Annuity Calculations

A $150,000 student loan at 5% over 20 years, the payment from a single annuity formula determines whether the graduate lives comfortably or barely makes rent.

The most common annuity problem. For an annuity-immediate:

$$P=\frac{PV}{a_{\overline{n|}}}$$

For due, replace with ${\ddot{a}}_{\overline{n|}}$. The due payment is smaller because earlier payments are more valuable.

HIGH-FREQUENCY: "Find the payment" is the single most common annuity question on FM.

$$PV=P\cdot a_{\overline{n|}},FV=P\cdot s_{\overline{n|}}$$

Direct calculations. Just match the annuity type to the problem.

$$v^n=1-\frac{i\cdot PV}{P},n=-\frac{\ln \left(1-\frac{i\cdot PV}{P}\right)}{\ln (1+i)}$$

This requires $P>i\cdot PV$, the payment must exceed the interest. Otherwise the loan never amortizes.

Non-integer $n$: the borrower makes $\lfloor n\rfloor$ full payments plus a smaller drop payment. The exam specifies the convention.

No closed form. The approach:

• Compute the target: $a_{\overline{n|}}=PV/P$.
• Test two candidate rates.

Common mistakes

• Wrong annuity factor for due vs. immediate. For beginning-of-year payments, use $\ddot{a}$. At 5%, 20 years: $\ddot{a}=13.08532$, payment = $11,463. Using $a=12.46221$ gives $12,036, $573 too high. Trap: $12,036 for a due problem.
• Using $a$ instead of $\ddot{a}$ in $(Ia)$. The formula is $(\ddot{a}-n{v}^n)/i$, not $(a-n{v}^n)/i$. At $n=10,i=0.06$: correct $(Ia)=36.96$, incorrect = 29.60. Trap: 29.60.
• Forgetting the deferral factor. First payment at time 6 (deferred 5): PV = $v^5\cdot P\cdot a_{\overline{n|}}$. Omitting $v^5$ overstates PV by factor $(1.05{)}^5=1.276$.

Bottom line

• Payment = PV / annuity factor. Match the factor to the type (immediate vs. due, annual vs. $m$-thly).
• Solve for $n$: $n=-\ln (1-i\cdot PV/P)/\ln (1+i)$. Requires $P>i\cdot PV$, or the loan never amortizes.
• Solve for $i$: no closed form. Compute $a_{\overline{n|}}=PV/P$, test two bracketing rates, then interpolate.
• Solve for $n$ on the AV side: $(1+i{)}^n=1+i\cdot FV/R$, then logs. A non-integer $n$ means a smaller final deposit, not a fractional one.

Exam shortcut

"Find the payment" (annuity factor first, then divide. "Find the term") log formula. "Find the rate" (target $a=PV/P$, two trials, interpolate. Arithmetic) always write the decomposition before computing. "Payment = PV over a." Arithmetic: "Level plus ramp", $(P-Q)$ level + $Q$ ramp. Geometric: "Ratio to the n minus 1, over the gap."

The full lesson (about 2,953 words, 20 min read) adds 4 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 2b

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