Interest and the Time Value of Money

Note

Key Point — Time-value principle

A rupee available today is normally worth more than the same nominal rupee in the future because present funds can earn a return, inflation reduces purchasing power, and future receipt carries risk and foregone opportunity.

Simple and Compound Interest

Caution

Key Formula — Simple interest

$$SI=Prt, \qquad A=P(1+rt),$$

where $P$ is principal, $r$ is the annual decimal rate, $t$ is years, and $A$ is the accumulated amount. Interest is earned only on the original principal.

[!IMPORTANT]

Key Formula — Compound interest

For $n_c$ compounding intervals per year over $t$ years,

$$A=P\left(1+\frac{r}{n_c}\right)^{n_ct}.$$

With annual compounding, $A=P(1+r)^t$. Compound interest earns interest on prior interest as well as principal.

Present and Future Value

Caution

Key Formula — Single-payment equivalence

For $n$ periods at effective rate $i$ per period,

$$\boxed{F=P(1+i)^n}, \qquad \boxed{P=\frac{F}{(1+i)^n}}.$$

Cash flows can be compared only after moving them to a common point in time at a consistent rate.

Present and future values as economically equivalent cash flows

Factor	Formula	Notation
Future value, single payment	$(1+i)^n$	$(F/P,i,n)$
Present value, single payment	$1/(1+i)^n$	$(P/F,i,n)$
Future value, uniform series	$[(1+i)^n-1]/i$	$(F/A,i,n)$
Present value, uniform series	$[(1+i)^n-1]/[i(1+i)^n]$	$(P/A,i,n)$
Capital recovery	$i(1+i)^n/[(1+i)^n-1]$	$(A/P,i,n)$
Sinking fund	$i/[(1+i)^n-1]$	$(A/F,i,n)$

Standard discrete-compounding factors

The uniform-series factors above describe an ordinary annuity, with equal end-of-period payments. An annuity due has payments one period earlier, so the ordinary-annuity value is multiplied by $(1+i)$.

Note

Example — Single-payment present and future value

An investment of Rs. 100,000 earns 10% annually for five years:

$$F=100{,}000(1.10)^5=\text{Rs.~}161{,}051.$$

At the same rate, the present value of Rs. 200,000 receivable in five years is

$$P=\frac{200{,}000}{(1.10)^5}=\text{Rs.~}124{,}184\text{ (approximately)}.$$

[!NOTE]

Example — Present value of an ordinary annuity

For Rs. 10,000 paid at each year-end for five years at 8%,

$$P=10{,}000\left[\frac{(1.08)^5-1}{0.08(1.08)^5}\right] =10{,}000(3.9927)=\text{Rs.~}39{,}927.$$

[!TIP]

Exam Focus — Rate-period consistency

The interest rate period must match the number and spacing of cash-flow periods. Convert nominal rates to the appropriate periodic or effective rate before using the factors, and place every cash flow at its actual timing point.