Sinking Fund: Saving Toward a Target

What is a sinking fund?

A sinking fund is a savings plan with regular equal deposits, all earning interest, accumulating to a target value at a future date. Used for: replacing equipment, retiring debt, funding expansions, building reserves.

It's the inverse of an annuity payment: instead of solving for the annuity payout, you solve for the contribution that reaches a target FV.

The formula

$PMT = \dfrac{FV \cdot r}{(1 + r)^n - 1}$

Derived from the ordinary-annuity FV formula ($FV = PMT \cdot \frac{(1+r)^n - 1}{r}$) solved for $PMT$.

where $PMT$ = periodic contribution, $FV$ = target, $r$ = periodic rate, $n$ = total periods.

Step-by-step solution

Setup: $FV = 100{,}000$, annual rate 5% ⟹ quarterly rate $r = 0.05/4 = 0.0125$, 10 years ⟹ $n = 40$ quarters.

Step 1 — Growth factor: $(1.0125)^{40} \approx 1.6436$

Step 2 — Annuity factor denominator: $(1.0125)^{40} - 1 \approx 0.6436$

Step 3 — Multiply by $r$: $0.6436 / 0.0125 = 51.489$

Wait — plug straight into the formula: $PMT = \dfrac{100{,}000 \cdot 0.0125}{0.6436} = \dfrac{1{,}250}{0.6436} \approx \boxed{1{,}941.70}$

So you deposit about $1,941.70 per quarter ($7,767/year) for 10 years.

Verification

Total deposits: $40 \cdot 1{,}941.70 \approx 77{,}668$. Interest earned: $100{,}000 - 77{,}668 \approx 22{,}332$. The interest covers 22% of the goal.

Sensitivity

• Higher rate: less contribution needed. At 8% quarterly compounding, PMT \approx \1,698/quarter$.
• Longer horizon: way less contribution. At 20 years @ 5%, PMT \approx \729/quarter$ — time is your best friend.
• More frequent compounding: small benefit. Monthly vs. quarterly @ 5% changes $PMT$ by ~1%.

Corporate use: bond sinking funds

When a corporation issues long-term debt, it often sets up a sinking fund to gradually redeem bonds over time rather than face a single giant payoff at maturity. The company's reserve accumulates from regular contributions plus interest, and redemptions happen at intervals.

Starting with a head start

If you already have $P$ today that's also growing, the required $PMT$ is smaller: $FV = P(1+r)^n + PMT \cdot \dfrac{(1+r)^n - 1}{r}$

Solve for $PMT$: $PMT = \dfrac{(FV - P(1+r)^n) \cdot r}{(1+r)^n - 1}$

For instance, if we already had $20K today:$(1.0125)^{40} \cdot 20{,}000 = 32{,}872$, so we'd need the annuity to cover only$100{,}000 - 32{,}872 = 67{,}128$, giving$PMT \approx $1{,}304/quarter$.

Common mistakes

• Using annual rate and number of years when payments are quarterly.
• Ignoring that the last payment earns no interest in an ordinary annuity (what this formula assumes).
• Starting with annuity-due assumptions. If payments are at period start, divide $PMT$ by an extra $(1 + r)$.

Try it in the visualization

Stacked columns grow quarter by quarter — each column adds the latest payment plus interest on the existing balance. A target line at $100K shows when the fund hits its goal.