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In financial mathematics, the cumulative discount factor is a variable related to the analysis of annuities. Annuities are constant cash flows that are fixed for a set period of time and arise from a single initial investment. The cumulative discount factor is slightly complex to understand in its mathematical form: [1 -- (1 + r)^(-n)]/r, where r is the interest rate and n is the number of periods the annuity exists. However, the true purpose of the cumulative discount factor is not to be understood but to be used. The cumulative discount factor has many practical uses throughout finance.

Compute the current value of a given annuity. Multiply your cash flow by the cumulative discount factor to yield the present value of the annuity to which the cash flow is related. For example, if your annuity with cash flow of 2,000 has been around 5 years and has a 10-percent cumulative discount factor, you can calculate its present value. Compute the cumulative discount factor: [1 -- (1 + .10)^(-5)]/(0.1), which is equal to 3.8. Multiply this value by the cash flow, 2,000. The result, 3.8 x 2,000 = 7,600, the present value of the annuity.

Calculate periodic savings. Set a goal for a future amount of money. Decide how long you wish to take to achieve that monetary goal. Compute the cumulative discount rate. Divide your goal by the cumulative discount factor to arrive at how much money you need to set aside yearly. For example, you might want to save £260,000 over seven years. If you know that the interest rate is stable at 1 per cent, the cumulative discount factor is [1 -- (1 + 0.1)^(-7)]/(0.1), or 4.87. Divide 400,000 by 4.87 to yield £53,388.0, the amount you must save each year.

Determine loan payments. Divide the loan amount by the cumulative discount factor to know how much you should pay back each period. For example, if you have taken out a £13,000 loan at a monthly interest rate of 1 per cent and plan to repay the loan in three years (36 months), your cumulative discount factor is [1 -- (1 + .1)^(-36)]/(0.1), which equals 9.68. Divide 20,000 by 9.68 to see that you must pay £1,342.90 every month to repay your loan in time.

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- "An Undergraduate Introduction to Financial Mathematics"; J. Robert Buchanan; 2006

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