Financial Management

Time value of money

Money today is worth more than the same money later because it can be invested now.

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Definition

The time value of money is the principle that a cash flow has different value depending on when it occurs. A sum received today can be invested and grow, so a future sum must be discounted to be compared fairly with today's money.

Compounding carries value forward and discounting brings it back. They are the same relationship read in opposite directions, linked by a rate and a number of periods.

Intuition

Waiting has a cost: the return you forgo by not having the money now. The discount rate is simply the price of that waiting.

When to use it

Comparing cash flows that arrive on different dates.
Pricing loans, savings goals, leases, and any repayment schedule.
As the engine behind NPV, bond pricing, and the dividend discount model.

When not to use it

Comparing cash flows that all occur today: no discounting is needed.
Mixing rates and periods without converting them first (a monthly rate with years).

Formula

Future value

FV = PV × (1 + r)ⁿ

Grows an amount forward by a constant rate per period.

PV: Present value
r: Rate per period
n: Number of periods

Formula

Present value

PV = FV ÷ (1 + r)ⁿ

Brings a future amount back to today.

FV: Future value
r: Discount rate per period
n: Number of periods

Worked example

Discounting a promised payment

You are promised 11,664 in two years and require an 8% annual return.

1
Identify the inputs
FV = 11,664, r = 8% (0.08), n = 2.
2
Discount to today
PV = 11,664 ÷ 1.08².
3
Simplify
1.08² = 1.1664, so PV = 11,664 ÷ 1.1664.

PV = 10,000. The promise is worth exactly today's 10,000 to you.

Common mistakes

Mismatching the rate basis and the period: 1% per month is not 12% per year when compounding.
Placing year-one cash flow at time zero instead of time one.
Discounting nominal cash flows with a real rate, or the reverse.