Question

Suppose that a two-factor model, where the factors are the market return (Factor 1) and the growth rate of industrial production (Factor 2), correctly describes the return generating processes of all assets and the corresponding two-factor APT correctly prices three well-diversified portfolios, A, B, and C.

Portfolio	Expected Return	Sensitivity to Factor 1	Sensitivity to Factor 2
A	15%	1	0
B	20%	1	1
C	5%	0	0

What are i) the risk premiums of the two factors and ii) the risk-free rate?

Another well-diversified portfolio D has sensitivities 0 and 1 to factor 1 and factor 2, respectively. What is the APT-consistent expected return on Portfolio D?

           

Suppose that Portfolio D’s expected return is 8%. Given your answers above, design an arbitrage strategy involving Portfolios A, B, C, and D. (Hint: an arbitrage strategy requires no initial investment, has no risk and yet generates a positive return.)

(Total for Question: 10 marks)

Answer 1

The 2 factor APT equation will be :

Expected Return E(R) = r_f? + b₁ f₁ + b₂ f₂ where r_?f is the risk free return and b is the sensitivity to respective factors.

We have 3 equations given to us:

Portfolio A : 15% = r_f? + 1 * f₁? + 0 * f?₂

Portfolio B : 20% = r_f? + 1 * f₁? + 1 * f?₂

Portfolio C : 5% = r_f? + 0 * f₁? + 0 * f?₂

_??From Portfolio C, we derive r_f? = 5%. Plugging this into Portfolio A, we get f₁? = 10% and finally into Portfolio B and we get f₂ = 5%.

Portfolio D : E(R)_?D? = 5% + 0 * 10% + 1 * 5% = 10%

Since there is a mismatch, we have arbitrage opportunity as below:

• ?Since we are given the expected returns, we arrive the current prices basis the 1 period hence returns given to us above (on a base of 100).
• Expected current prices should be Portfolio A = 100 /(1+15%) = 86.96; Portfolio B = 100/(1+20%) = 83.33; Portfolio C = 100/(1+5%) = 95.24; Portfolio D_{APT consistent} ?= 100/(1+10%) = 90.91 and    Portfolio D_?expected ?= 100/(1+8%) = 92.59
• Now we see that as per APT Portfolio D should be priced at 90.91 but it is priced at 92.59. Hence we should short at 92.59 and create a synthetic Portfolio D by buying Portfolio B & C and selling Portfolio A. The net cash flows will be : (92.59-83.33-95.24+86.96) = 0.98
• At the end of the period: We will receive 100 each for all the 4 portfolio and the net cash flow would be zero since we are short on 2 and long on 2 portfolios. Thus we would have made an arbitrage profit of 0.98 in this scenario.