10 The Mathematics of Money

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

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

Installment Loan - Deferred Annuity The most important distinction between an installment loan and a fixed deferred annuity is that an installment loan has apresent 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 addingthe future value of each payment.

Example 10.24 Financing That Red Mustang You’ve just landed a really good job and decide to buy the car of your dreams, abrand-new red Mustang. You negotiate a good price ($23,995 including taxesand license fees). You have $5000 saved for a down payment, and you can get acar loan from the dealer for 60 months at 6.48% annual interest. If you take outthe loan from the dealer for the balance of $18,995, what would the monthlypayments 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 morecomfortable doing the calculation yourself? Now you can,and here is how it goes. Every time you make a future payment on an installmentloan, that payment has a present value, and the sum of the present values of

Example 10.24 Financing That Red Mustang all the payments equals the present value of theloan–in this case, the $18,995 that you are financing. Although eachmonthly loan payment of F has a different present value, each of these presentvalues can be computed using the general compounding formula: The present valueP of a payment of $F paid T months in the future isP = F(1 + p)T, where p denotesthe monthly interest rate.

Example 10.24 Financing That Red Mustang 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

Example 10.24 Financing That Red Mustang Adding all the above present values gives

Example 10.24 Financing That Red Mustang Notice that the left-hand side of the above equation is a geometric sum ofT = 60terms with initial term (F/1.0054) and common ratio (1/1.0054). Usingthe geometric sum formula, the equation can be rewritten as

Example 10.24 Financing That Red Mustang From here on there is a bit of messy arithmetic to take care of, but we cansolve for F and get F = $371.48. Although you may have negotiated a good price for the car, when you throwin 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 morethan the $18,995 present value of the loan). At the end, you decide to look aroundfor a better deal.

AMORTIZATION FORMULA If an installment loan of P dollars is paid off in T payments of F dollars at aperiodic interest of p (written in decimal form), then where q = 1/(1 + p).

Example 10.24 Financing That Red Mustang: Part 2 During certain times of the year automobile dealers offer incentives in the formof cash rebates or reduced financing costs (including 0% APR), and often thebuyer can choose between those two options. Given a choice between a cashrebate or cheap financing, a savvy buyer should be able to figure out which of thetwo 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 havenegotiated a price of $23,995 (including taxes and license fees) for a brand-newred Mustang, and you have $5000 for a down payment. The big break for you isthat this dealer is offering two great incentives: a choice between a cash rebate of$2000 or a 0% APR for 60 months.

Example 10.24 Financing That Red Mustang: Part 2 If you choose the cash rebate, you will have abalance 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 60months on a balance of $18,995. Which is a better deal? To answer this question we will compare the monthly payments under bothoptions.

Example 10.24 Financing That Red Mustang: Part 2 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

Example 10.24 Financing That Red Mustang: Part 2 Option1 (continued) Solving the above equation for F gives the monthlypayment under the rebate option: F = $332.37 (rounded to the nearest penny)

Example 10.24 Financing That Red Mustang: Part 2 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 (rounded to the nearest penny)

Example 10.24 Financing That Red Mustang: Part 2 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 situationis different, and it can also be true that the rebate offer is better than the cheapfinancing option.

Winning The Lottery 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% ofthe jackpot (the lump-sum option), or take the full jackpot but paid out in 25equal annual payments (the annuity option). Most people opt for the “instantgratification” 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 optionalways the right choice? Many factors that come into play here (life expectancy,family and friends, current financial situation, etc.), but at least from a purelymathematical point of view, we can answer this question by comparing the lump-sum amount to the present value of the annuity.

Example 10.26 The Lottery Winner’s Dilemma Imagine the unimaginable–you own the only winning ticket to a $9 millionlottery jackpot. After taxes are taken out, your winnings are $6.8 million, paid outin 25 annual installments of $272,000 a year (the annuity option). You can alsochoose 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 theseoptions 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 moneyyou gave up. You can then compute the present value of this installment loanusing the amortization formula and compare this present value with the presentvalue of the lump-sum option, which is $3.75 million.

Example 10.26 The Lottery Winner’s Dilemma Before we continue, two important observations are in order. First, sincepayments are made annually, the annual interest rate equals the periodic interestrate. Second, when a lottery prize is paid off under the annuity option, the firstinstallment check is handed out immediately, and all future installments aremade at the beginning of the year. So, the version of the amortization formulathat applies in this case is

Example 10.26 The Lottery Winner’s Dilemma In this particular example, F = $272,000 and T = 25.All we need now is theannual interest rate.

Example 10.26 The Lottery Winner’s Dilemma A reasonable assumption is that for a safeinvestment made over 25 years one should expect an annual rate of return in therange between 4% and 6%, with 4% representing a very conservative approachand 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 returnof 4%, 4.5%, 5%, 5.5%, and 6%, respectively.

Example 10.26 The Lottery Winner’s Dilemma Case 1. APR = 4% (p = 0.04).In this case the present value of the annuity is

Example 10.26 The Lottery Winner’s Dilemma Case 2. APR = 4.5% (p = 0.045).In this case the present value of the annuity is

Example 10.26 The Lottery Winner’s Dilemma Case 4. APR = 5.5% (p = 0.055).In this case the present value of the annuity is

Example 10.26 The Lottery Winner’s Dilemma 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-sumoption. For the less conservative assumptions (cases 4 and 5), the lump-sumoption is about the same or slightly better than the annuity option. This analysisgives us a good picture of the mathematical part of the story.

Example 10.26 The Lottery Winner’s Dilemma There are, however,many nonmathematical reasons people tend to choose the lump-sum option overthe annuity–control of all the money, the ability to spend as much as we wantwhenever we want, and the realistic observation that we may not be around longenough to collect on the annuity.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma 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. (Aswith 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 offeris worth pursuing by comparing it with your current mortgage–a30-year mortgage for $180,000 with a 6.75% APR.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma Obviously, a6% APR is a lot better than a 6.75% APR, but do the savingsjustify your up-front expenses for taking out the new loan? Besides, you have made 30 monthly payments on your current loanalready (you have lived in the house for 2 1/2 years). Will all thesepayments be wasted? For a fair comparison between the two options (take out anew loan or keep the current loan), we will compare the monthly payments onyour current loan with the monthly payments you

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma would be making if you tookout the new loan for the balance of what you owe on your current loan. (This waywe are truly comparing apples with apples.) We can then determine if themonthly savings on your payments justify the up-front loan origination costs ofthe new loan. The computation will involve several steps, but fortunately for us,each step is based on an application of the amortization formula.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma Step 1.Compute the monthly payment F on the current 30-year mortgagewith the 6.75% APR; amortization formula withP =$180,000, T = 360,and p = 0.0675/12 = 0.005625. Solve: F= $1167.48

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma Step 3.Compute the monthly payment F* on a new mortgage for $174,951with a 6% APR. Here P = $174,951, T = 330,and p = 0.06/12 = 0.005. Solving: $F* = $1083.75.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma 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), youwould be saving$1167.48 – $1083.75 = $83.73a month. How much are thesemonthly savings worth over time? This is an immediate annuity question, essentiallythe 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?

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma The answer tothese 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 monthsover 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 isgiven in Step 4a.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma Here is a case in which your refinancing savings are much less than yourup-front loan origination costs. In this case, refinancing is a bad idea.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma Step 4a.Compute the present value P* of 330 monthly payments of F = $83.73at an APR of 6%(p = 0.005).

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma Now the up-front loan origination costs are about $3250: $1500 for closingcosts 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 loanfor the long haul, by all means you should refinance! The benefit ($13,516.70)by far outweighs the cost ($3250).

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma If you are getting a good offer now, though, you might be getting an even betteroffer in one, two, or three years from now. If you think that you might be refinancingagain in the near future, the present value of your monthly savings (were you torefinance now) is considerably less. For example, let’s imagine that you refinanceagain 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.

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma Step 4b.Present value P** of 24 monthly payments of F = $83.73at an APRof 6%(p = 0.005).

Example 10.27 To Refinance or Not to Refinance: A Homeowner’s Dilemma So, what should you do? Refinance or not refinance? As we have seen fromthe preceding discussion, for a definitive answer you would need a crystal ball. Ifyou think that an even better refinancing offer might be coming your way withinthe next few years, you should not refinance. On the other hand, if you think thatonce you refinance you are likely to keep the new loan for a long time, youshould definitely do so.

10 The Mathematics of Money