Duration & Convexity

Understanding Duration

Compare the four bonds (all semi-annual coupon payers) Bond

A

B

C

Z

Coupon

6

6

10

0

Maturity(yrs)

10

20

20

20

Price @ 6% YTM

100

100

146.2295

30.6557

Price @ 5% YTM

107.7946 112.5514

162.7569

37.2431

Change

7.7946%

12.5514% 11.3024%

21.4883%

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Higher the maturity of the bond, greater is the change in price in response to change in yield. Price of a zero-coupon bond is more sensitive to yield changes than a coupon bond. A bond with higher coupon is less sensitive than a bond with lower coupon. [As bond with higher coupon gets the cash flows earlier than the bond with lower coupon, assuming both bonds have same maturity period] We have established the following: – Giv en equ al sens itiv e to – Giv en equ al sens itiv e to

coup on s, lo nger m atu rity bon ds are m ore yield chan ges. matu rities, h igh er cou pon bonds ar e less yield chan ges.

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Expectation of Portfolio Manager based on interest rate movement – – In a declining interest rate scenario portfolio manager will prefer • Longer Maturity bonds • Lower coupon bonds

– But if bonds differ by both maturity and coupon – and the impacts of the two on price sensitivity are in opposite directions, as is the case with bonds A and C – a judgment about relative price sensitivities is more difficult. This is where Duration comes in.

Duration • • • •

Duration measures the average maturity of a bond’s total cash flows – coupon as well as principal. The average is weighted by the present value dollars paid in each period. Thus, duration encompasses all the parameters of a bond – maturity, coupon and yield. Duration is the tool we are looking for: the one measure of bond price sensitivity to yield changes.

Duration Parameters • • • •

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Duration rises with maturity Higher coupon means lower Duration [Duration of a zero-coupon bond is equal to its maturity as it has only one cash flow] Duration is inversely related to yield Higher duration bond is more sensitive to interest rate movements.

A sizable increase/decrease in yield is necessary for considering duration as a measure of price sensitivity to yield changes. Small changes in yield hardly change the bond’s duration.

Duration… •

Absolute Price Change ∆P Dur --- = - ------- X P ∆y (1+y) •

Percentage Price Change ∆P /P Dur --= - ------∆y (1+y) •

Duration

- Duration = ∆P (1+y) / (P * ∆y)

For semi-annual coupon bonds divide y by 2, for quarterly coupon bonds divide y by 4.

Macaulay Duration • Macaulay duration is calculated by adding the results of multiplying the present value of each cash flow by the time it is received and dividing by the total price of the security.

Modified Duration … • - Duration = ∆P (1+y) / (P * ∆y) • •

The presence of the 1+y term (or 1+y/2 for semi-annual bonds) in the denominators makes comparison difficult To address this we have Modified Duration

Mod. Duration = Duration/(1+y) [or Mac.Duration/(1+y)] ∆P/P = − Mod.Dur x ∆y Mod. Duration = - ∆P/(P x ∆y) •

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Armed with the modified durations of all the bonds currently in the portfolio or with the potential to be there, a trader can easily and quickly estimate the impacts of expected yield changes on bond prices. For example, a half-percent increase in yield to maturity on the bond (duration=5½ years) results in a 2.75% drop in its price. If a bond’s duration is 12 years, a ten basis point drop in yield produces a 1.2% increase in price

Portfolio Duration • •

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Duration of portfolio is calculated by assigning weights to bonds. Using portfolio duration as a measure of price sensitivity is only relevant when the assumed interest rate change is the same (or nearly the same) for all the bonds in the portfolio. If, for Bond1 (maturity 4 yrs) the four-year yield rises by half a percent and for Bond2 (maturity 10 yrs) the ten-year by one and a half percent, even though the average increase is one percent, the resulting price change of the portfolio will be quite different. In other words, portfolio duration as a measure of price sensitivity is only accurate for a parallel shift in the yield curve.

Zero Duration •

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Duration reflects the degree of sensitivity of a bond – or a portfolio of bonds – to interest rate changes. A zero duration means no sensitivity. Why is there no sensitivity? Because the effect of a change in interest rates on the bond owned should be matched – in the opposite direction – by its effect on the bond sold short. Hedging, therefore, represents a situation where a portfolio is insulated from interest rate shifts in the market. Note again, though, that this assumes a parallel shift in the yield curve. If the yields on the four and ten-year notes move differently, the hedge will not remove the interest rate risk. The ten-year’s dollar duration is nearly double that of the four-year. Therefore, only about half of a ten-year needs to be sold for every one four-year held to achieve zero duration.

Problem with Duration •

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When interest rates fall, Duration understates positive effect on bond’s price. – For e.g. if interest rates fall by 0.5%, bond price may actually rise by $2.053, but duration may predict a price rise of $2.033. Similarly when interest rates rise, Duration overstates the drop in bond’s price. – For e.g. if interest rates rise by 0.5%, bond price may actually fall by $1.99, but duration may predict a price fall of $2.01.

Convexity • • •

As yield-to-maturity declines, the rise in the bond’s price becomes larger and larger. Conversely, as the yield increases, each successive fall in price is smaller than the one before. This relationship between yield and price is called its convexity. It exists because the shape of the price-yield curve is convex [and not linear].

A better explanation on next slide …

Convexity … Why duration is inaccurate in measuring the effect of yield changes on price? • Duration and YTM are inversely related. • As yields rise, duration falls. Thus, the next yield increase has less of a negative effect on price since duration is lower. • As yields fall, duration rises. Thus, the next yield decline has more of a positive effect on price since duration is higher. • Using the duration method to calculate the new price of a bond following a yield change – negative or positive – will, therefore, be inaccurate (and always too low) since the bond’s original duration is used in the calculation and the duration itself changes.

Convexity … •

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In short, duration is only an estimate of the effect of yield-to-maturity changes on bond price, as it effectively assumes the price-yield relationship is linear (mathematically, it has a constant slope). Since, in fact, it is not linear (the slope declines as yield rises), using duration will over or understate the effect. Convexity is a measure of the difference between what this estimate of price change is, and the actual price change.

Price

Duration

Yield

Convexity

The fact that the positive effects of successive yield declines accelerate and the negative effects of yield increases decelerate is obviously an attractive feature of bonds that display convexity. The more convex a bond, the more attractive.

Convexity … • •

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During times of volatility in rates, investors prefer bonds with higher convexity. Thus, when an increase in volatility is expected, for example, a less convex bond should be sold along with the purchase of the more convex security in a combination that produces a zero duration. Thus, if volatility is feared, the convexity characteristic makes more sense. In other words, the more risk averse the investor, combined with a fear of volatility, the more preferable convexity is to yield. Conversely, investors with greater tolerance for risk or less anxiousness about future volatility will opt for higher yield and not need the convexity. A more convex portfolio, in a sense, may be considered a defensive market position.

Negative Convexity • • • •

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For a non-callable bond, there is an inverse relationship between duration and yield Negative convexity means that as market yields decrease, duration decreases as well. Since this is an unfavorable characteristic of a bond, investors demand a higher yield. Negatively convex bonds (such as callable corporates and mortgages) thus yield more than otherwise equivalent non-callable, or positively convex, bonds. Investors expecting stability in yields are attracted to such bonds.

Determinants of Convexity •

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Duration: convexity increases with duration, since a higher duration means more of the bond’s cash flows occur later in time, which enhances the “curvature” of the yield-price relationship Maturity: unlike duration, whose increase will decelerate as maturity lengthens, convexity tends to accelerate (mathematically, this is because convexity is related to the second derivative – the “square” of time) Coupon: a higher coupon will reduce convexity, as the importance of earlier cash flows to the bond’s value increases Dispersion of cash flows: for the same duration, a security whose cash flows are more dispersed around a mean will have greater convexity since the addition to convexity supplied by the later cash flows more than offsets the subtraction from convexity due to the earlier cash flows Yield: a higher yield will also reduce convexity, since it shifts the relative weightings to earlier cash flows Any call features attached to the bond.