Finite Life Streams

Of more practical importance is the case where the payment stream has a finite lifetime Suppose that the stream consists of n periodic payments of amount A, starting at the end of the current period and ending at period n. The pattern of periodic cash flows together with the time indexing system is shown in Figure 3 1

The present value of the finite stream relative to the interest rate r per period is

n Periods

FIGURE 3.1 Time indexing., Time is indexed from 0 lo n. A period is a span between time points, with the first period being the time from 0 to 1 A standard annuity has a constant cash flow at the end of each period

This is the sum of a finite geometric series If you do not recall the formula for this sum, we can derive it easily by a simple trick. The value can be found by considering two perpetual annuities. Both pay an amount A each year, but one starts at time 1 and the other starts at time n -f 1. We subtract the second from the first The result is the same as the original stream of finite life This combination is illustrated in Figure .3,2 for the case of a stream of length 3,

The value of the delayed annuity is found by discounting that annuity by the factor (1 + r)~" because it is delayed n periods Hence we may write r r(l +r)n

We now highlight this important result:

Annuity formulas Consider an annuity that begins payment one period from the present, paying an amount A each period for a total of n periods The present value P, the one-period annuity amount A, the one-period interest rate r, and the number of periods n of the annuity are related by

or, equivalently,

FIGURE 3,2 Finite stream from two perpetual annuities. The top line shows a perpetuity starting at time 1, the second a negative perpetuity starting at time 4 The sum of these two is a finite-life annuity with payments starting at time 1 and ending at time 3

Although these formulas are simple in concept and quite easy to derive, they are sufficiently complex that they cannot be evaluated easily by hand It is for this reason that financial tables and financial calculators are commonly available. Professional tables of this type occupy several pages and typically give P/A as a function of / and n, For some purposes A/P (just the reciprocal) is more convenient, and there are tables written both ways

It is important to note that in the formulas of this section, r is expressed as a per-period interest rate If the period length is not equal to 1 year, this / will not be equal to the yearly rate; so care must be exercised

The annuity formula is frequently used in the reverse direction; that is, A as a function of P This determines the periodic payment that is equivalent (under the assumed interest rate) to an initial payment of P This process of substituting periodic payments for a current obligation is referred to as amortization. Hence one may amortize the cost of an automobile over 5 years by taking out a 5-year loan.

Example 3.2 (Loan calculation) Suppose you have borrowed $ 1,000 from a credit union. The terms of the loan are that the yearly interest is 12% compounded monthly You are to make equal monthly payments of such magnitude as to repay (amortize) this loan over 5 years How much are die monthly payments?

Five years is 60 months, and 12% a year compounded monthly is 1% per month Hence we use the formula for n = 60, r — 1%, and P = $1,000 We find that the payments A tire $22.20 per month.

Example 3.3 (APR) A typical advertisement from a mortgage broker is shown in Table 3.1. In addition to the interest rate, term of the loan, and maximum amount, there are listed points and the annual percentage rate (APR), which describe fees and expenses. Points is the percentage of the loan amount that is charged for providing the mortgage. Typically, there are additional expenses as well. All of these fees and


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Max amt


7 625


30 yr


7 883

7 875


30 yr


8 083


2 25

30 yr




1 00

15 yr



7 500


15 yr



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APR is the rote of interest that implicitly includes the fees associated with a mortgage expenses are added to the loan balance, and the sum is amortized at the stated rate over the stated period This results in a fixed monthly payment amount A,

The APR is the rate of interest that, if applied to the loan amount without fees and expenses, would result in a monthly payment of A, exactly as before.

As a concrete example, suppose you took out a mortgage corresponding to the first listing in Table .3.1. Let us calculate the total fees and expenses Using the APR of 7.88.3%, a loan amount of $203,150, and a 30-year term, we find a monthly payment of A =$1,474,

Now using an interest rate of 7.625% and the monthly payment calculated, we find a total initial balance of $208,267, The total of fees and expenses is therefore $208,267 - $203,150 = $5,117 The loan fee itself is 1 point, or $2,03.2 Hence other expenses are $5,117 - $2,0.3.2 - $3,085

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