Download APPENDIX 11A AND APPENDIX 11B

APPENDIX 11A AND APPENDIX 11B
APPENDIX 11A
DERIVATION OF THE FORMULA FOR n FROM THE
FORMULA FOR FV
The formula for the future value of an ordinary annuity is
FV ⫽ PMT c
11 ⫹ i2 n ⫺ 1
d
i
Multiplication of both sides of the equation by
(10-1)
i
gives
PMT
i ⫻ FV
⫽ 11 ⫹ i2 n ⫺ 1
PMT
Rearranging this equation to isolate (1 ⫹ i)n on the left-hand side, we obtain
11 ⫹ i2 n ⫽ 1 ⫹
i ⫻ FV
PMT
Now, take (natural) logarithms of both sides and use the rule that ln(ak) ⫽ k(ln a).
n ln11 ⫹ i2 ⫽ ln a 1 ⫹
i ⫻ FV
b
PMT
Therefore,
n⫽
APPENDIX 11B
ln a 1 ⫹
i ⫻ FV
b
PMT
ln11 ⫹ i2
(10-1n)
THE TRIAL-AND-ERROR METHOD FOR CALCULATING THE
INTEREST RATE PER PAYMENT INTERVAL
The trial-and-error method is a procedure for beginning with an estimate of the solution to
an equation and then improving upon the estimate. The procedure may be repeated as often
as needed to obtain the desired degree of accuracy in the answer. Each repetition of the procedure is called an iteration.
To describe the procedure in general terms, let us consider an annuity where FV, PMT,
and n are known, and we need to calculate the value for i. Since FV is known, the appropriate formula to work with is
FV ⫽ PMT c
11 ⫹ i2 n ⫺ 1
d
i
(10-1)
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Appendix 11A and Appendix 11B
The steps in the procedure are listed in the following table. (The steps are the same if we are
given PV and start with formula (10-2) instead of (10-1).)
Step
Comment
1. Using the most recent estimate
for i, evaluate the right-hand side
(RHS).
2. Compare the value obtained in
Step 1 to the left-hand side (LHS).
3. Choose a new value for i that will
make the RHS’s value closer to the
LHS’s value.
For the very first estimate, make an educated guess at a reasonable value.
For example, try i ⫽ 3% for a three-month payment interval, or i ⫽ 1% for a onemonth payment interval.
Is the calculated RHS larger or smaller than the LHS?
Use some intuition here. A larger i makes the future value and the RHS larger.
(On the other hand, a larger i makes the present value and the RHS of formula (10-2)
smaller.)
Example 11.3A, previously solved using the financial calculator method, will now be
repeated as Example 11B to illustrate the trial-and-error method.
EXAMPLE 11B | FINDING THE RATE OF RETURN ON FUNDS USED TO PURCHASE AN ANNUITY
A life insurance company advertises that $50,000 will purchase a 20-year annuity paying $341.13 at the end of
each month. What nominal rate of return and effective rate of return does the annuity investment earn?
(Calculate these rates to the nearest 0.1%.)
SOLUTION
The purchase price of an annuity equals the present value of all of its payments. Hence, the rate of return on the
$50,000 purchase price is the discount rate that makes the present value of the payments equal to $50,000. The
payments form an ordinary annuity with
PV ⫽ $50,000
PMT ⫽ $341.13
Substitute these values into formula (10-2).
$50,000 ⫽ $341.13 c
and
n ⫽ 12(20) ⫽ 240
1 ⫺ 11 ⫹ i2 ⫺240
d
i
The general trial-and-error procedure suggested an initial guess of i ⫽ 1% for each month in the payment
interval. The payment interval here is just one month. Therefore, choose i ⫽ 1% as the initial guess.
The results of successive iterations should be tabulated.
Iteration
number
Estimate
of i
(1 ⴙ i )ⴚ240
Right-hand
side (RHS) ($)
1
2
3
4
0.01
0.005
0.004
0.0045
0.0918
0.3021
0.3836
0.3404
30,981 (Note 1)
47,615 (Note 2)
52,566 (Note 3)
50,001 (Note 4)
Note 1:
We have missed our target value ($50,000) by a country mile. Present values are larger for lower
discount rates. Try i ⫽ 0.5% next.
Note 2:
Now we are getting warm. Since we want a larger present value, reduce i some more. Try
i ⫽ 0.4%.
Appendix 11A and Appendix 11B
Note 3:
At this point, we can see that our target value ($50,000) for the RHS is about midway between
$47,615 and $52,566. Therefore, the next estimate for i should be midway between 0.005 and
0.004. Try i ⫽ 0.0045.
Note 4:
It is pure luck that we happen to hit the target value with our fourth estimate for i. In general,
you will need two or three more iterations to obtain a good estimate of the value for i that satisfies the equation. If you hit the target value to three-figure accuracy, your interest rate will be
accurate to two figures.
The nominal interest rate is
j ⫽ mi ⫽ 12(0.45%) ⫽ 5.4% compounded monthly
and the effective interest rate is
f ⫽ (1 ⫹ i)m ⫺ 1 ⫽ 1.004512 ⫺ 1 ⫽ 0.0554 ⫽ 5.5%
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