**Duration & Convexity – Full Understanding**

The wide impact that interest rate changes have on business performance, the fact that all market participants are, to a higher or lesser degree, exposed to interest rate risk, as well as high volatility in interest rates during the recent years, make interest rate risk one of the most significant risks. Therefore, it is of the utmost importance to manage this kind of risk adequately. It is difficult to completely neutralize interest rate risk, however, in regard to the great impact that interest rate changes have on business performance, it is necessary to reduce it to a minimum.

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In order to manage it effectively, exposure to interest rate risk must be identified and measured. __Two basic methods to measure Interest rate risk are Duration and Convexity.__

**Duration:**

- It is used to measure interest rate risk and it is less computationally intensive.
- Duration is the weighted average of the present value of the bond’s payments.
- The longer the duration, the longer is the average maturity, and, therefore, the greater the sensitivity to interest rate changes.
- Mathematically, duration is the 1st derivative of the price-yield curve, which is a line tangent to the curve at the current price-yield point.
- Although the effective duration is measured in years, it is more useful to interpret duration as a means of comparing the interest rate risks of different securities. Securities with the same duration have the same interest rate risk exposure. For instance, since zero-coupon bonds only pay the face value at maturity, the duration is equal to its maturity.
- It also follows that any bond of a certain duration will have an interest rate sensitivity equal to a zero-coupon bond with a maturity equal to the bond’s duration.

Duration is also often interpreted as the percentage change in a bond’s price for a small change in its yield to maturity (YTM). It should not be surprising that there is a relationship between the change in bond price and the change in duration when the yield changes, since both the bond and duration, depend on the present values of the bond’s cash flows.

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**The formula for Duration is as follows:**

- C = Coupon payment per period
- M= Face or Par value
- r =Effective periodic rate of interest
- n = Number of periods to maturity

**Duration has several simple properties:**

- Duration is proportional to the maturity of the bond, since the principal repayment is the largest cash flow of the bond and it is received at maturity;
- Duration is inversely related to the coupon rate, since there will be a larger difference between the present values for the earlier payments over the lesser value for the principal repayment;
- Duration decreases with increasing payment frequency, since half of the present value of the cash flows is received earlier than with less frequent payments, which is why coupon bonds always have a shorter duration than zeros with the same maturity.

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**Example:** a bond with annual coupon payments. Let us assume that company Reliance Ltd has issued a bond having face value of $100,000 and maturing in 4 years. The prevailing market rate of interest is 10%. Calculate the bond duration for the following annual coupon rate: (a) 8% (b) 6% (c) 4%

**Given, M = $100,000, n = 4, r = 10%**

**Calculation for Coupon Rate of 8%**

Coupon payment (C) = 8% * $100,000 = $8,000

Bond Price will be 88,196.16

Numerator of the formula will be 311,732.81

Duration = 311,732.81/ 88,196.16 = 3.53 years

**Calculation for Coupon Rate of 6%**

Coupon payment (C) = 6% * $100,000 = $6,000

Bond Price will be 83,222.46

Numerator of the formula will be 302,100.95

Duration = 311,732.81/ 88,196.16 = 3.63 years

**Calculation for Coupon Rate of 4%**

Coupon payment = 4% * $100,000 = $4,000

Bond Price will be 78,248.75

Numerator of the formula will be 292,469.09

Duration = 311,732.81/ 88,196.16 = 3.74 years

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From the example, it can be seen that the duration of a bond increases with the decrease in coupon rate. In case investors are seeking benefit from fall in interest rate, the investors will intend to buy bonds with a longer duration which is possible in case of bonds with lower coupon payment and long maturity. On the other hand, investors who want to avoid the volatility in interest rate, the investors will be required to invest in bonds that have a lower duration or short maturity and higher coupon payment.

Duration is only an approximation of the change in bond price. For small changes in yield, it is very accurate, but for larger changes in yield, it always underestimates the resulting bond prices for non-callable, option-free bonds. This is because duration is a tangent line to the price-yield curve at the calculated point, and the difference between the duration tangent line and the price-yield curve increases as the yield moves farther away in either direction from the point of tangency.

**Convexity** is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation for the duration and the 2nd derivative to the equation for the price-yield function. Convexity is always positive for vanilla bonds. Furthermore, the price-yield curve flattens out at higher interest rates, so convexity is usually greater on the upside than on the downside, so the absolute change in price for a given change in yield will be slightly greater when yields decline rather than increase. Consequently, bonds with higher convexity will have greater capital gains for a given decrease in yields than the corresponding capital losses that would occur when yields increase by the same amount.

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__S ome additional properties of convexity include the following:__

- Convexity increases as yield to maturity decreases, and vice versa. Convexity decreases at higher yields because the price-yield curve flattens at higher yields, so modified duration is more accurate, requiring smaller convexity adjustments. This is also the reason why convexity is more positive on the upside than on the downside.
- Among bonds with the same YTM and term length, lower coupon bonds have a higher convexity, with zero-coupon bonds having the highest convexity. This result because lower coupons or no coupons have the highest interest rate volatility, so modified duration requires a larger convexity adjustment to reflect the higher change in price for a given change in interest rates.

**The formula for Convexity is as follows:**

- P = Bond price
- Y = yield to maturity
- T = Maturity in years
- CFt = Cash flow at time t

In the below graph Bond A is more convex than Bond B even though they both have the same duration and hence Bond A is less affected by interest rate changes.

Convexity is a risk management tool used to define how risky a bond is as more the convexity of the bond; more is its price sensitivity to interest rate movements. A bond with a higher convexity has larger price change when the interest rate drops than a bond with lower convexity. Hence when two similar bonds are evaluated for investment with similar yield and duration the one with higher convexity is preferred in a stable or falling interest rate scenarios as price change is larger. In a falling interest rate scenario again, a higher convexity would be better as the price loss for an increase in interest rates would be smaller.

Convexity can be positive or negative. A bond has positive convexity if the yield and the duration of the bond increase or decrease together, i.e. they have a positive correlation. The yield curve for this typically moves upward. This typical is for a bond which does not have a call option or a prepayment option. Bonds have negative convexity when the yield increases the duration decreases i.e. there is a negative correlation between yield and duration and the yield curve moves downward.

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**Conclusion:**

- Duration and convexity are two metrics used to help investors understand how the price of a bond will be affected by changes in interest rates. How a bond’s price responds to changes in interest rates is measured by its duration, and can help investors understand the implications for a bond’s price should interest rates change. The change in a bond’s duration for a given change in yields can be measured by its convexity.
- If rates are expected in increase, consider bonds with shorter durations. These bonds will be less sensitive to a rise in yields and will fall in price less than bonds with higher durations.
- If rates are expected to decline, consider bonds with higher durations. As yields decline and bond prices move up, higher duration bonds stand to gain more than their lower duration counterparts.

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