An annuity is a series of payments of equal value made at regular intervals in the future. They are most commonly seen in pensions. A standard personal pension works by taking contributions from an individual over the course of their working life. These contributions are then invested to produce a sum of capital known as a pension pot. When the person retires, they use this pension pot to buy an annuity, which pays them a sum of money every year until they die. As such, pension companies need to know the present value of annuities in order to know what size annual pension can be bought for a certain size of pot.
For a simple annuity over three years, the present value would be equal to the sum of three consecutive identical cash flows received at the end of year 1, year 2 and year 3. If C is the value of each cash flow then the present value of the annuity is equal to:
Present Value = C / ( 1 + i ) + C / ( 1 + i ) ^ 2 + C / ( 1 + i ) ^ 3
In general, for an annuity which will last for t years:
Present Value = C / ( 1 + i ) + C / ( 1 + i ) ^ 2 + ... + C / ( 1 + i ) ^ t
This produces a series where every term is a multiple of the previous term. This series can be solved by dividing each side by the multiplicative term, in this case (1+i):
PV / ( 1 + i ) = C / ( 1 + i ) ^ 2 + C / ( 1 + i ) ^ 3 + ... + C / ( 1 + i ) ^ t+1
If this second equation is subtracted from the first, all the terms on the right hand side of the equation will cancel each other out, with the exception of the first and last terms. Therefore:
PV - PV / ( 1 + i ) = C / ( 1 + i ) - C / ( 1 + i ) ^ t+1
Multiplying each side by 1 + i gives:
PV (1 + i) – PV = C – C / (1+i) ^ t
This simplifies to:
i * PV = C (1 – 1 / (1+i) ^ t)
Dividing by i gives:
PV = C (1 – 1 / (1 + i) ^ t) / i
To find the cash flow, simply multiply by i and divide by (1 – 1 / (1+i) ^ t):
C = PV * i / (1 – 1 / (1 + i) ^ t)
This demonstrates that the present value of an annuity will rise as interest rates rise. This helps to explain the ‘pension crisis’ that the media referred to a lot in the second half of 2008 and the start of 2009. As many central banks reduced their interest rates to fight the global recession, so the interest rates used to calculate pensions fell. Therefore, many people coming to retirement age found that their pension would no longer buy them as large an annuity as they expected.
As with the future and present values, it is possible to create an annuity table of the value of (1 – 1 / (1 + i) ^ t) / i for a variety of values of i and t, and this will again often be provided for examinations.
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