CHAPTER 5

Mathematics of Individual Finance

Determining Payments

Often, we know the present or future value, the interest rate and the number of periods, but need to determine the size of the annuity payment. This is not difficult to accomplish.

Example 5-13

Tony and Gayle are looking to purchase a home. They found one that they like that costs $150,000. They can get a 30 year mortgage at 8% and plan to make a down payment of 20% of the selling price. What will be their monthly mortgage payment?

Solution 5-13

Conventional mortgage payments, such as this, are amortizing meaning that both principal and interest are paid off, in equal monthly payments, over the term of the loan. Since the payments are equal, we are dealing with an annuity. The amount of the loan, which is 80% of the selling price, or $120,000, is the present value of the annuity. The term of the loan is 30 years times 12 months or 360 months while the interest rate per period is 8%/12 or 0.6667% per month.

On a financial calculator [30 x 12 = N 8/12 = %i 120000 PV CPT PMT] the answer is $880.52 per month. On LOTUS you can use the payment function @PMT(principal ,interest ,term) which, in this case, would be @PMT(120000,.006667,360) because everything is stated in months.

You can also solve directly for the Cash flow payment C by using the following formula:

C = (P x r)/[1-1/(1+r)N]

where

C is the cash flow payment

P is the principal (amount borrowed)

r is the interest rate

N is the number of payments

In the example given above, P = $120,000, r = .0066667, and N = 360. Therefore, according to

C = (P x r)/[1-1/(1+r)N]

= (120,000 x .0066667)/[1-1/(1.006667)360]

= (800)/[1-1/10.9487]

= (800)/[1-.09133]

= (800)/.90867

= $880.41

This is very close to the answer found on the calculator, subject to inevitable rounding errors.

Example 5-14

In Example 5-13, when Tony and Gayle go to the bank, the are offered an annual percent rate of 7.5% if they take a 15 year loan rather than one for 30 years. Tony and Gayle are skeptical because they can't afford to make twice the payment calculated for 30 years. In actual fact, how much would their payment be if they repaid the mortgage in 15 years?

Solution 5-14

Here, there are 15 times 12 or 180 payments and the interest rate is 7.5 divided by 12 or 0.625%. The monthly payment becomes $1,112.42 which is only $231.89 per month more than the 30 year payment.

Example 5-15

Gary is 22 years old and wants to be a millionaire by the time he is 45. He is planning to put aside a sum of money at the end of each year sufficient to accumulate a million dollars in 23 years using an interest rate of 10%. How much must he put aside?

Solution 5-15

In this example, there are 23 periods, an interest rate of 10 percent and a future value of $1,000,000. Gary must put aside $12,571.81 each year.

Example 5-16

Suppose Gary can only put aside $10,000 per year. How high a rate of return must he realize to achieve his goal?

Solution 5-16

Here, we use the $10,000 as the payment (which may have to be entered as a negative number on some calculators) and solve for the interest rate which is 11.674%. This interest rate is called the "internal rate of return" and cannot be solved for directly. Rather, an iterative process, using trial and error, must be employed. If you look at Equation 5-7 for future value, it consists of a series of payments divided by (1+r) raised to successively higher powers. As you can see, solving for r cannot be done algebraically.

While many financial calculators can find the internal rate of return, others cannot, and a process of successive estimation must be used. If you tried 10% and calculated the future value, you would come up with $795,430.20 which is too low. Trying 11% you would reach a future value of $911,478.83 and 12% would overshoot the $1 million by giving a future value of $1,046,028.90. Note that you can keep substituting interest rates without changing the other keys with most calculators. This is a laborious process, but it will eventually work.

Using the tables in the Appendix will give you an approximate answer. Table A-4 for 23 years and 10% will give an interest factor of 79.543. Multiplying that by $10,000 will give $795,430 which is too low. Trying 12% will give an interest factor of 104.60 which, multiplied by $10,000 gives $1,046,000 which is too high. This implies that the correct rate is just below 12%.

Interestingly, LOTUS does not do internal rates of return as well as most small calculators. Payments must be individually specified in a range and a guess is needed for the rate to activate the @IRR function. If the guess is not close enough, the function will return an error message.

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