**Table of Contents**

Chapter 1: Categories of Bonds

Chapter 2: Pricing of Bonds

Chapter 3: Calculating Yield and Understanding Yield Curve

Chapter 4: Duration of Bonds

Chapter 5: Relationship Between Price, Yield and Duration

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Chapter 4: Duration of Bonds

– Factors that Affect Duration

– Duration of the Two Basic Bond Types

– Zero Coupon Bond

– Vanilla Bond

– Types of Duration

– Macaulay Duration

– Modified Duration

– Effective Duration

– Key-Rate Duration

Duration of a bond is a measure often used to assess the risk involved with the bond instrument. It is the effective duration in years or maturity time, in which the bond price is repaid by the internal cash flows. The greater the duration, the more is the sensitivity to interest rate changes, which in turn means higher risk with the instrument. This measure is very useful in comparing the interest rate risks of various securities.

Duration is many times understood as the % change in the price of the bond for a unit percent change in yield to maturity. The duration can be approximated using the following formula:

Duration of bond = (Bond Price when interest rate increases – Bond Price when interest rate decreases) / (2 x Initial Bond Price x Change in interest rate)

Duration of bond can be visualized as a seesaw with a fulcrum whose position when changed balances the payments’ present values and the bond’s principal payment.

The duration of a bond changes as the coupon payments keep getting made. If we visualize a seesaw with the length of plank equal to the maturity period, fulcrum placed at the duration of bond and money bags representing repayment cash flows placed on the plank all through the length to the maturity time, then every coupon payment is take off the plank as they get paid, causing the fulcrum (duration) to move forward at future point in time (representing the new duration of bond).

The duration value keeps on changing throughout the bond period. However, the value of duration in years, keep decreasing with the progress towards the maturity date. It also increases momentarily on the date of coupon payment. The duration value keeps varying in this fashion till it eventually becomes zero and merges with the maturity of the bond.

Other factors that affect the bond duration are the bond’s yield and coupon rate. Bonds that have high coupon rates, which in turn means high yields, usually have smaller durations than the vice versa case. A higher coupon rate or yield means that the bondholder is receiving the invested money at a faster rate, which is indicated by lower durations and hence, lower perceived risks.

**Zero Coupon Bond**: For a zero coupon bond, duration is the same as its maturity period. For a zero coupon bond, the fulcrum on the seesaw would be placed right under the bond’s future value money bag at the maturity period (right most end of the plank), balancing its load right under. This is because the complete cash flow for a zero coupon bond comes through at the time of maturity.**Vanilla Bond**: For a vanilla bond, the duration of bond is less than the maturity time. If we now again visualize the seesaw, then in the case of vanilla bond, the money bags would be placed all over the plank. Smaller moneybags representing smaller repayments / cash flows before the maturity time (right most end of plank) and a bigger money bag placed at the right most end indicating the last big payment at the end of bond maturity.The fulcrum in this case will be positioned somewhere between the initial and final point of time (beginning and end of bond period). The fulcrum balances the plank at the point in time when the total cash flow till that time equals the bond price.

**Duration of the Two Basic Bond Types**

Calculation of duration can be done in more than one ways, each giving rise to a slightly different type of duration. These types differ in the way they account for the changes in interest rate and redemption features and embedded bond options. Four key types of duration are discussed here:

**Macaulay Duration**- This is the most common way of calculating a bond’s duration and it became popular starting 1970s. This duration derives its name from its creator – Frederick Macaulay. This duration is calculated as the sum of multiplications of cash flow present values and corresponding time in which they are paid, divided by the bond’s total price.

The formula of this duration is:

Macaulay duration = [ ∑(t=1 to n) {(t x C) / (1 + Yi)^n + (n x Par) / (1 + Yi)^n } ] / Bond Price

Where Yi is the required yield, Par is the maturity (par) value, C is cash flow, t is the bond’s maturity time and n is the no. of cash flows.

**Modified Duration**- This is a modified version of Macaulay duration and takes into account the interest rate changes. Changes in interest rates affect duration as the yield gets affected each time the interest rate varies.
In regular bonds, the interest rates and bond price move in opposite directions. An approximate unit percent change in yield shares an inverse relationship with modified duration. This type of duration is very well suited for the purpose of gauging a particular bond’s volatility.

The formula for modified duration is as follows:

Modified duration = Macaulay Duration / (1 + (yield to maturity / no. of coupon periods per year))

**Effective Duration**- In above formula, cash flows are considered to be constant. However, in case of bonds with redemption features and embedded bond options, cash flows also change as the rate of interest changes. For such bonds, the duration calculation has to take into account this variability. Effective duration does just that. It uses binomial trees to estimate the option adjusted spread. These calculations can get a bit complex.

**Key-Rate Duration**- Key rate duration calculates the eleven ‘key’ maturities’ spot durations. These maturities are positioned at 3 months, 1 year, 2 years, 3 years, 5 years, 7 years, 10 years, 15 years, 20 years, 25 years and 30 years on the spot rate curve.
The key rate duration allows portfolio duration to be calculated for a change equivalent to 1 basis point in rate of interest, while keeping the yield for maturities other than those on the 11 points constant. This duration is most commonly used for portfolios, which comprise of securities with differing maturity periods. It should be noted that addition of the various key rate durations along the spot rate curve is the same as calculating effective duration.

The formula for key rate duration is:

{Bond price after yield decrease of 1% – Bond price after yield increase of 1%} / {2 x Initial bond price x 1%}