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Statistics · Chapter 7 · Concept lesson · 7 min read

Why money now and money later are different numbers

Interest makes a dollar today worth more than a dollar later. One formula links the amount, the rate, the number of periods and the payment, and every function from PMT to Goal Seek just solves it for a different unknown.

The idea

A dollar today is worth more than a dollar in a year, because today's dollar can earn interest, and a borrowed dollar costs interest. Everything in this chapter is that one fact turned into arithmetic. There are four ingredients: the amount, the rate, the number of periods and the payment. Fix three and the fourth is determined.

The questions finance asks are the same relationship solved for different unknowns. "What is the payment on this loan?" (PMT). "What will my savings be worth?" (FV). "How many months to repay?" (NPER). "How big can the loan be?" (Goal Seek). In a spreadsheet each is a button. Here you write the formula and see what the button does.

Figure 1

Balance of a $48,000 loan at 7.5% a year, paid monthly over 5 years

$0 $12,500 $25,000 $37,500 $50,000 Still owed, $ Start: $48,000Year 1: $39,779Year 2: $30,921Year 3: $21,374Year 4: $11,086Year 5: $0 StartYear 1Year 2Year 3Year 4Year 5

Step 1 of 4 The supplier's racking costs $48,000. The bank charges 7.5% a year and wants 60 equal monthly payments. The payment works out to $961.82.

  1. The supplier's racking costs $48,000. The bank charges 7.5% a year and wants 60 equal monthly payments. The payment works out to $961.82.
  2. After the first year you have paid $11,541.86, but you owe $39,779.37, only $8,220.63 less. The other $3,321.23 was interest, charged on the large balance.
  3. Each payment is the same, but the interest part shrinks as the balance falls, so the balance drops faster every year. After 60 payments it reaches exactly zero.
  4. Over five years you pay $57,709.29 for a $48,000 loan: $9,709.29 of interest. That is the price of having the racking now.

Why it exists

Harbor & Pine's operations lead wants new racking for $48,000, and the bank offers 7.5% a year over five years. Whether that is affordable is a question about money over time: not "can we find $48,000" but "what does the monthly payment do to our cash flow, and how much extra do we pay for the convenience?" With $2,300,169 of completed-order revenue in 2025, the $11,541.86 paid yearly is 0.5% of it. That context is what a finance team needs.

There is a second reason to work this out by hand once. A decision about a loan usually has a limit attached: the payment must stay under a cap, or the total interest must stay under a budget. Those limits turn the question around, from "what does this loan cost?" to "what is the most we can borrow?". You can only answer that if you know which way the formula runs and which inputs can move.

Spreadsheet functions give an answer in a second and no way to see whether it is sensible. Understanding the formula lets you check it, spot when it is fed the wrong units, and explain it to someone who does not trust a number they cannot see.

How it works under the hood

The loan payment

A loan repaid in equal payments is an annuity. Each month the bank charges interest on what is still owed, you pay a fixed amount, and what is left over repays principal. With a monthly rate r and n payments, the payment that exactly clears the loan is:

payment = principal * r / (1 - (1 + r) ** -n)

For $48,000 at r = 0.075 / 12 = 0.625% and n = 60: $961.82. The total paid is 60 x $961.82 = $57,709.29, so the interest is $9,709.29. The early payments are mostly interest, the late ones mostly principal, as the first chart shows.

Rate and period must match

The formula has no idea what unit you use, only that r and n use the same one. An annual rate with monthly payments needs r = annual / 12 and n = years x 12. Mix them and the answer is wrong by a factor of about twelve with no error message.

Figure 2

The rate and the period must be measured in the same unit

  1. 7.5% a year, 5 years the bank's quote
  2. r = 0.075 / 12, n = 60 monthly rate, monthly periods
  3. r = 0.075, n = 5 annual rate, annual periods
  4. PMT $961.82 a month

Step 1 of 4 A quote is stated in years, but the payments are monthly. Before using any formula, decide the period of one payment: here, a month.

  1. A quote is stated in years, but the payments are monthly. Before using any formula, decide the period of one payment: here, a month.
  2. Convert both to months. The monthly rate is the annual rate divided by 12, 0.625%, and there are 5 x 12 = 60 payments. This gives $961.82 a month.
  3. Leave the quote in years and the formula still runs and returns $11,863.91. That is a yearly payment. Read as a monthly one it is 12.3 times too big.
  4. Same loan, same formula, an answer that is off by a factor of twelve, and nothing in Python or a spreadsheet raises an error. Check the unit of the answer against the unit of the question.

Savings, repayment time and the other unknowns

The same relationship gives the future value of regular saving: put aside $1,000 a month at 4% a year for 60 months and you hold 1000 * ((1 + r) ** n - 1) / r with r = 0.04 / 12, which is $66,298.98: $60,000 of deposits and $6,298.98 of interest. Rearranged for time, paying $1,500 a month on the $48,000 loan takes about 35.8 months, which in practice means 36 payments with a smaller last one. Rearranged for the rate, there is no neat formula, and that is where search comes in.

Goal seek by bisection

When no formula solves for what you want, you can search. If the payment rises with the loan size, guess the middle of a range, see which side of the target you landed on, and discard the other half. That is bisection, and it is what Excel's Goal Seek does under a friendlier name.

Figure 3

Goal seek by halving: monthly payment for each guessed loan size

$0 $625 $1,250 $1,875 $2,500 Monthly payment, $ $100,000: $2,004$50,000: $1,002$75,000: $1,503$87,500: $1,753$93,750: $1,879 $100,000$50,000$75,000$87,500$93,750 payment cap $2,000

Step 1 of 4 Finance can pay at most $2,000 a month. What is the biggest loan that fits? Start with a range known to contain the answer: $0 to $200,000.

  1. Finance can pay at most $2,000 a month. What is the biggest loan that fits? Start with a range known to contain the answer: $0 to $200,000.
  2. Try the middle, $100,000. Its payment is $2,003.79, just over the cap, so the answer is below $100,000. The upper end becomes $100,000.
  3. Try $50,000: $1,001.90, well under the cap, so the answer is above $50,000. Then $75,000, $87,500 and $93,750, each time keeping the half that still contains $2,000.
  4. Each guess halves the uncertainty. After about 25 halvings the guess is within a cent of $99,810.62, the loan whose payment is exactly $2,000.

It works for any quantity that moves in one direction as you change the input, such as the highest interest rate a budget can bear, and it needs no algebra. Always check the answer by plugging it back into the original formula.

What-if tables

A finance meeting rarely wants one number. A table of payments across several rates and terms lets people see how sensitive the answer is. Build it by running the same formula over a grid, and label the units in the headers so nobody reads a yearly figure as monthly. Rows for 6%, 7.5% and 9%, columns for 3, 5 and 7 years, make a small grid that answers most of the questions people ask in the room.

Common misconceptions

"Interest is just the rate times the loan times the years"

That is simple interest, and a loan with equal payments does not work that way. 7.5% of $48,000 for five years would be $18,000. The real interest is $9,709.29, because the balance falls with every payment and interest is only charged on what is still owed.

"A fifth of the payments made means a fifth of the loan repaid"

Early payments are mostly interest. After the first year, a fifth of the schedule, you have paid $11,541.86 but cleared only $8,220.63 of the $48,000, 17% of it. The balance falls slowly at the start and faster at the end.

"The spreadsheet gave a number, so it must be right"

PMT, FV and NPER do exactly what you tell them. Feed a yearly rate with monthly periods and you get a confident, precise and wrong answer. Check the answer against the question: is a monthly payment of $11,863.91 plausible on a $48,000 loan? Compare with a rough estimate before trusting it.

Check yourself

One question. Answering it ticks this lesson off on the map; it never locks a level.

The bank quotes 7.5% a year over 5 years, paid monthly, on $48,000. A colleague plugs 0.075 as the rate and 5 as the number of periods into the payment formula and gets $11,863.91. What went wrong?

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