• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

Stochastic Rate Of Return

A

asmkdas

Member
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.
 
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).
 
Back
Top