRP
=>RF=16% -1.2*0.075
=16% -0.09
=0.07 or 7%

Answer: The risk-free rate that makes both the securities correctly priced is 7%