This post presents a working example of Macaulay & Modified duration calculations. Earlier we had considered the importance of the Duration risk metric to Asset Liability Management (ALM) and managing interest rate risk. In this post, we will look at the specific mechanics of the Macaulay Duration and Modified Duration calculations. For application to US Treasuries also see the duration calculation example for US treasuries.

## 1. Fixed income term sheet

Let us consider a fixed income instrument with the following terms:

Issue Date = 10^{th} December 2010

Face Amount = 100

Tenor = 1 year

Coupon = 10% per annum

Coupon payment frequency, k = semi-annual = 2

Yield to Maturity (YTM) = 12% per annum

The schedule of payments will, therefore, be as follows:

Date |
| t= Tenor * Frequency | Cash Flow |

10-June 2011 | 0.5 | 1 | (10%/2) * 100 = 5 |

10-December 2011 | 1 | 2 | (10%/2) * 100 + 100 = 105 |

## 2. Price

The price of the instrument on the issue date will be the discounted value of the cash flows, i.e. their present value (PVCF). This is equal to:

PVCF_{t} = Cash flow/(1+YTM/2)^(t)

t |
| Present Value of Cash Flow (PVCF) |

1 | 5 | 5/(1+12%/2)^1 = 4.7170 |

2 | 105 | 105/(1+12%/2)^2 = 93.4496 |

The price of the instrument at issue is, therefore, the sum of these two present values, i.e. the present value of total cash flow, PVTCF = 4.7170 + 93.4496 = 98.1666

The price may also be obtained directly using the excel formula “PRICE” with the following input parameters:

Settlement = Issue Date = 10^{th} December 2010

Maturity = Maturity Date = 10^{th} December 2011

Rate = Coupon Rate = 10%

Yld = YTM = 12%

Redemption = Face Amount =100

Frequency = k =2

Basis = Type of day count basis = 1 (Actual/ Actual)

## 3. Macaulay Duration

Macaulay Duration is given by the following formula:

For this instrument, Macaulay Duration works out to:

(1*4.7170+2*93.4496)/(2*98.1666) = 0.9760

The Macaulay Duration may also be obtained directly using the excel formula “DURATION” with the following input parameters:

Settlement = Issue Date = 10^{th} December 2010

Maturity = Maturity Date = 10^{th} December 2011

Coupon = Coupon Rate = 10%

Yld = YTM = 12%

Frequency = k =2

Basis = Type of day count basis = 1 (Actual/ Actual)

## 4. Modified Duration

Modified Duration is given by the following formula:

Therefore modified duration for the following formula will be:

0.9760/(1+12%/2) = 0.9207

The Modified Duration may also be obtained directly using the excel formula “MDURATION” with the following input parameters:

Settlement = Issue Date = 10^{th} December 2010

Maturity = Maturity Date = 10^{th} December 2011

Coupon = Coupon Rate = 10%

Yld = YTM = 12%

Frequency = k =2

Basis = Type of day count basis = 1 (Actual/ Actual)

In this post, we saw how price, Macaulay Duration and Modified duration were calculated for the sample instrument using both first principles as well as the EXCEL worksheet formulas. In the next post, we will see how Effective Duration is calculated.

Also see **Asset Liability Management for Board members – 7 posts, 60 minutes.**