1. Two electronic traded funds (ETF’s) which we call ETF A and ETF B have the fo

Question

1. Two electronic traded funds (ETF’s) which we call ETF A and ETF B have the following distribution for their annual rate of return. Let X and Y be the rate of returns for the ETFs.

                            ETF X                                                      ETF Y

Rate of Return x,   Probability                   Rate of Return, y            Probability

        +30%                    .25                                          +15%                          .40

        +10%                    .30                                          +5%                             .25

        +5%                     .20                                             0%                            .20

        -10%                    .25                                            -5%                            .15

           

a)Assuming the ETF return probabilities are independent, find the joint probability distribution f(x,y).

b)If $40,000 is invested in ETF X, what is the probability of losing money? What if it is all $40,000 is invested in ETF Y? Which investment is safer? Explain.

c)Find the 4 by 4 payoff matrix when $10,000 is invested in ETF X and $30,000 is invested in ETF Y.

d)What is the probability of a loss when $10,000 is invested in ETF X and $30,000 is invested in ETF Y?

Explanation / Answer

b) If $40000 is invested in ETF X then probability of losing money = 0.25 as given with -10% return , that is 10% loss with probability 25%. On the other hand if the same amount is invested in ETF Y then probability of losing money is 0.15 with 5% loss.

That is loss in ETF X= 10% of 40000 = $4000

loss in ETF Y = 5% of 40000 = $ 2000

Therefore,ETF Y is better than ETF X.

c)

Above figures are in dollors. For example, 30% of 10000 + 15% of 30000 = 3000 + 4500 = $7500 etc.

d) The probability of a loss when $10,000 is invested in ETF X and $30,000 is invested in ETF Y = 0.05 + 0.045 + 0.03 +0.0375 = 0.1625 ( BOLD FIGURES ARE LOSSES IN THE PAYOFF MATRIX. RESPECTIVE PROBABILITIES IN ABOVE JOINT PROBABILITY DISTRIBUTION TABLE ARE FOR LOSSES WHICH ARE ADDED)

Y=15% Y=5% Y=0% Y= -5% Total X=30% 0.25*0.4 = 0.1 0.25*0.25= 0.0625 0.25 *0.20= 0.05 0.25*0.15=0.0375 0.25 X=10% 0.30*0.4= 0.12 0.30*0.25= 0.075 0.30* 0.20= 0.06 0.30 * 0.15 = 0.045 0.3 X=5% 0.20*0.4=0.08 0.20*0.25=0.05 0.20*0.20=0.04 0.20*0.15= 0.03 0.2 X= -10% 0.25*0.4 = 0.1 0.25*0.25= 0.0625 0.25 *0.20= 0.05 0.25*0.15=0.0375 0.25 Total 0.4 0.25 0.2 0.15 1

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