Annuity Definitions

Beginning-of-month vs. end-of-month on a 25-year pension: same $4,000, same 300 payments, but the timing shift swings PV by $17,600 at 5%.

Pays at the end of each period. PV one period before the first payment:

$$a_{\overline{n\vert}} = \frac{1 - v^n}{i}$$

Accumulated value at the time of the last payment:

$$s_{\overline{n\vert}} = \frac{(1+i)^n - 1}{i}$$

Relationship: $s_{\overline{n\vert}} = a_{\overline{n\vert}} \cdot (1+i)^n$.

The name "immediate" is misleading, payments come at the end, not immediately.

Pays at the beginning of each period. Double-dot notation:

$$\ddot{a}_{\overline{n\vert}} = \frac{1 - v^n}{d}$$

$$\ddot{s}_{\overline{n\vert}} = \frac{(1+i)^n - 1}{d}$$

HIGH-FREQUENCY: The only difference between immediate and due is the denominator, $i$ vs. $d$. Equivalently:

$$\ddot{a}_{\overline{n\vert}} = (1+i) \cdot a_{\overline{n\vert}}$$

Multiplying by $(1+i)$ shifts every payment one period earlier.

A perpetuity has $n \to \infty$. Since $v^n \to 0$:

$$a_{\overline{\infty\vert}} = \frac{1}{i}, \quad \ddot{a}_{\overline{\infty\vert}} = \frac{1}{d}$$

Common mistakes

• Using $i$ when the annuity is due. For due, the denominator is $d$. At 7%, 15-year: immediate $a = 9.1079$, due $\ddot{a} = 9.1079 \times 1.07 = 9.7455$. Using 9.1079 for due undervalues by 7%. Trap: $18,216 instead of $19,491 for $2,000 annuity-due.
• Swapping $i^{(m)}$ and $d^{(m)}$. End-of-month = $i^{(12)}$. Beginning-of-month = $d^{(12)}$. Using the wrong one changes PV.
• Forgetting to scale to annual. For $500/month, the annual rate is $6,000. The formula assumes total 1 per year. Multiplying $500 \times a^{(12)}$ instead of $6000 \times a^{(12)}$ understates PV by factor 12.

Bottom line

• Numerator is always $1 - v^n$; the denominator sets the type: $i$, $d$, $i^{(m)}$, $d^{(m)}$, or $\delta$.
• Due = immediate $\times (1+i)$; the denominator switches from $i$ to $d$.
• Perpetuity: $1/i$ (immediate) or $1/d$ (due).
• For $m$-thly annuities paying $P$ per period, the annual rate is $mP$ (the formula assumes a total of 1 per year).

Exam shortcut

Before computing, answer two questions: (1) beginning or end? (2) how many payments per year? Write the denominator first, it anchors the calculation. "Denominator determines everything." "Due means double-dot, due means d", both start with "d." Immediate-to-due shortcut: multiply by $(1+i)$.

The full lesson (about 2,441 words, 16 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2a

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