# 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:

$$ A=P\times\frac{{(1+\frac{r_s}{m})}^{n\times m}-1}{\frac{r_s}{m}} $$

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:

$$ P=\frac{A}{\frac{{(1+\frac{r_s}{m})}^{n\times m}-1}{\frac{r_s}{m}}} $$

## 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

$$ Periodic\ contribution\ to\ sinking\ fund=\frac{$500,000,000}{\frac{{(1+\frac{5\%}{2})}^{15\times2}-1}{\frac{5\%}{2}}}=$11,388,820 $$