Stochastic Rate Of Return

[asmkdas]

An individual wishes to make an investment that will pay out £200,000 in twenty years’ time. The interest rate he will earn on the invested funds in the first ten years will be either 4% per annum with probability of 0.3 or 6% per annum with probability 0.7. The interest rate he will earn on the invested funds in the second ten years will also be either 4% per annum with probability of 0.3 or 6% per annum with probability 0.7. However, the interest rate in the second ten year period will be independent of that in the first ten year period.

(i) Calculate the amount the individual should invest if he calculates the investment using the expected annual interest rate in each ten year period.

(ii) Calculate the expected value of the investment in excess of £200,000 if the amount calculated in part (i) is invested.

(iii) Calculate the range of the accumulated amount of the investment assuming the amount calculated in part (i) is invested.

Answer:
I got the j=0.054,s^2=0.000084
Hence answer of (i) P(1.054^20)=200000 => P=69858.26
But completely unable to get the answers of (ii) & (iii)
Please help.

 

[Mark Mitchell]

This is Q7 on the UK September 2012 paper.

In (ii), the three possible accumulated values at time 20 are:

(A) 69,858.26*(1.04)^20, which occurs with probability 0.3*0.3
(B) 69,858.26*(1.04)^10*(1.06)^10, which occurs with probability 2*0.3*0.7
(C) 69,858.26*(1.06)^20, which occurs with probability 0.7*0.7

Using the standard approach to calculating an expectation, the expected amount in excess of 200,000 is:

(A - 200,000)*0.3*0.3 + (B - 200,000)*2*0.3*0.7 + (C - 200,000)*0.7*0.7

= 1,336.54

In (iii), the range is just the highest accumulated value that could be obtained (C), minus the lowest accumulated value that could be obtained (A).