## What is the Duration of a Bond?

The duration of a bond expresses the sensitivity of the bond price to changes in the interest rate. In other words, the bond duration measures the movement in the price of the bond for every 1% change in the interest rate.

The unit of bond duration is expressed in years. Also, the price of the bond and the interest rates are inversely related. Therefore, if a bond has a duration of 5 years, it signifies that fo^{1}r every 1% increase in the interest rate, the price of the bond will fall by 5% and vice-a-versa. The greater is the bond duration, the greater will be the amplification in the movement of bond price for every single unit of change of the interest rates.

## Bond Duration Vs Bond Maturity

The maturity of the bond simply states the period by which the last cash flow arising from the bond will be received.

Duration of a bond, on the other hand, is a slightly technical and advanced spin on bond maturity. It is a weighted average period of time until all the cash flows from the bond are received. Weights are given to the present value of each cash flow (coupon payment) at the applicable interest rate for the life of the bond. In other words, it conveys the time period for which the investor must hold position to recover the present value of the bonds.

The duration of a bond will always be less than its maturity period. This is because when the present value of coupons is incorporated into the equation, it reduces the recovery period of the cash flows.

## How to Find Duration of a Bond?

### Macaulay Duration

This method was devised by an economist named Frederick Macaulay. He approached the formula with a very simple rationale. The present value of all cash flows is compared to the market price of the bond. The higher is the resultant number, more is the remaining life of the bond and therefore more are the number of payments awaiting to be received. As the bond approaches maturity, the gap between the duration and maturity reduces but never becomes zero.

Where, = present value of the coupon payments and final redemption amount discounted at YTM rate.

MP of the bond = Current Market Price of the Bond

### Modified Duration

If the Macaulay Duration states the time period within which the PV of the bond shall be realized, the modified duration expresses the sensitivity of bond price to interest rates. It is expressed in percentage. Modified duration is derived from the Macaulay duration itself making adjustments for the YTM.

Modified Duration = Macaulay Duration/(1+YTM)

## Calculation of Duration of a Bond

Let us calculate the duration of a 5-year bond, Face value = $100 traded at par, Coupon Rate= 9% p.a., YTM= 6%

Year | Cash Flow @ 9% (A) | Product (Year * CF) | Discount Factor @ 6% (B) | PV of CF |

1 | 9 | 9 | 0.94 | 8.49 |

2 | 9 | 18 | 0.89 | 16.02 |

3 | 9 | 27 | 0.84 | 22.67 |

4 | 9 | 36 | 0.79 | 28.52 |

5 | 109 | 545 | 0.75 | 407.26 |

Sum | 482.95 |

Therefore, the Macaulay bond duration = 482.95/100 = 4.82 years

And Modified Duration= 4.82/ (1+6%) = 4.55%

The above calculations roughly convey that a bondholder needs to be invested for 4.82 years to recover the cost of the bond. Also, for every 1% movement in interest rates bond price will move by 4.55% in the opposite direction.

## Duration of a Bond Portfolio

### What is a Bond Portfolio?

A bond portfolio is nothing but a basket of various bonds and fixed income securities. They do not enjoy as much limelight as equity stock portfolios. However, the fund manager highly counts on bonds to provide their fund with the necessary hedge and stability. Bond portfolios are also a go-to investment type for various conservative and risk-averse investors such as managers of pension and retirement funds.

The duration of a bond portfolio enables a fund manager to study the behavior of various bonds in conjunction with one another. Often, the income arising from the bond universe can be exploited when bonds are pooled together. This enables the harnessing of gains more effectively arising from their varying coupon rates, maturity and market values.

### Computation of Bond Portfolio Duration

Duration of Bond Portfolio is the weighted average of the duration of bonds comprising the portfolio.

= w1D1 + w2D2+ …wnDn

W= Weights (Market Value of Bond/ Market Value of Portfolio)

Di= Duration of Bond i

n= Number of Bonds in a portfolio

Let us go through the following example.

Consider the following Bond Portfolio. A YTM rate of 10% has been assumed.

Bond | Coupon Rate | Maturity | Market Price ($) | Units Held | Market Value ($) | Weights |

A | 10% | 5 years | 100 | 1,75,000 | 17500000 | 0.51 |

B | 15% | 10 years | 130 | 50,000 | 6500000 | 0.19 |

C | 8% | 30 years | 81 | 1,25,000 | 10125000 | 0.30 |

34125000 |

The duration of the bond has been computed by the Macaulay Duration formula. The so computed duration is then multiplied with the market value weights. The resultant sum product: 6.49 is the duration of a bond portfolio.

Bonds | Weights | Computed Duration | Product |

A | 0.51 | 4.16 | 2.13 |

B | 0.19 | 6.28 | 1.20 |

C | 0.30 | 10.64 | 3.16 |

6.49 |

Thus, owing to Bond A & B, the investor can recoup the present value of bonds A, B & C in 6.49 years. This is a valuable saving in time of (30-6.49) 23.51 years. As is evident, holding bonds with shorter maturity periods (A &B) significantly reduce the holding period requirement of bonds with a longer maturity (C).

Not to mention, the liquidity benefit of clubbing bonds with varying maturity. For example, in this case, the fund manager can recover the costs of a 30 year bond within 6.49 years. He is at liberty at any time thereafter to square off his position. He can thus enter into seemingly more profitable investments without losing the initial capital invested.^{2}

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