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Sinking funds

The borrower pays the lender interest only and saves separately to repay the principal in one lump - two rates, two cash flows, one total outlay.

The formulas

Sinking fund deposit

$$D = \dfrac{L}{s_{\overline{n}|j}}$$

j is the rate the fund earns

Total outlay

$$\text{Total} = Li + D$$

Equivalent amortisation

$$Li + \dfrac{L}{s_{\overline{n}|j}} = \dfrac{L}{a_{\overline{n}|i}} \iff i = j$$

Net amount owed

$$L - D\,s_{\overline{t}|j}$$

Where it comes from

1. The lender is kept whole each period by the interest payment $Li$, so the principal $L$ is still owed at time $n$.
2. The fund must accumulate to $L$: level deposits $D$ accumulate to $D\,s_{\overline{n}|j}$, so $D = L/s_{\overline{n}|j}$.
3. When $j = i$ the total outlay is exactly the amortisation payment, because $\frac{1}{a_{\overline n|i}} = i + \frac{1}{s_{\overline n|i}}$ - the identity to remember.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• 1/aₙ = i + 1/sₙ. That single identity is the bridge between amortisation and sinking funds.
• Two rates: the loan rate goes to the lender, the fund rate grows your savings. Never mix them.

Traps

• Using aₙ instead of sₙ for the deposit - the fund ACCUMULATES.
• Assuming the sinking-fund method is always more expensive; it is cheaper exactly when j > i.

Related

• Loan amortisation
• Outstanding loan balance

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