Annuities and Perpetuity

Standard cash flows
Annuities (series of equal payments) and perpetuities (infinite flows) are standard cash flow patterns with simple calculation formulas. They simplify calculations for loans, pensions, valuation, and many financial instruments.

Ordinary Annuity

Ordinary annuity: a series of equal payments at the end of each period. The most common type (loans, bonds with fixed coupon).

Present Value:
$PV = PMT \times \left[ \frac{1 - (1+r)^{-n}}{r} \right]$.
Where PMT is payment per period, r is the rate per period, n is the number of periods.

Future Value:
$FV = PMT \times \left[ \frac{(1+r)^n - 1}{r} \right]$.
The accumulated sum at the end of n periods.

Example: $100 at the end of each year, 5 years, 10%.
$PV = 100 \times \left[\frac{1 - 1.10^{-5}}{0.10}\right] = 100 \times 3.791 = $379.08$.

Annuity Due

Annuity due: payments at the beginning of each period (rent, insurance premiums). One period earlier vs ordinary annuity.

Adjustment:
$PV \text{ (annuity due)} = PV \text{ (ordinary annuity)} \times (1+r)$.

FV is similarly adjusted.

Payments are received earlier — higher PV.

Example: same $100, but at the beginning of the year.
$PV = 379.08 \times 1.10 = $416.99$.
The difference — one year of additional compounding.

Perpetuity

Perpetuity: infinite series of equal payments. Theoretical concept, but useful for approximating very long-lived streams.

Present Value:
$PV = \frac{PMT}{r}$.

The simplest formula — infinite geometric progression.

Example: $100 annually forever, 10% rate.
$PV = 100 / 0.10 = $1,000$.

Check: $1,000 \times 10% = $100$ annual income, principal preserved.

Application: preferred stock valuation (fixed dividend forever); real estate (stable rent stream); console bonds (perpetual bonds).

Growing Perpetuity

When payments grow at a constant rate $g$:
$PV = \frac{PMT_1}{r - g}$.

Where $PMT_1$ is the payment at the end of the first period, $g$ is the growth rate.

Condition: $g < r$.

Gordon Growth Model: application for stock valuation.
$P = \frac{D_1}{r - g}$.

Where $D_1$ is the expected dividend next year, $g$ is the dividend growth rate.

Example: dividend $2 next year, growing 5% forever, required return 12%.
$P = 2 / (0.12 - 0.05) = $28.57$.

Growing Annuity

Finite series of growing payments.

Formula is more complex:
$PV = PMT_1 \times \frac{1 - \left( \frac{1+g}{1+r} \right)^n}{r - g}$.

Application: salaries (growing with inflation), graduated payment mortgages.

Tip: can be calculated as the difference between two perpetuities with different start dates.

Loan Amortization

Loan with equal payments — ordinary annuity.
Knowing PV (loan amount), r, n — find PMT.

$PMT = PV \times \left[ \frac{r}{1 - (1+r)^{-n}} \right]$.

This is the payment needed to repay the loan with interest.

Amortization schedule: each payment includes interest on remaining balance + principal repayment. Interest decreases, principal increases over time.

Example: mortgage $300,000, 30 years, 6%.
Monthly rate = 0.5%, n = 360.
$PMT = 300000 \times \left[ \frac{0.005}{1 - 1.005^{-360}} \right] = $1,798.65$.

Applications in Corporate Finance

Bond valuation: PV of coupon annuity + PV of principal at maturity = bond price.

Lease analysis: PV of lease payments = present value of lease obligation.

Project evaluation: many projects have annuity-like cash flows — constant revenue stream.

Retirement planning: target retirement income as a perpetuity, accumulation via annuity contributions.

Deferred Annuities

Annuity starting not immediately, but after $k$ periods. First payment at end of period $k+1$.

$PV = [\text{Standard Annuity PV}] \times (1+r)^{-k}$.

Discount the annuity value back $k$ additional periods.

Example: pension plans — start of payments after 20 years of work.

Calculator and Spreadsheet Tips

Financial calculator keys: N (periods), I/Y (interest per period), PV, PMT, FV. Solve for one given others.

Excel functions: PV, FV, PMT, NPER, RATE.

Consistent sign convention: outflows negative, inflows positive.

Common mistake: forgetting to adjust for payment timing (ordinary vs due). Check mode settings.