As shown in Exhibit 16.1, the graph of bond prices versus yield is convex. For market participants who own a bond, collect the coupon and hold it to maturity, market volatility is irrelevant; principal and interest are received according to a predetermined schedule.
But participants who buy and sell bonds before maturity are exposed to many risks, most importantly changes in interest rates. When interest rates increase, the values of existing bonds fall since new issues pay a higher yield. Likewise, when interest rates decrease, the values of existing bonds rise since new issues pay a lower yield. This is the fundamental concept of bond market volatility: changes in bond prices are inverse to changes in interest rates. Fluctuating interest rates are part of a country's monetary policy and bond market volatility is a response to expected monetary policy and economic changes.
Duration and convexity are factor sensitivities that describe exposure to parallel shifts in the spot curve. They can be applied to individual fixed income instruments or to entire fixed income portfolios. The idea behind duration is simple. Suppose a portfolio has a duration of 3 years. Then that portfolio's value will decline about 3% for each 1% increase in interest rates - or rise about 3% for each 1% decrease in interest rates. Such a portfolio is less risky than one which has a 10-year duration. That portfolio is going to decline in value about 10% for each 1% rise in interest rates. Convexity provides additional risk information.
If we fit a tangent line to the curve in Exhibit 16.1, it will capture the direction and magnitude of the portfolio's sensitivity to interest rates. For small changes in interest rates, the line and the curve almost overlap. Duration is defined to be the slope of that tangent line, multiplied by negative one.
Tangent lines are the province of calculus, so we turn to calculus for the formal definition. Duration is a weighted partial derivative:
This leads to the approximation
For example, suppose a portfolio has a duration of 5 years. That portfolio will appreciate about 5% for each 1% decline in rates. It will depreciated about 5% for each 1% rise in rates. It is as simple as that.
Suppose a portfolio has a duration of —2 years. The portfolio's value will rise about 2% for every 1% rise in rates. It will decline about 2% for each 1% decline in rates.
Typically, a bond's duration will be positive. However, instruments such as interest only mortgage-backed securities have negative durations. You can also achieve a negative duration by shorting fixed income instruments or paying fixed for floating on an interest rate swap. Inverse floaters tend to have large positive durations. Their values change significantly for small changes in rates. Highly leveraged fixed-income portfolios tend to have very large (positive or negative) durations.
For portfolios whose cash flows are all fixed (e.g., a portfolio of non-callable bonds) there is a particularly simple way to calculate duration. For such portfolios, duration is just the average maturity of the of the cash flows. Specifically, assume a portfolio has fixed cash flows ci, each occurring at some time Ti years from time 0. Let 0pv(ci) denote the present value at time 0 of the cash flow ci, then the duration is
Macaulay Duration =
When duration is calculated in this way, it is called Macaulay duration. The Macaulay formula for duration is correct only if interest rates are continuously compounded.
Take, for example, a 5-year zero-coupon note. Because it pays no coupon, its average maturity is precisely 5 years. Hence, based on the Macaulay formula for duration, the bond's duration will be 5 years. This means that a 5-year zero will appreciate about 5% in value for each 1% decline in continuously compounded interest rates based on approximation.
In formula, all present values should be calculated using the spot interest rate for the maturity of the cash flow it is discounting.In practice, people often calculate all present values with a non-continuously compounded yield to maturity y for the entire portfolio. If this is done, formula must be modified slightly. It becomes
where m is the frequency of compounding for the yield to maturity. For example, if the yield to maturity is compounded quarterly, m = 4. This formula is called modified duration.
For portfolios containing instruments that do not pay fixed cash flows, such as callable bonds, mortgage-backed securities or interest rate caps, the Macaulay or modified formulas for duration will not work. For these portfolios, other means must be employed for calculating duration.
Now let us consider convexity. If duration summarized the most significant piece of information about a bond or a portfolio's sensitivity to interest rates, convexity summarizes the second-most significant piece of information. Duration captured the fact that the graph in Exhibit 16.1 was downward sloping. It did not, however, capture its upward curvature. Convexity describes curvature.
Convexity is defined as a weighted second partial derivative
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