Value of money over time

From CEOpedia
Value of money over time
See also


Value of money over time (the time value of money, TVM) - is the principle that a sum of money available today is worth more than the same nominal sum received at a later date, because money held now can be invested to earn a return, is not exposed to the risk of non-payment and keeps its present purchasing power[1]. A promise of 1,000 euros in five years and 1,000 euros in your pocket are two different things. The interest rate is the exchange rate between them.

Every financial decision that spreads cash over several dates relies on this idea: a savings plan, a mortgage, a lease contract, a pension, a bond, a factory investment. The arithmetic itself is old and fairly simple. Choosing the right rate is where things get hard.

Why money today is worth more

Three separate forces push in the same direction.

  • Opportunity cost. Cash received today can be deposited or invested at once. Waiting means giving up the interest it could have earned in the meantime. That forgone income is the opportunity cost of delay, and it exists even in a world with stable prices and perfectly reliable debtors.
  • Inflation. Rising prices erode what a fixed amount can buy. 100 units of currency next year will buy fewer goods than 100 units today if prices go up in between. The effects of inflation and the distinction between nominal amounts and real value follow directly from this.
  • Risk. A future payment may simply not arrive. The borrower can default, the project can fail, the law can change. Lenders ask for a premium over the risk-free return to cover that chance, and the riskier the claim, the bigger the premium.

There is also a fourth, more psychological reason: people are impatient. Most of us prefer to consume now rather than later, and need a reward to postpone spending. Irving Fisher built his whole theory of interest on that impatience combined with the opportunities to invest[2].

History

Charging interest and compounding it are ancient practices. The theory came much later.

Some popular accounts name Martín de Azpilcueta (1491-1586) of the School of Salamanca as the discoverer of the time value of money, pointing to his writings on money, exchange and usury from the 1550s. The claim deserves caution. A 2020 study by Pedro J. Caranti argues that Azpilcueta condemned usury throughout his life and that his views on time preference were "much more nuanced, and, at times, inconsistent" than the popular story suggests[3]. Fair to say that the Salamanca scholars discussed the changing value of money, but not that they produced the modern doctrine.

Practical discounting got a mathematical footing in the 17th century, mostly through the pricing of annuities. In 1693 Edmond Halley published a life table based on births and burials in Breslau and used it to estimate the price of annuities upon lives, combining mortality data with discounting at a given rate of interest[4]. Recent work on the history of financial economics traces present-value methods back to such valuations of life and joint-life annuities by Johan de Witt, Halley and their successors[5].

The modern economic framework arrived with Irving Fisher's The Theory of Interest (1930), subtitled "As Determined by Impatience to Spend Income and Opportunity to Invest It". Fisher treated interest as the price that balances people's preference for present goods against the productivity of investment, and he separated the money rate of interest from the real rate, adjusted for changes in purchasing power[2]. That separation survives today as the Fisher equation.

Formulas

The notation below is standard: PV - present value, FV - future value, r - interest rate per period, n - number of periods, C - a regular payment.

Future value and present value

A sum invested today grows by compound interest:

\[ FV = PV \times (1 + r)^n \]

Reversing the formula gives the present value of a future amount, a process called discounting:

\[ PV = \frac{FV}{(1 + r)^n} \]

The factor \( 1/(1+r)^n \) is the discount factor. It shrinks as either the rate or the horizon grows. Details and more examples are in the article on future value.

Compounding frequency

Banks often quote a nominal annual rate but add interest monthly or quarterly. With m compounding periods per year:

\[ FV = PV \times \left(1 + \frac{r}{m}\right)^{m \cdot n} \]

The more often interest is added, the more interest earns interest. In the limit, with compounding every instant, the formula becomes continuous compounding:

\[ FV = PV \times e^{r \cdot n} \]

Effective annual rate

To compare offers with different compounding schedules, convert each one to the effective annual interest rate (EAR)[6]:

\[ EAR = \left(1 + \frac{r}{m}\right)^{m} - 1 \]

For continuous compounding, \( EAR = e^{r} - 1 \). A nominal 5% rate gives an EAR of 5.0945% with quarterly compounding, 5.1162% with monthly compounding and 5.1271% with continuous compounding. Small gaps. On a large loan over many years they are not small at all.

Annuities

An annuity is a series of equal payments at equal intervals. In an ordinary annuity payments fall at the end of each period (typical for loans); in an annuity due they fall at the beginning (typical for rent and leases)[7].

Present value of an ordinary annuity:

\[ PV = C \times \frac{1 - (1 + r)^{-n}}{r} \]

Future value of an ordinary annuity:

\[ FV = C \times \frac{(1 + r)^{n} - 1}{r} \]

For an annuity due, each payment comes one period earlier, so both values are multiplied by \( (1 + r) \):

\[ PV_{due} = C \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r) \]

Perpetuity and growing perpetuity

A perpetuity pays the same amount forever. Its value is finite, because distant payments are worth almost nothing today:

\[ PV = \frac{C}{r} \]

If payments grow at a constant rate g (with \( g < r \)) and the first payment C arrives at the end of the first period, the formula becomes:

\[ PV = \frac{C}{r - g} \]

The growing perpetuity is the core of dividend discount models and of terminal value calculations in company valuation[7]. A payment of 1,000 a year forever at 5% is worth 20,000; a payment starting at 1,000 and growing 3% a year, discounted at 8%, is worth the same 20,000.

Nominal and real rates (Fisher equation)

The nominal rate of return includes compensation for inflation; the real rate of return does not. With nominal rate i, real rate r and inflation π, the exact Fisher relation is[2]:

\[ (1 + i) = (1 + r) \times (1 + \pi) \qquad \Rightarrow \qquad r = \frac{1 + i}{1 + \pi} - 1 \]

The popular shortcut \( r \approx i - \pi \) works only at low rates. A deposit paying 7% with inflation at 3% gives a real return of 3.88%, not 4%. Because future inflation is unknown, decisions are made on the basis of expected inflation; whether people form those expectations rationally is the subject of rational expectations theory.

Examples

All figures below are hypothetical and were checked numerically.

Example 1: savings

Anna deposits 10,000 at a nominal 5% a year for 10 years.

Compounding Value after 10 years Effective annual rate
Simple interest (no compounding) 15,000.00 -
Annual 16,288.95 5.0000%
Quarterly 16,436.19 5.0945%
Monthly 16,470.09 5.1162%
Daily (365) 16,486.65 5.1267%
Continuous 16,487.21 5.1271%

Compounding alone adds almost 1,300 compared with simple interest. Notice how the gain from going beyond monthly compounding is tiny: under 20 over ten years.

Turned the other way round: to have 10,000 in ten years at 5% (annual compounding), Anna needs to deposit only 6,139.13 today. That number is the present value.

A regular saver works with the annuity formula. Paying 3,000 at the end of each year for 30 years at 6% builds 237,174.56, although only 90,000 was paid in. Make the same payments at the start of each year (annuity due) and the result rises to 251,405.03. One year of extra interest on every payment.

Example 2: loan instalment

A company borrows 200,000 for 20 years at a nominal 6% a year, repaid in equal monthly instalments (an ordinary annuity with \( r = 0.5\% \) per month and \( n = 240 \)). Solving the annuity formula for C:

\[ C = \frac{PV \times r}{1 - (1 + r)^{-n}} = \frac{200{,}000 \times 0.005}{1 - 1.005^{-240}} \approx 1{,}432.86 \]

Total payments reach 343,886.91, so interest costs 143,886.91. In the first month 1,000.00 of the instalment is interest and only 432.86 reduces the debt; the proportion reverses gradually, as a loan amortization schedule shows month by month.

Example 3: comparing two offers

A seller of a small business gets two offers:

  • Offer A: 50,000 paid today.
  • Offer B: 12,000 at the end of each year for 5 years (60,000 in total).

Which is better? Depends on the rate.

Discount rate PV of offer B Better choice
5% 51,953.72 B
8% 47,912.52 A

The two offers are equal at about 6.40%, which is the internal rate of return of offer B against offer A. If the seller can invest money at more than 6.4%, or if the buyer's credit looks shaky and calls for a higher rate, the cash today wins even though it is 10,000 less in nominal terms. Change offer B so that the instalments come at the start of each year and the break-even rate jumps to roughly 10.05%. Timing alone moved the answer.

Applications

The principle sits underneath most of finance.

  • Investment appraisal. Net present value, internal rate of return and the discounted version of the payback period are all built on discounting. See also investment appraisal techniques.
  • Loans and credit. Instalments, effective rates of consumer credit and early repayment amounts come from annuity formulas.
  • Pensions and savings. The size of contributions needed to reach a target, and the value of a stream of pension payments, are future and present values of annuities. Defined-contribution schemes such as a money-purchase pension plan depend on compounding over decades.
  • Leasing. Lease payments are usually made in advance, so lessors and lessees value them as annuities due; the implicit rate in a long-term lease is the rate that equates payments with the asset's value.
  • Bonds. The price of a bond equals the present value of its coupons (an annuity) plus the present value of the face value. When market rates rise, prices fall. See bonds in finance.
  • Company valuation. Discounted cash flow models project future cash flows and a terminal value, usually a growing perpetuity, and discount them at the required rate of return.
  • Personal finance. Comparing a lump-sum prize with an annual payout, or a discount for cash with an instalment plan, is a TVM problem. Public tools, for example the compound interest calculator run by the U.S. Securities and Exchange Commission at Investor.gov, let anyone run the numbers[8].

Advantages

  • Puts cash flows from different dates on a common footing, so they can be added and compared.
  • Makes the cost of credit visible. An effective rate tells more than a "low monthly payment".
  • Works the same way for a household, a firm or a government.
  • Formulas are simple and available in every spreadsheet and financial calculator.
  • Lets inflation and risk be included explicitly through the choice of rate.
  • Supports sensitivity analysis: change the rate, see how the decision moves.

Limitations

  • Constant-rate assumption. Textbook formulas assume one rate for all periods. Real interest rates change, yield curves are rarely flat, and a 20-year projection at a single rate is a simplification.
  • Choosing the discount rate. The result can swing from positive to negative with a one-point change in the rate, as Example 3 shows. There is no single "correct" rate: it may be the cost of borrowing, the return on an alternative investment, or a rate including a market risk premium for the specific risk of the investment. Disagreements here are often bigger than any arithmetic error.
  • Uncertain cash flows. Discounting is precise; forecasts are not. Garbage in, garbage out.
  • Inflation forecasts. Mixing nominal cash flows with real rates (or the other way round) is a common and costly mistake.
  • Very long horizons. At normal rates, anything beyond 30-50 years is discounted to almost nothing, which raises ethical questions in public projects and climate policy.
  • Behaviour. People do not always discount consistently. Many prefer a small reward today over a slightly larger one tomorrow, yet choose the larger one when both are a year away. Standard formulas ignore such patterns.

Related approaches

References

Footnotes

  1. ↑ Dahlquist J., Knight R. et al., Principles of Finance, OpenStax, 2022, section 7.1 "Now versus Later Concepts"
  2. ↑ Fisher I., The Theory of Interest: As Determined by Impatience to Spend Income and Opportunity to Invest It, Macmillan, New York, 1930
  3. ↑ Caranti P.J., Martín de Azpilcueta: The Spanish Scholastic on Usury and Time-Preference, Studia Humana, Vol. 9, No. 2, 2020, pp. 28-36
  4. ↑ Halley E., An Estimate of the Degrees of the Mortality of Mankind..., Philosophical Transactions of the Royal Society of London, Vol. 17, No. 196, 1693
  5. ↑ Poitras G., More on the Origin of Financial Economics: Early Contributions to Joint Life Annuity Valuation, Journal of the History of Economic Thought, Vol. 47, No. 3, 2025, pp. 339-362
  6. ↑ Dahlquist J., Knight R. et al., Principles of Finance, OpenStax, 2022, section 8.4 "Stated versus Effective Rates"
  7. ↑ Dahlquist J., Knight R. et al., Principles of Finance, OpenStax, 2022, sections 8.1 "Perpetuities" and 8.2 "Annuities"
  8. ↑ U.S. Securities and Exchange Commission, Compound Interest Calculator, Investor.gov
Value of money over time — recommended articles
Future value — Net present value — Cost of money — Real rate of return — Effective annual interest rate — Opportunity cost — Bonds in finance
Retrieved from CEOpedia