Question

If the client was investing the money for 10 years in investment type Y , one way to think about this would be that the return in the first year is Y1 ∼ N(5, 3^2 ) and the return in the 2nd year is Y2 ∼ N(5, 3^ 2 ), . . ., and finally the return in the 10th year is Y10 ∼ N(5, 3^2) We could then define the average return as AY = (Y1+Y2+...+Y10)/10 , assuming the returns are independent from year to year calculate the distribution of this average return for AY . Would you change your advice for the risk averse and risk taking clients that you had in (c) regarding investment Y if their investment was for 10 years? [6 marks] Note: The geometric average would probably be a better indicator of average return but that is beyond the scope of this course.

Previous questions:

Your client is planning to invest some money with your investment company. The client narrows their options down to 2 investment strategies X and Y , and from past experience the annual percentage returns are as follows: X ∼ N(3, 1 2 ) and Y ∼ N(5, 3 2 ).

(a) For each investment strategy, calculate the probability of a negative return. [4 marks]

(b) For each investment strategy, calculate the probability of return greater than 5.5%. [4 marks]

(c) Describe the advice you might give to your client about these two investments depending on whether they were risk averse or a risk taker. [2 marks]

(d) Find k such that P(X > k) = P(Y > k) Do not simply solve this by trial and error, you need to show working out as to how your proved this. [4 marks] (

e) Assume that within the same year Corr(X, Y ) = 2 3 and that X and Y are bivariate normal. Remember X and Y being bivariate normal just means that sums of X and Y will still follow a normal distribution. Then answer the following questions: (i) If your client invests half of their money in each investment type, what are the mean and variance of their annual return? [6 marks] (ii) If your client invest a fraction k of their money in investment type X and the rest in Y (ie 0 ≤ k ≤ 1), what are the mean and variance of their annual return in terms of k? [5 marks] (iii) If your client invest a fraction k of their money in investment type X and the rest in Y (ie 0 ≤ k ≤ 1), what value of k makes the probability of their investment achieving a negative return equal to 1%. [6 marks] Note: this is a very difficult question and will require working with a quadratic function. Just do your best.

Answer 1

P(z<Z) table :

4.

a.

for X P(return < 0) :

for X P(return < 0) = 0.0013

for Y P(return < 0) :

for Y P(return < 0) = 0.0478

b.

for X P(return > 5.5%) :

for X P(return > 5.5%) = 0.0062

for Y P(return > 5.5%) :

for Y P(return > 5.5%) = 0.4338

c.

for risk averse :

take strategy that has least probability of negative return

for X P(return < 0) = 0.0013

for Y P(return < 0) = 0.0478

therefore for risk averse they should take strategy X

for risk taker:

take strategy that has highest probability of return > 5.5%

for X P(return > 5.5%) = 0.0062

for Y P(return > 5.5%) = 0.4338

therefore for risk taker they should take strategy Y

d) P(X>k) = P(Y>k)

=> P( (3+Z*1)>k) =P( (5+Z*3)>k)

So, basically 3+Z*1 = 5+Z*3

So, Z = -1

For Z = -1 , X = 3+(-1)*1 = 2

So, k = 2

So, the probability of getting a higher return than 2% is the same for X and Y as both has the same value of Z =-1 for this probability and the probability is 0.8413 or 84.13%

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