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:

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

For due, replace with $\ddot{a}_{\overline{n\vert}}$. 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.

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

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

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

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: $a_{\overline{n\vert}} = PV/P$.
• 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 $(\ddot{a} - nv^n)/i$, not $(a - nv^n)/i$. 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 = $v^5 \cdot P \cdot a_{\overline{n\vert}}$. 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 $a_{\overline{n\vert}} = PV/P$, 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 3,620 words, 24 min read) adds 7 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 2b

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