Modified Duration of Bonds

There is an inverse relationship between price and yield. What should be the change in price with a change in yield? It can be answered with the help of duration concept and especially with modified duration. Modified duration is a modified version of the Macaulay model that accounts for changing interest rates. Modified formula shows how much the duration changes for each percentage change in yield. There is an inverse relationship between modified duration and an approximate 1% change in yield. The calculation of the same is as follows:

Where ‘f’ is frequency of payment of coupon

Illustration:

Consider a bond with a YTM of 12% and duration is 5 years. If the interest rate increases by 50 basis points , change in the price of the bond will be..

Modified duration of the bond is 5/1.12 = 4.46 years

Change is price can be calculated as follows:

%ΔPrice = -Dmod ×%Δyield

                  = -4.46 ×0.5 = -2.23%

Hence price will decline by 2.23% for a given change in yield.

As duration is an easy approach to calculate the price change but it also suffers from its own limitations. As we have already understood that for the same change is yield either side, price change is not same but duration assumes that there is a linear movement between price and yield. Hence duration is useful for smaller changes in interest rates but should not be used in isolation when larger changes are predicted. The error caused by duration can be eliminated using the concept of ‘convexity’. Learning convexity is out the scope for the CFP aspirants. So will discuss the same on other time.

I hope you have enjoyed learning bonds.

Thank You.

Prashant V Shah

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Bond Duration Analysis

What is Duration of the bond?

The very first thing to remember is, duration of the bond is not maturity of the bond as both are different.

• Duration is defined as a weighted average of the maturities of the individual payments
• It is a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows
• It is an important measure for investors to consider, as bonds with higher durations carry more risk and have higher price volatility than bonds with lower durations
• Duration is also a point where the investor faces no interest rate risk

Terminology:

Zero coupon Bond: Issued at discount to face value and redeemed at par. Does not pay any interest

Vanilla bond/Straight: A normal interest bearing bond

 

Macaulay Duration:
The formula usually used to calculate a bond’s basic duration is the Macaulay duration

Where,

n = number of cash flows
t = time to maturity
C = cash flow
r = required yield (YTM)
M = maturity (par) value

Alternate Equation:

Lets understand the same with an illustration:

Consider a 12.5% bond with annual coupons, redeemable after 5 years at a premium of 5%. If the current interest rate is 15%, calculate duration of the bond.

Hence Duration of the bond = 375.11/94.11 = 3.99 Years

Remember duration of a zero coupon bond is equal to its maturity.

Duration and Bond Characteristics:

 

•   The lower the coupon, the higher the duration.

•   The higher the coupon, the lower the duration.

•   The longer the term to maturity, the higher the duration.

•   The shorter the term to maturity, the lower the duration.

•   The smaller the duration, the smaller the price volatility of the bond.

•   The greater the duration, the greater the price volatility of the bond.