Excursions in Modern Mathematics, 7e: 10.6 - 2Copyright © 2010 Pearson Education, Inc. 10 The Mathematics of Money 10.1Percentages 10.2Simple Interest

Excursions in Modern Mathematics, 7e: 10.6 - 2Copyright © 2010 Pearson Education, Inc.

10 The Mathematics of Money

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Geometric Sequences

10.5 Deferred Annuities: Planned Savings for the Future

10.6 Installment Loans: The Cost of Financing the Present


An installment loan (also know as a fixed immediate annuity) is a series of equal payments made at equal time intervals for the purpose of paying off a lump sum of money received up front. Typical installment loans are the purchase of a car on credit or a mortgage on a home.

Installment Loan


The most important distinction between an installment loan and a fixed deferred annuity is that an installment loan has a present value that we compute by adding the present value of each payment, whereas a fixed deferred annuity has a future value that we compute by adding the future value of each payment.

Installment Loan - Deferred Annuity


You’ve just landed a really good job and decide to buy the car of your dreams, a brand-new red Mustang. You negotiate a good price ($23,995 including taxes and license fees). You have $5000 saved for a down payment, and you can get a car loan from the dealer for 60 months at 6.48% annual interest. If you take out the loan from the dealer for the balance of $18,995, what would the monthly payments be?

Example 10.24 Financing That Red Mustang


Can you afford them? Typically, buyers blindly trust the finance department at the dealership to provide this information accurately, but as an educated consumer, wouldn’t you feel more comfortable doing the calculation yourself? Now you can, and here is how it goes.

Every time you make a future payment on an installment loan, that payment has a present value, and the sum of the present values of



all the payments equals the present value of the loan–in this case, the $18,995 that you are financing. Although each monthly loan payment of F has a different present value, each of these present values can be computed using the general compounding formula: The present value P of a payment of $F paid T months in the future is P = F (1 + p)T, where p denotes the monthly interest rate.



In this example, the monthly interest rate isp = 0.0648/12 = 0.0054. Thus,


■ Present value of the first payment of F:F/1.0054

■ Present value of the second payment of F:F/(1.0054)2

■ Present value of the third payment of F:F/(1.0054)3

■ Present value of the last payment of F:F/(1.0054)60

…


Adding all the above present values gives


F

1.0054

F

1.0054 2

F

1.0054 3K

F

1.0054 60

$18,995


Notice that the left-hand side of the above equation is a geometric sum of T = 60 terms with initial term (F/1.0054) and common ratio (1/1.0054). Using the geometric sum formula, the equation can be rewritten as


F

1.0054

11.0054

60

1

11.0054

1

$18,995


From here on there is a bit of messy arithmetic to take care of, but we can solve for F and get F = $371.48 . Although you may have negotiated a good price for the car, when you throw in the financing costs this may not be such a great deal after all (60 payments of $371.48 equals $22,288.80, and even when adjusting for inflation that is a lot more than the $18,995 present value of the loan). At the end, you decide to look around for a better deal.



If an installment loan of P dollars is paid off in T payments of F dollars at a periodic interest of p (written in decimal form), then

where q = 1/(1 + p).

AMORTIZATION FORMULA

P Fq

qT 1

q


During certain times of the year automobile dealers offer incentives in the form of cash rebates or reduced financing costs (including 0% APR), and often the buyer can choose between those two options. Given a choice between a cash rebate or cheap financing, a savvy buyer should be able to figure out which of the two is better, and we are now in a position to do that.

Example 10.24 Financing That Red Mustang: Part 2


We will consider the same situation we discussed in Example 10.24. You have negotiated a price of $23,995 (including taxes and license fees) for a brand-new red Mustang, and you have $5000 for a down payment. The big break for you is that this dealer is offering two great incentives: a choice between a cash rebate of $2000 or a 0% APR for 60 months.



If you choose the cash rebate, you will have a balance of $16,995 that you will have to finance at the dealer’s standard interest rate of 6.48%. If you choose the free financing, you will have a 0% APR for 60 months on a balance of $18,995. Which is a better deal?

To answer this question we will compare the monthly payments under both options.



Option 1: Take the $2000 rebate. Here the present value is $P = $16,995, amortized over 60 months at 6.48% APR. The periodic rate is p = 0.0054

Applying the amortization formula we get


F

1.0054

11.0054

60

1

11.0054

1

$16,995


Option1 (continued)

Solving the above equation for F gives the monthly payment under the rebate option:

F = $332.37

(rounded to the nearest penny)



Option 2: Take the 0% APR. Here the present value is P = $18,995, amortized over 60 months at 0% APR. There is no need for any formulas here: With no financing costs, your monthly payment amortized over 60 months is


F

18,995

60

$316.59

(rounded to the nearest penny)


Clearly, in this particular situation the 0% APR option is a lot better than the $2000 rebate option (you are saving approximately $16 a month, which over 60 months is a decent piece of change). Of course, each situation is different, and it can also be true that the rebate offer is better than the cheap financing option.



If you were to win a major lottery jackpot, you would immediately face a critical decision: Take a single lump-sum payment up front for about 50% to 60% of the jackpot (the lump-sum option), or take the full jackpot but paid out in 25 equal annual payments (the annuity option). Most people opt for the “instant gratification” approach and choose the lump-sum option without giving any consideration to the financial value of the annuity option.

Winning The Lottery


But is the lump-sum option always the right choice? Many factors that come into play here (life expectancy, family and friends, current financial situation, etc.), but at least from a purely mathematical point of view, we can answer this question by comparing the lump-sum amount to the present value of the annuity.

Winning The Lottery


Imagine the unimaginable–you own the only winning ticket to a $9 million lottery jackpot. After taxes are taken out, your winnings are $6.8 million, paid out in 25 annual installments of $272,000 a year (the annuity option). You can also choose to take your winnings as a single, tax-free lump-sum payment of $3.75 million (the lump sum option). From a purely financial point of view, which of these options is better?

Example 10.26 The Lottery Winner’s Dilemma


You can answer this question by thinking of the annuity option as an installment loan in which you are the lender getting monthly payments for the money you gave up. You can then compute the present value of this installment loan using the amortization formula and compare this present value with the present value of the lump-sum option, which is $3.75 million.



Before we continue, two important observations are in order. First, since payments are made annually, the annual interest rate equals the periodic interest rate. Second, when a lottery prize is paid off under the annuity option, the first installment check is handed out immediately, and all future installments are made at the beginning of the year. So, the version of the amortization formula that applies in this case is



In this particular example, F = $272,000 and T = 25. All we need now is the annual interest rate.


P F

qT 1

q


A reasonable assumption is that for a safe investment made over 25 years one should expect an annual rate of return in the range between 4% and 6%, with 4% representing a very conservative approach and 6% representing the less conservative end of the spectrum. To get a good picture of the situation, let’s consider five separate cases using annual rates of return of 4%, 4.5%, 5%, 5.5%, and 6%, respectively.



Case 1.APR = 4% (p = 0.04). In this case the present value of the annuity is


P $272,000

11.04

25

1

11.04

1

$4,419,170


Case 2.APR = 4.5% (p = 0.045). In this case the present value of the annuity is


P $272,000

11.045

25

1

11.045

1

$4,214,770




P $272,000

11.05

25

1

11.05

1

$4,025,230


Case 4.APR = 5.5% (p = 0.055). In this case the present value of the annuity is


P $272,000

11.055

25

1

11.055

1

$3,849,260




P $272,000

11.06

25

1

11.06

1

$3,685,700


The bottom line is that under the more conservative assumptions (cases 1, 2, and 3), the annuity option appears to be significantly better than the lump-sum option. For the less conservative assumptions (cases 4 and 5), the lump-sum option is about the same or slightly better than the annuity option. This analysis gives us a good picture of the mathematical part of the story.



There are, however, many nonmathematical reasons people tend to choose the lump-sum option over the annuity–control of all the money, the ability to spend as much as we want whenever we want, and the realistic observation that we may not be around long enough to collect on the annuity.



Imagine that you are a homeowner and have just received an offer in the mail to refinance your home loan. The offer is for a 6% APR on a 30-year mortgage. (As with all mortgage loans, the interest is compounded monthly.) In addition, there are loan origination costs: $1500 closing costs plus 1 point (1% of the amount of the loan). You are trying to decide if this offer is worth pursuing by comparing it with your current mortgage–a 30-year mortgage for $180,000 with a 6.75% APR.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma


Obviously, a 6% APR is a lot better than a 6.75% APR, but do the savings justify your up-front expenses for taking out the new loan? Besides, you have made 30 monthly payments on your current loan already (you have lived in the house for 2 1/2 years). Will all these payments be wasted? For a fair comparison between the two options (take out a new loan or keep the current loan), we will compare the monthly payments on your current loan with the monthly payments you



would be making if you took out the new loan for the balance of what you owe on your current loan. (This way we are truly comparing apples with apples.) We can then determine if the monthly savings on your payments justify the up-front loan origination costs of the new loan. The computation will involve several steps, but fortunately for us, each step is based on an application of the amortization formula.



Step 1. Compute the monthly payment F on the current 30-year mortgage with the 6.75% APR; amortization formula with P = $180,000, T = 360, and p = 0.0675/12 = 0.005625.


$180,000 F

1.005625

11.005625

360

1

11.005625

1

Solve: F = $1167.48


Step 3. Compute the monthly payment F* on a new mortgage for $174,951 with a 6% APR. Here P = $174,951, T = 330, and p = 0.06/12 = 0.005.


$174,951F *

1.005

11.005

330

1

11.005

1

Solving: $F* = $1083.75.


At this point, we know that if you refinance and take out a comparable loan (i.e., the new loan picks up exactly where you were on the original loan), you would be saving$1167.48 – $1083.75 = $83.73 a month. How much are these monthly savings worth over time? This is an immediate annuity question, essentially the same as asking for the present value of a series of monthly payments of $83.73. But how many payments are we talking about? And at what APR?



The answer to these two questions will determine the present value of your monthly savings.First, let’s assume that you will be keeping the new loan for the full 330 months over which it is amortized (T = 330), and let’s further assume an APR of 6%. Under these assumptions, the computation of the present value of your savings is given in Step 4a.



Here is a case in which your refinancing savings are much less than your up-front loan origination costs. In this case, refinancing is a bad idea.



Step 4a. Compute the present value P* of 330 monthly payments of F = $83.73 at an APR of 6% (p = 0.005).


P* $83.73

1.005

11.005

330

1

11.005

1

$13,516.70


Now the up-front loan origination costs are about $3250: $1500 for closing costs plus 1 point (1% of $174,951 is $1749.51, but for simplicity we’ll call it $1750). Thus, the loan origination costs of $3250 are more than offset by the long-term savings of refinancing. If you think you would keep the new loan for the long haul, by all means you should refinance! The benefit ($13,516.70) by far outweighs the cost ($3250).



If you are getting a good offer now, though, you might be getting an even better offer in one, two, or three years from now. If you think that you might be refinancing again in the near future, the present value of your monthly savings (were you to refinance now) is considerably less. For example, let’s imagine that you refinance again in 24 months and assume as before an APR of 6%. Under these assumptions, the present value of refinancing now is given in Step 4b.



Step 4b. Present value P** of 24 monthly payments of F = $83.73 at an APR of 6%(p = 0.005).


P * * $83.73

1.005

11.005

24

1

11.005

1

$1889.19


So, what should you do? Refinance or not refinance? As we have seen from the preceding discussion, for a definitive answer you would need a crystal ball. If you think that an even better refinancing offer might be coming your way within the next few years, you should not refinance. On the other hand, if you think that once you refinance you are likely to keep the new loan for a long time, you should definitely do so.


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Excursions in Modern Mathematics, 7e: 10.6 - 2Copyright © 2010 Pearson Education, Inc. 10 The Mathematics of Money 10.1Percentages 10.2Simple Interest