Interest Rates and the Time Value of Money

Confuse a nominal rate for an effective rate on a $10 million liability and the present value error exceeds $120,000. Every FM calculation starts with the right rate.

Under simple interest, one dollar at time 0 grows to $1 + it$ at time $t$:

$$a(t) = 1 + it$$

Under compound interest, growth is exponential:

$$a(t) = (1 + i)^t$$

Simple interest applies to short-term instruments like T-bills. Compound interest dominates everything else. Unless a problem says "simple interest," assume compound.

Under simple interest the effective rate per period falls each year, because the fixed interest $i$ is earned on a growing base:

$$i_n = \frac{i}{1 + i(n-1)}$$

The simple and compound curves cross at $t = 1$, where both give $1 + i$. Before $t = 1$ simple interest is larger. After $t = 1$ compound interest pulls ahead and the gap widens.

Common mistakes

• Using the nominal rate as the effective rate. With $i^{(12)} = 0.06$, the annual effective rate is $(1.005)^{12} - 1 = 0.06168$, not 0.06. Plugging 6% directly gives $5000(1.06)^{10} = 8954.24$. Trap: $8,954.24 instead of the correct $9,096.98.
• Forgetting to integrate a variable force. When $\delta(t) = 0.04 + 0.002t$, the accumulation is $e^{\int_0^5 \delta(t)\,dt}$, not $e^{5 \cdot \delta(5)} = e^{0.25} = 1.284$. Trap: $25,680 instead of $25,046.
• Confusing discount rate and interest rate. Given $d = 0.05$, the interest rate is $i = 0.05/0.95 = 0.05263$, not 0.05. Over 20 years, this error compounds to more than 1% of the final answer.

Bottom line

• Compound interest is the FM default, $a(t) = (1+i)^t$; use simple interest $a(t) = 1+it$ only when explicitly stated.
• Six rate symbols: $i$, $d$, $v$, $i^{(m)}$, $d^{(m)}$, $\delta$. Converting between them is the most tested mechanical skill.
• Discount factor $v = 1/(1+i) = 1 - d$ appears in virtually every FM formula.
• Under simple interest the effective rate declines, $i_n = i/[1+i(n-1)]$; the simple and compound curves cross at $t=1$.

Exam shortcut

When you see a nominal rate, immediately compute the effective rate per period, write $i^{(m)}/m$ first. For variable force, write the integral before anything else. "DIVE": Discount factor $v = 1/(1+i)$, Interest-discount link $d = iv$, Variable force: integrate then exponentiate, Effective from nominal: $(1 + i^{(m)}/m)^m = 1 + i$. Force of interest: "Force is the log."

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Learning objectives

• 1a

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