Citation: Ghimire, Binam, Gautam, Rishi, Karki, Dipesh and Sharma ...
1 Duration and Convexity by Binam Ghimire. Learning Objectives Duration of a bond, how to compute...
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Transcript of 1 Duration and Convexity by Binam Ghimire. Learning Objectives Duration of a bond, how to compute...
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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