Question

3. Nancy and Dave currently have $700,000 in assets and $260,000 in liabilities. Their average cost of debt is fixed at 7%. Their consumption and tax rates are 40% and 30%, respectively. Taxes are based on returns to assets less interest expense. Suppose the rate of return on assets will be one of five values. Each of the values has a known probability. The five values and respective probabilities are: Return to assets in Percent (%) Probability of Return Being Received

a.

What are the expected value and standard deviation for the rate of return on assets?

b.

What is the expected rate of growth under risk?

c.

What are the standard deviation (risk) and coefficient of variation of the expected rate of growth?

Answer 1

Ans a.

Using the values from the table given, we can calculate the expected rate of return on assets as below:

Expected rate of return = sum of products of probability of return being received and the return being received in that situation.

= r_{​​​​​​1*}p_{​​​​​​1}​​​+ r_​​2*p_​2+ ...+rn*p_{​​​​​​n}

​​​​Expected rate of return = (0.06*0.3 + 0.09*0.2 + 0.12*0.1 + 0.15*0.15 + 0.18*0.25)*100

= 0.1155*100= 11.55%

Here, 11.55% is the value of mean return(r_{​​​​​​m})

Now, Standard deviation for the rate of return on asset =

square root of [(( r₁​​​- r_{​​​​​​m})² +(r_​​​2 - r_{​​​​​​m})²+ ...+ (r_{​​​​​​n} - r_​​​m​​​)²)]/(n-1)

Here, r_{​​​​​​n} is the rate of return at nth observation

n is the number of observation

Standard deviation of the expected rate of return = square root[(0.06-0.1155)^2 + (0.09- 0.1155)^2 + (0.12-0.1155)^2 + (0.15-0.15)^2 + (0.18 - 0.15)^2]/(5-1)

= 0.00465/4

​​​​​​= 0.001162

= 0.1162%

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