CHAPTER 5

Mathematics of Individual Finance

[Mathematics of Individual Finance Page 1] [Time Value Of Money][Future Values] [More Frequent Compounding] [Effective Annual Interest Rate] [Present Value]
[Annuities] [Present Value Of An Annuity] [Annuities Due] [Future Value Of An Annuity] [Determining Payments] [Bond Valuation]
[Uneven Cash Flows] [Amortization Schedule] [Household Capital Budgeting] [Duration] [Summary] [Discussion Questions]
[Problems]

Present Value of an Annuity

The present value of an annuity cannot be easily calculated arithmetically as can the present value of a future sum. Rather, the present value of each payment must be calculated by dividing each by 1 plus the discount rate raised to the power of the number of periods involved. This is most easily seen in an example.

Example 5-7

Brian has won a scholarship which pays him $5,000 per year for 3 years beginning a year from today. Brian wants to know the present value of the scholarship using a discount rate of 7%.

Solution 5-7

We will solve this using a variant of Equation 5-6:

PV0= FV1/(1+r)1 + FV2/(1+r)2 + FV3/(1+r)3
= $5,000/(1.07) + $5,000/(1.07)2 + $5,000/(1.07)3
= $4,672.90 + $4,367.19 + $4,081.49
= $13,121.58

Since in an annuity, the future values are all equivalent payments, the general form of the formula is:

PV0= PMT/(1+r)1 + PMT/(1+r)2 + ...+ PMT(1+r)n
= PMT[1/(1+r)1 + 1/(1+r)2 +...+ 1/(1+r)n]

The portion of the formula in brackets, [1/(1+r)1 + 1/(1+r)2 +...+ 1/(1+r)n] is called the present value interest factor for the annuity which is found in Appendix page A-4. Because of the complexity of this calculation, it is generally done on a financial calculator or by computer or, if the discount rate and periods match, with the aid of a table.

In a financial calculator, the following buttons must be pushed: [3 N 7 %i 5000 PMT CPT PV]. The answer is $13,121.58 which is identical to that given above.

The present value interest factor for the annuity of 3 periods and 7% is 2.6243. Multiplying this by the payment amount of $5,000 gives us $13,121.50.

On a LOTUS spreadsheet, the formula would be @PV(payments,interest,term). In this case it would be @PV(5000,.07,3). Note that in LOTUS the interest rate is entered as a decimal.

Example 5-8

Bob has just retired and will receive an annuity of $1,000 per month for 20 years. Payments are made at the end of each month. If Bob uses an discount rate with an annual return of 7%, what is the present value of his annuity?

Solution 5-8

The number of periods are 20 years times 12 months or 240 months. The interest rate is 7% divided by 12 months or 0.583% per month. Payments are $1,000 per period. The present value [20 x 12 = N 7/12 = %i 1000 PMT CPT PV] is $128,982.51. On a LOTUS spreadsheet the formula would be @PV(1000,.00583,240).

[Annuities Due]

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