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1. Introduction

Economic trends vary from time to time. No one ensures today’s gain will lead to the consequence that I will still profit tomorrow. Therefore, it is truly crucial to generate scenarios for various economic variables and asset returns using Monte Carlo technique across multiple time periods. Moreover, many economic variables such as equity returns, credit transitions and exchange rates are stochastic processes. As a simple illustration for equity scenarios, we can use the basic Geometric Brownian Motion, which is given by: , which is unpredictable ‘random shocks’. In order to deal with the randomness, Economic Scenario Generators produce the forward-looking situations for a specified set of risk
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Using Time Value of Money calculation， , which is the present value, DF is the discount factor and is the future value.

Under the continuously compounded hypothesis, we get , where can be either stochastic or deterministic processes.

2.2 Annuity valuation

When the policyholders retire, they can choose to buy an annuity from an insurance company using the previous lump sum. The annuity will offer 20 annual payments paid at the end of each year. The payment amounts depend on interest rate at retirement and can be formulated by solving: for the unknown , where represents the lump sum and means the price of a zero coupon bond maturing at time t as seen at time 15. Prove that if the interest rates at retirement are 6% per year, then the annual payment are approximately £43,600, and then calculate the annual payments if the interest rates were to fall to 1%.

Firstly, we can demonstrate using the given annual payment £43,600 to calculate the lump sum. The lump sum is £500,088.5651 £500,000.

Next, we will use the £500,000 to calculate the value of X.

As known before, the lump sum is £500,000, which is the present value at time 15.

Using the annuity paid annually in arrears equation to find the accumulated value of all cash flows received, , where is the annual cash flow, is the interest rate and denotes years. Substituting with what we have already known, , we get

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