duration and convexity by binam ghimire

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Duration and Convexityby Binam Ghimire

Learning Objectives Duration of a bond, how to compute it Modified duration and the relationship between a bond’s

modified duration and its volatility Convexity for a bond, and computation Under what conditions is it necessary to consider both

modified duration and convexity when estimating a bond’s price volatility?

Excel computation

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Duration Developed by Frederick Macaulay, 1938 It combines the properties of maturity and

coupon

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Duration Example Two 20 – year bonds, one with an 8% coupon and

the other with a 15% coupon, do not have identical life economic times. An investor will recover the original purchase price much sooner with the 15% coupon bond.

Therefore a measure is needed that accounts for the entire pattern (both size and timing) of the cashflows over the life of the bond – the effective maturity of the bond. Such a concept is called Duration

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Duration

Where: t = time period in which the coupon or principal payment

occursCt = interest or principal payment that occurs in period t i = yield to maturity on the bond

price

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Duration Duration is the average number of years an

investor waits to get the money back. Duration is the weighted average, on a present

value basis, of the time to full recovery of the principal and interest payment on a bond.

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Duration Calculation of Duration depends on 3 factors

The Coupon PaymentsTime to MaturityThe YTM

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Duration The Coupon of Payments

Coupon is ………….related to duration. This is logical because higher coupons lead to …………….. recovery of the bond’s value resulting in a ………… duration, relative to lower coupons

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Duration The Coupon of Payments

Coupon is inversely related to duration. This is logical because higher coupons lead to quicker recovery of the bond’s value resulting in a shorter duration, relative to lower coupons

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Duration Time to Maturity

Duration ………………. with time to maturity but a decreasing rate

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Duration Time to Maturity

Duration expands with time to maturity but a decreasing rate

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Duration Time to Maturity

Note that for all coupon paying bonds, duration is always less than maturity.

For a zero coupon bond, duration is equal to maturity

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Duration YTM

YTM is inversely related to duration

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Characteristics of Duration Duration of a bond with coupons is always less

than its term to maturity because duration gives weight to these interim paymentsA zero-coupon bond’s duration equals

its maturity There is an inverse relation between duration

and coupon

Characteristics of Duration There is a positive relation between term to

maturity and duration, but duration increases at a decreasing rate with maturity

There is an inverse relation between YTM and duration

Sinking funds and call provisions can have a dramatic effect on a bond’s duration

Modified Duration and Bond Price VolatilityAn adjusted measure of duration can be used to

approximate the price volatility of a bond

mYTM1

durationMacaulay duration modified

Where:

m = number of payments a year

YTM = nominal YTM

Duration and Bond Price Volatility Bond price movements will vary proportionally

with modified duration for small changes in yields

An estimate of the percentage change in bond prices equals the change in yield time modified duration

iDPP

mod100Where:P = change in price for the bondP = beginning price for the bondDmod = the modified duration of the bondi = yield change in basis points divided by 100

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Convexity

The equation above generally provides an approximate change in price for very small changes in required yield. However, as changes become larger, the approximation becomes poorer.

Modified duration merely produces symmetric percentage price change estimates using equation when, in actuality, the price-yield relationship is not linear but curvilinear. (pls see price-yield graph already covered)

Hence, Convexity is the term used to refer to the degree to which duration changes as the YTM changes.

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Convexity

Convexity is largest for low coupon bonds, long-maturity bonds, and low YTM.

Convexity

The convexity is the measure of the curvature and is the second derivative of price with resect to yield (d2P/di2) divided by price

Convexity is the percentage change in dP/di for a given change in yield

PdiPd2

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Convexity

Convexity

Inverse relationship between coupon and convexity

Direct relationship between maturity and convexity

Inverse relationship between yield and convexity