Time value of money

A dollar received now is worth more than a dollar received later because today’s dollar can be invested, earns a return and remains available for use. This principle—the time value of money—allows cash flows occurring at different dates to be compared on a common basis.

When to use it

• Compare personal investments or financing choices.

• Evaluate the economic return of a project or acquisition.

• Translate amounts received or paid at different points in time into equivalent values.

Origins

The intuition is older than modern finance: lending, interest and the preference for earlier payment have existed for centuries. Accounting historian R. H. Parker reported in 1968 that surviving interest-rate tables date to 1340. Over time, commercial practice and mathematics formalised the intuition into present-value, future-value, annuity and discounted-cash-flow techniques.

What it is

An interest rate is the price paid for using money over time. With simple interest, each period’s interest is calculated only on the original principal. If the principal is $100 and the annual rate is 10 per cent, each year adds $10:

Year 1: 10 per cent of $100 = $10 + $100 = $110 Year 2: 10 per cent of $100 = $10 + $110 = $120 Year 3: 10 per cent of $100 = $10 + $120 = $130 Year 4: 10 per cent of $100 = $10 + $130 = $140 Year 5: 10 per cent of $100 = $10 + $140 = $150

With compound interest, earned interest is reinvested, so later returns are earned on both the original principal and accumulated interest. Assuming no tax, the same investment begins as follows:

Year 1: 10 per cent of $100.00 = $10.00 + $100.00 = $110.00

The next compounded balance is $121.00. The following period adds $12.10 to $121.00, producing $133.10. The sequence then continues:

Year 4: 10 per cent of $133.10 = $13.31 + $133.10 = $146.41 Year 5: 10 per cent of $146.41 = $14.64 + $146.41 = $161.05

Across 20 years, simple interest produces linear growth while compound interest produces geometric growth. The underlying chart uses the scale and periods shown here: 800 700 600 500 400 300 200 100 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20. The gap widens as the investment horizon increases.

Compound value can be written as:

Pn = P0(1 + I)n

where Pn is the ending value after n periods, P0 is the starting value and I is the interest rate per period.

How to use it

The core task is to choose a valuation date and move every relevant cash flow to that date with a rate consistent with its timing and risk. A quick sensitivity view can also compare a base index of 100 under a 10 per cent assumption with alternative rates and horizons.

Net present value (NPV)

Net present value converts future project cash flows into today’s money and subtracts the initial investment. Consider an opportunity requiring $100 now, with a cost of capital of 10 per cent and expected cash receipts of $30 annually for 10 years. The nominal receipts total $300, but later receipts are worth less today.

Discount each payment at 10 per cent. The $30 received in Year 2 has a factor of 0.83 and a present value of $24.79. The final $30 in Year 10 uses a factor of 0.39, giving a present value of $11.57. The complete illustration appears below:

After deducting the initial $100, the project has a positive NPV of $84.34 and therefore creates value under these assumptions. That result is below the undiscounted gain of $200—the difference between nominal receipts of $300 and the $100 investment—because NPV recognises when the cash arrives.

Present value of an annuity

An annuity is a series of equal or regular cash flows, such as rent or pension payments. Discounting each payment at an appropriate rate converts the stream into a single present value. Inflation may inform that rate when the objective is to express future purchasing power, while investment decisions usually require a risk-appropriate discount rate.

Future value

Future value moves a current amount forward by compounding it at an assumed growth or return rate. An asset worth $100 today that compounds at 10 per cent annually for five years reaches $161.05.

Top practical tip

Lay out the cash-flow dates before selecting a formula. Identify the number of periods, periodic rate, present value, regular payment and future value, and make sure rate and period use the same time unit. Once the known variables are consistent, solve for the missing one.

Top pitfall

Long-horizon valuations are highly sensitive to the discount rate. Do not create false precision around cash-flow estimates while treating the rate as an afterthought. Test plausible rates and recognise that risk and the cost of capital may change during the project.

Further reading

• Fisher, I. (nineteen thirty). The Theory of Interest. Macmillan.
• Brealey, R.A., Myers, S.C. and Allen, F. (twenty twenty-three). Principles of Corporate Finance. McGraw Hill.