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【正文】 11Present Value of an Annuity Due (cont.) ? Therefore, the present value of our example annuity due is: ? ? ? ?PV AD ??????????????? ??1001 11 100 10100 416 985 1...? Note that this is higher than the PV of the, otherwise equivalent, regular annuity Future Value of an Annuity Due ? To calculate the FV of an annuity due, we can treat it as regular annuity, and then take it one more period forward: ? ? ? ?FV PmtiiiADN?? ??????????1 110 1 2 3 4 5 Pmt Pmt Pmt Pmt Pmt Future Value of an Annuity Due (cont.) ? The future value of our example annuity is: ? ? ? ?FV AD ???????????1001 10 10 101 10 671 565... .? Note that this is higher than the future value of the, otherwise equivalent, regular annuity Deferred Annuities 遞延年金 ? A deferred annuity is the same as any other annuity, except that its payments do not begin until some later period ? The timeline shows a fiveperiod deferred annuity 0 1 2 3 4 5 100 100 100 100 100 6 7 PV of a Deferred Annuity ? We can find the present value of a deferred annuity in the same way as any other annuity, with an extra step required ? Before we can do this however, there is an important rule to understand: When using the PVA equation, the resulting PV is always one period before the first payment occurs PV of a Deferred Annuity (cont.) ? To find the PV of a deferred annuity, we first find use the PVA equation, and then discount that result back to period 0 ? Here we are using a 10% discount rate 0 1 2 3 4 5 0 0 100 100 100 100 100 6 7 PV2 = PV0 = PV of a Deferred Annuity (cont.) ? ?PV 251001 11 100 10379 08???????????????...? ?PV 0 2379 081 10313 29? ?...Step 1: Step 2: FV of a Deferred Annuity ? The future value of a deferred annuity is calculated in exactly the same way as any other annuity ? There are no extra steps at all Uneven Cash Flows ? Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity ? We can find the present or future value of such a stream by using the principle of value additivity Uneven Cash Flows: An Example (1) ? Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment? 0 1 2 3 4 5 100 200 300 ? ? ? ? ? ?PV ? ? ? ?1001 072001 073001 07513 041 2 3. . ..Uneven Cash Flows: An Example (2) ? Suppose that you were to deposit the following amounts in an account
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