Timedora

3 Time Value of Money

Learn how compounding, discounting, and payment timing let you compare financial cash flows at a common date.

Why timing changes value

The means that money available now can be invested or used sooner, so its value differs from the value of the same amount received later. It provides a way to compare cash flows that occur at different times.

A calculation depends on the amount, the interest or discount rate, the number of periods, and when payments occur. To compare alternatives, express their cash flows at a common date, often today.

and

means earning a return on the original amount and on returns already earned. For a single amount invested at a fixed rate per period, is calculated as:

FV=PV(1+r)nFV = PV(1+r)^n

Here, PVPV is , FVFV is , rr is the rate per period, and nn is the number of periods. For example, $1,000\$1{,}000 invested for five years at 5%5\% compounded annually grows to:

FV=1,000(1.05)5=$1,276.28FV = 1{,}000(1.05)^5 = \$1{,}276.28

More frequent changes the result. If an annual nominal rate is compounded mm times per year for tt years, use:

FV=PV(1+rm)mtFV = PV\left(1 + \frac{r}{m}\right)^{mt}

Match the rate and number of periods: a monthly rate must be paired with the number of months.

and

reverses : it converts a future amount into its equivalent value today using a discount rate. For one future cash flow:

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

At a 5%5\% annual discount rate, $1,276.28\$1{,}276.28 received in five years has a of about $1,000\$1{,}000. A higher discount rate or a longer wait generally lowers .

The discount rate represents the return required for waiting and may reflect opportunity cost and the risk of receiving the cash flow.

Valuing equal periodic payments

An is a series of equal payments made at regular intervals. An pays at the end of each period, while an pays at the beginning. Because each payment occurs at a different time, each must be valued according to when it occurs.

For an with payment PMTPMT, periodic rate rr, and nn payments, the present and future values are:

PVannuity=PMT(1−(1+r)−nr)PV_{annuity} = PMT\left(\frac{1-(1+r)^{-n}}{r}\right)
FVannuity=PMT((1+r)n−1r)FV_{annuity} = PMT\left(\frac{(1+r)^n-1}{r}\right)

For an , multiply the corresponding ordinary- value by (1+r)(1+r), because every payment occurs one period earlier. These formulas assume the rate and payment interval match.

For example, saving $1,000\$1{,}000 at the end of each year for three years at 5%5\% yields a of about $3,152.50\$3{,}152.50.

Using TVM in financial decisions

TVM helps compare alternatives whenever their cash flows occur on different dates. Before comparing options, identify the timing of every cash flow, choose an appropriate rate, use consistent periods, and make assumptions explicit.

  • Saving and borrowing: Estimate how an investment balance may grow, or calculate the current value of future loan payments.

  • Payment choices: Compare a lump sum today with installments or a lump sum later by valuing each option at the same date and rate.

  • Investments: Discount expected project cash flows to today. is the of future cash inflows minus the initial investment and any other relevant outflows. A positive NPV means projected inflows exceed costs at the chosen discount rate; it does not guarantee that the project will succeed.

For uneven cash flows, discount each amount separately and add the results:

PV=∑t=1nCFt(1+r)tPV = \sum_{t=1}^{n}\frac{CF_t}{(1+r)^t}

The result is a decision aid, not a prediction that uncertain future cash flows will occur exactly as estimated. In summary, moves money forward through time, moves it back, single-payment formulas value lump sums, and formulas value equal periodic payments.