Sinking fund factor

The sinking fund factor $(A/F,i,N)$ converts a future amount $F$ to the equal periodic deposit $A$ needed to accumulate to it after $N$ periods at periodic rate $i$:

$A=F\cdot (A/F,i,N),(A/F,i,N)=\frac{i}{(1+i{)}^N-1}.$

The name comes from the historical instrument: a sinking fund is a savings account a corporation or government feeds with regular deposits to retire a long-term debt (a bond, say) when it comes due. Deposits accumulate with interest; the size needed to hit a target $F$ on schedule is $A$.

Derivation. The future value of an annuity of $A$ over $N$ periods is $F=A\cdot \frac{(1+i{)}^N-1}{i}$ (see Uniform series compound amount factor). Solving for $A$ gives the sinking fund factor as its reciprocal.

Engineering applications:

• Estimating the equal annual deposit needed to build up replacement capital for an asset.
• Funding a future major maintenance event (overhaul, refit) by spreading the cost across the working life.
• Bond sinking funds, though these have mostly given way to call provisions in modern bond markets.

For the reverse direction (annuity to future amount) see Uniform series compound amount factor.