Chapter 4 of 22 · Unit 1 · Foundations · 9 min read
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After this chapter you can
A dollar today is worth more than a dollar promised in ten years. You could invest today's dollar and have more than a dollar later; you could spend it now; and the promise might not be kept. Every price in finance, from a bond to a share to a pension, comes from that one idea applied with care. This chapter builds the tools and uses them on two long stretches when rates told savers something unpleasant.
For every reader: the idea in plain words, no formulas.
If a bank pays 5 % a year, $100 today becomes $105 next year. So $105 next year and $100 today are worth the same to someone who can earn that 5 %. Turning a future sum into today's money is called discounting, and the rate used is the discount rate. A higher rate makes future money worth less today, which is why bond prices fall when rates rise.
When you leave earnings in place, next year's return applies to a larger sum. Over long periods that compounding does most of the work. Ten thousand dollars at 7 % a year becomes $76,123 after thirty years, and more than half of that growth arrives in the last ten years.
Not every promise deserves the same rate. A payment from the US Treasury is close to certain, so markets discount it at the Treasury yield. A payment from a young company might not arrive, so investors demand a higher rate, and the same $100 in ten years is worth less today. The discount rate always holds two parts: the reward for waiting, and an extra reward for the risk that the money does not come. Most of the disagreement about what a share or a building is worth comes down to that second part.
Prices rise over time, so a dollar later buys less than a dollar now even if the bank keeps every promise. The return that matters is the real interest rate, the rate after inflation. A savings account paying 3 % while prices rise 4 % loses buying power every year, even as the balance grows.
For readers who want the mechanism, with the diagrams.
Beginner path stops here
Small differences in the rate turn into large differences in the ending sum. We grow $10,000 for thirty years at three rates.
| Yearly rate | Value after 30 years |
|---|---|
| 4 % | $32,434 |
| 7 % | $76,123 |
| 10 % | $174,494 |
Going from 7 % to 10 % more than doubles the result. Costs work the same way in reverse: a fee of one percentage point a year comes off the rate and compounds against you for the whole period.
| Year | At 4 % ($) | At 7 % ($) | At 10 % ($) |
|---|---|---|---|
| 0 | 10,000 | 10,000 | 10,000 |
| 5 | 12,167 | 14,026 | 16,105 |
| 10 | 14,802 | 19,672 | 25,937 |
| 15 | 18,009 | 27,590 | 41,772 |
| 20 | 21,911 | 38,697 | 67,275 |
| 25 | 26,658 | 54,274 | 108,347 |
| 30 | 32,434 | 76,123 | 174,494 |
Source: computed: FV = 10,000 x (1 + r) to the power of the years, as of 2026-10-28.
Discounting runs the same arithmetic backward. To have $10,000 in thirty years at 7 %, you need $1,314 today. A timeline helps: put each future payment on it and pull each one back to today by dividing by the growth factor for its date. Adding the pulled-back values gives the price today of any stream of payments.
| Year | Cash ($) | Present value at 5 % ($) |
|---|---|---|
| 1 | 100 | 95.24 |
| 2 | 100 | 90.7 |
| 3 | 100 | 86.38 |
| 4 | 100 | 82.27 |
| 5 | 100 | 78.35 |
| Total | 500 | 432.95 |
Source: computed: PV = cash / (1 + 0.05) to the power of the year, as of 2026-10-28.
A mortgage with level payments charges interest on the balance still owed. Early on the balance is large, so most of each payment is interest and the loan shrinks slowly. Later the balance is small, interest takes less, and principal falls fast. On a 30-year loan at 6 %, about $334,743 of a $400,000 loan is still owed after ten years and $216,014 after twenty. Borrowers who expect to move after a few years have repaid less than they might think.
Banks quote an annual percentage rate on loans and an annual percentage yield on deposits. The yield counts compounding within the year; the rate does not. A 6 % rate compounded monthly gives a yield of 6.17 %. When comparing two offers, compare yields with yields, or rates with rates compounded the same way.
Dividing 72 by the rate in percent gives a quick estimate of the years needed to double money. The rule works best near 8 %.
| Rate | Rule of 72 | Exact doubling time |
|---|---|---|
| 2 % | 36.0 years | 35.0 years |
| 7 % | 10.3 years | 10.2 years |
| 20 % | 3.6 years | 3.8 years |
The same rule works on inflation. At 3 % inflation, prices double in about 24.0 years, so a retiree's spending power halves over a long retirement unless income rises with prices.
A nominal rate is what the bank quotes. A real rate subtracts inflation. The Fisher relation, after the economist Irving Fisher, ties them together: one plus the nominal rate equals one plus the real rate times one plus expected inflation. For small numbers the real rate comes close to the nominal rate minus inflation. Treasury Inflation-Protected Securities quote a real yield directly, which lets investors read the market's inflation expectations by comparing them with ordinary Treasuries.
For Series 65 and CFA readers: formulas and worked examples.
Series 65 path stops here
For a sum invested at rate per period for periods, we write the two directions of the same relation.
In words, the future value equals today's sum multiplied by one plus the rate, once for every period.
In words, the present value equals the future sum divided by one plus the rate, once for every period.
At 7 % for thirty years the growth factor is 7.61, which turns $10,000 into $76,123. Dividing by the same factor brings $10,000 due in thirty years back to $1,314.
An annuity pays the same amount at the end of each of periods. Adding the present values of all the payments gives a closed form.
In words, the value of a level stream is the payment times one minus the discount factor for the last date, divided by the rate.
Worked example A $400,000 mortgage at 6 % for 30 years
Monthly payments mean and . Solving the annuity formula for gives a payment of $2,398.20 a month. Over thirty years the borrower pays $863,353, of which $463,353 is interest. In the first month, $2,000.00 of the payment goes to interest and only $398.20 to principal; the split shifts toward principal each month as the balance falls.
Saving works the same way forward. Putting $500 a month away at 7 % a year for thirty years builds $609,985, from $180,000 of deposits. Growth on the deposits supplies the rest.
A perpetuity pays every period forever. Letting grow without limit in the annuity formula leaves a short result, and a perpetuity growing at rate has one too.
In words, a level payment forever is worth the payment divided by the rate.
In words, a payment that grows at g forever is worth the first payment divided by the rate minus the growth rate.
At 8 %, $4 a year forever is worth $50.00. If the $4 grows at 3 % a year, the value rises to $80.00. That second formula, with dividends in place of payments, is the Gordon growth model for valuing a share in Chapter 10.
The net present value of a project adds the present values of all its cash flows, with the cost today entered as a negative number.
In words, net present value is the sum of every cash flow, each divided by one plus the discount rate once for every year until it arrives.
The internal rate of return is the rate that makes the NPV zero. A project is worth doing when its NPV at the investor's required rate is positive, which for a normal project means its IRR beats that rate.
Worked example Pay $1,000, receive $300 a year for five years
At an 8 % required return, the five payments are worth $1,197.81 today, so the NPV is $197.81. Searching for the rate that sets the NPV to zero gives an IRR of 15.24 %. Both tests say yes. They can disagree when comparing projects of different sizes or timing, and then the NPV gives the better answer because it measures dollars of value added.
Taking logarithms of the future value formula gives the time needed to reach a target.
In words, the number of years equals the log of the target over today's sum, divided by the log of one plus the rate.
Growing $10,000 into $100,000 takes 34.0 years at 7 % and 24.2 years at 10 %. Three extra points of return cut about a decade off the wait. The rule of 72 is a quick version of the same formula for a doubling.
A 7 % rate compounded monthly grows a little faster than one compounded yearly, and compounding continuously is the limit. Continuous compounding uses the exponential function.
In words, with continuous compounding the future value is today's sum times e raised to the rate times the number of years.
| $10,000 at 7 % for 30 years | Ending value |
|---|---|
| compounded yearly | $76,123 |
| compounded monthly | $81,165 |
| compounded continuously | $81,662 |
A continuous 7 % equals a yearly rate of 7.25 %, its effective annual rate. Option pricing and most academic finance use continuous rates because they add across time: two years at a continuous 7 % gives a continuous 14 %.
In words, one plus the nominal rate equals one plus the real rate times one plus inflation.
With a 7 % nominal return and 3 % inflation the real return is , which is 3.88 %. The shortcut of subtracting gives 4.00 %; the gap grows when inflation runs high.
For readers who want past episodes run through the Level 3 math.
1974 Treasury bills lose to inflation
In 1974 the three-month Treasury bill yield averaged 7.8 % while consumer prices rose 12.3 % from December 1973 to December 1974 (FRED series TB3MS and CPIAUCNS). A saver rolling bills that year ended with −4.0 % in buying power.
Treasury bills carry no credit risk and almost no price risk, so many savers in the 1970s thought of them as safe. We can test that with the Fisher relation. Suppose a saver held bills from the start of 1973 to the end of 1980, earning each year's average bill yield. Her nominal balance grew 77 %. Over the same eight years consumer prices rose 103 %, so prices more than doubled. Dividing one growth factor by the other gives a real loss of −12.7 %, about −1.7 % a year. In the worst single year, 1974, buying power changed by −4.0 %.
So the safe asset was safe in dollars and unsafe in what dollars buy. This is inflation risk from Chapter 2 in its plainest form, and it is the reason a long-horizon saver cannot hold only cash. The episode ended when the Federal Reserve under Paul Volcker pushed short rates far above inflation from late 1979, which Chapter 7 takes up.
1999 The Bank of Japan sets its policy rate near zero
In February 1999 the Bank of Japan cut its overnight rate to about zero, and short rates stayed close to zero for most of the next quarter century. It moved the rate below zero in early 2016 and ended negative rates in March 2024.
Near-zero rates turn the discounting formula upside down for savers and pension funds. Take a promise to pay 100 in thirty years. At a 4 % discount rate its value today is 30.8; at 0.5 % it is 86.1. A pension fund that owes such payments needs almost three times as much money set aside today when rates sit near zero. Japanese savers met the same problem from the other side: a deposit earning almost nothing does not grow, so reaching a goal needed far larger contributions.
Low rates also lift the present value of distant cash flows from shares and property, which helps explain high asset prices in long low-rate periods. The same arithmetic runs in reverse when rates rise, as bondholders found in 2022.
Both cases come down to the gap between the rate money earns and the rate the world asks of it. In the 1970s that gap ran against savers because inflation outpaced bill yields. In Japan's long low-rate period it ran against anyone with a fixed future target, because money set aside could not grow toward it. The tools of Level 3 measure both: the Fisher relation for the first case, the present value of a distant liability for the second. We find it useful to check any savings plan against both: what if inflation runs two points above the plan, and what if returns stay two points below it for a decade?
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