Basics of the Time Value of Money

Fundamental Concepts of TVM

Time Value of Money (TVM) is the concept that money today is worth more than the same amount in the future. This is a fundamental principle of finance that underpins all investment decisions, valuation, and financial planning.

Why Money Has a Time Value

Opportunity cost: Money today can be invested to earn a return. Delayed receipt means foregone income.
Inflation: The purchasing power of money decreases over time. $100 today will buy more than $100 in a year.
Risk: Future money is less certain than money today. Uncertainty requires compensation.
Consumption preference: People prefer to consume sooner rather than later. Deferred consumption requires a reward (interest).

Future Value (FV)

Future Value is the value of a present amount at a future date at a given rate of return.

Simple interest formula:
$ FV = PV \times (1 + r \times n) $.
Where $ PV $ is present value, $ r $ is the rate per period, $ n $ is the number of periods.
Interest is accrued only on the principal.
Compound interest formula:
$ FV = PV \times (1 + r)^n $.
Interest is accrued on principal plus accumulated interest.
Compounding is a more realistic model.

Example: $1,000 at 10% annual interest for 3 years.
Simple: $ FV = 1000 \times (1 + 0.10 \times 3) = $1,300 $.
Compound: $ FV = 1000 \times (1.10)^3 = $1,331 $.

Present Value (PV)

Present Value is the current worth of a future sum. The reverse operation of FV.

Formula:
$ PV = \dfrac{FV}{(1 + r)^n} $.
Discounting: Bringing future money to today's value.
Discount rate ($r$): Reflects opportunity cost, risk, time preference.
Choice of rate is a critical decision in valuation.

Example: $1,331 in 3 years at 10% rate.
$ PV = \dfrac{1331}{(1.10)^3} = $1,000 $.
We are indifferent between $1,000 today and $1,331 in 3 years.

Compounding Frequency

Annual compounding: Interest is credited once per year.
Semi-annual: Twice per year.
$ FV = PV \times (1 + r/2)^{2n} $.
Each period is half a year.
Quarterly: Four times a year.
$ FV = PV \times (1 + r/4)^{4n} $.
Continuous compounding: Interest is credited infinitely often.
$ FV = PV \times e^{r \times n} $.
The mathematical limit of discrete compounding.
Effective Annual Rate (EAR): The actual annual yield considering compounding.
$ EAR = (1 + r/m)^m - 1 $, where $ m $ is the compounding frequency.
$ EAR > $ stated rate if $ m > 1 $.

Discount Factor

Discount Factor = $ \dfrac{1}{(1 + r)^n} $.

The multiplier to convert FV to PV.
The discount factor is always less than one.

$ \text{Present Value} = \text{Future Value} \times \text{Discount Factor} $.

Useful for calculating PV of multiple cash flows.

Timeline Approach

Visualizing cash flows on a timeline is an essential tool.

Time 0 — today.
Positive cash flows go up, negative go down.

Example:
Investment of $1,000 today (outflow at $ t=0 $), receipt of $300 in years 1, 2, 3, and $400 in year 4.

All cash flows must be brought to a single point in time for comparison.

Usually $ t=0 $ (present value analysis).

Rule of 72

Quick estimate: at what rate will money double? Approximately $ 72 / r $ (%).

At 8% — about 9 years.
At 12% — about 6 years.

Alternatively: how many years to double at a given rate? $ 72 $ divided by annual percent.

Useful for quick mental calculations without a calculator.

Implications for Investing

Power of compounding: Small differences in the rate dramatically affect long-term results.
10% vs 8% over 30 years: $1,000 becomes $17,449 vs $10,063.
Early investing: Starting earlier is more important than investing more later.
Time is the most valuable resource in investing.
Cost of delay: Delayed investing has an opportunity cost.
Each year of delay reduces final wealth.

TVM in Corporate Finance

Capital budgeting: Project evaluation via NPV (discounting future cash flows).
Valuation: DCF company valuation — discounting expected free cash flows.
Bond pricing: $ PV $ of coupon payments and principal = bond price.
Lease vs buy: Comparison of PV cash outflows for alternatives.