# Sinking Fund

A sinking fund is a fund required to be setup by the bond issuer to which it must contribute money each period to repurchase a certain portion of the bond issue. It can also be referred to a fund created by a company to accumulate money for replacement of a large asset or any other major expenditure.

In case of bonds, the sinking fund is a credit enhancement because it is a separate custodial account whose funds are earmarked for repurchase of bonds. In an accelerated sinking fund, the issuer is entitled to retire a higher percentage of the bond. An accelerated sinking fund provision has an embedded call option because it allows the bond issuer to retire bonds earlier than their scheduled retirement.

## Formula

Let A be the money accumulated and P be the periodic contribution, if the interest rate is r, there are n number of years and m number of payments per year, we can use the future value of annuity formula to find the accumulated amount A:

$$\text{A}=\text{P}\times\frac{{(\text{1}+\frac{\text{r} _ \text{s}}{\text{m}})}^{\text{n}\times \text{m}}-\text{1}}{\frac{\text{r} _ \text{s}}{\text{m}}}$$

Dividing both sides by the second term on the right hand side in the above equation and flipping it, we get the formula for calculation of periodic contribution needed:

$$\text{P}=\frac{\text{A}}{\frac{{(\text{1}+\frac{\text{r} _ \text{s}}{\text{m}})}^{\text{n}\times \text{m}}-\text{1}}{\frac{\text{r} _ \text{s}}{\text{m}}}}$$

## Example

Goliath Infrastructures (GI) just issued 5 million $100-par bonds payable carrying 8% coupon rate and maturing in 15 years. The bond indenture requires GI to set up a sinking up to pay off the bond at the maturity date. Semi-annual payments are to be made to the fund which is expected to earn 5% per annum. Find the amount of required periodic contributions. The future value required to be accumulated equals$500 million (=5,000,000*\$100)

Since the payments are semi-annual, the periodic interest rate = 5%/2 = 2.5%

Number of periods = 2*15 = 30

$$\text{Periodic contribution to sinking fund}=\frac{\text{\500,000,000}}{\frac{{(\text{1}+\frac{\text{5%}}{\text{2}})}^{\text{15}\times\text{2}}-\text{1}}{\frac{\text{5%}}{\text{2}}}}=\text{\11,388,820}$$