Economics Dynamics Problems 124

# Economics Dynamics Problems 124 - FV = A ³(1 r n − 1 r...

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108 Economic Dynamics Example 3.7 An investor makes an initial deposit of £ 10,000 and an additional £ 250 each year. The market interest rate is 5% per annum. What are his accumulated savings after five years? For this problem, Y 0 = £ 10 , 000 , a k = £ 250 for all k and b = (1 + r ) = 1 . 05. Hence Y 5 = 250 1 (1 . 05) 5 1 1 . 05 + (1 . 05) 5 (10000) = £ 14 , 144 . 20 3.7 Discounting, present value and internal rates of return Since the future payment when interest is compounded is P t = P 0 (1 + r ) t , then it follows that the present value , PV , of an amount P t received in the future is PV = P t (1 + r ) t and r is now referred to as the discount rate . Consider an annuity. An annuity consists of a series of payments of an amount A made at constant intervals of time for n periods. Each payment receives interest from the date it is made until the end of the n th-period. The last payment receives no interest. The future value, FV , is then FV = A (1 + r ) n 1 + A (1 + r ) n 2 + · · · + (1 + r ) A + A Utilising a software package, the solution is readily found to be
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Unformatted text preview: FV = A ³ (1 + r ) n − 1 r ´ On the other hand, the present value of an annuity requires each future payment to be discounted by the appropriate discount factor. Thus the payment A received at the end of the Frst period is worth A / (1 + r ) today, while a payment A at the end of the second period is worth A / (1 + r ) 2 today. So the present value of the annuity is PV = A (1 + r ) + A (1 + r ) 2 + ··· + A (1 + r ) n − 1 + A (1 + r ) n with solution PV = A ³ 1 − (1 + r ) − n r ´ Example 3.8 £ 1,000 is deposited at the end of each year in a savings account that earns 6.5% interest compounded annually. (a) At the end of ten years, how much is the account worth? (b) What is the present value of the payments stream?...
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