The link between duration and volatility of prices is clear because of the first derivative that we used in obtaining the duration formula. There are two alternative measures that are worth investigating. The first is called modified duration. This is derived by dividing the duration measure by one plus the current market yield. The elasticity measure will be very close in practice to the modified duration measure.
The closeness is dependent upon the size of the movement in interest rates. Hence, we see the link between duration, modified duration, and elasticity. Now lets consider another example. We will look at bonds of different maturity. For ease of calculation, we assume annual compounding.
Note that the price elasticity measure is very close to the modified duration measure. Note also that the closeness of this measure depends upon the size of the interest rate move. We would like to use the modified duration to approximate bond price movements for a given change in interest rates.
The approximation will only be accurate for small changes in the interest rate. This is because the bond price is convex in the yields. We have seen this convexity when we looked at the plot of the bond price against various yields to maturity.
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Below is a figure that illustrates the error. Also, if stated interest rates are compounded semi-annually, then the same procedure is used to calculate duration in half years. The duration in half years is converted to years by dividing by 2 and then transformed into modified duration by dividing by the effective periodic semi-annual rate.
One might wonder why we divide by the semi-annual rate rather than 1 the annual percentage rate or 2 the true annualized rate. The answer, unfortunately, has to do with convention. Most top investment houses calculate the modified duration by dividing by the semi-annual rate - even if coupons are paid quarterly or monthly.
In most of my examples, I have annual compounding and don't to worry about this convention. Below is a numerical example using annual rates. Bond A has a maturity of 5 years and Bond B has a maturity of 10 years. The reason for this error is the bond price convexity. In large interest rate moves, this convexity is increasingly important. Convexity is measure of the way the slope of the bond price -- yield curve is changing. Duration gives us a linear approximation. But if the curve is convex this linear approximation may not be very high quality.
If this is not familiar, you can review your undergraduate calculus text. It should have a section on Taylor series. The Taylor Series is a way to use derivatives to approximate a function with a polynomial. Take f x to be the pricing function for the bond. The price is a function of the yield x. Consider the change in the yield to be the h variable. I cut off the approximation after the second term. The first order part, f' x , is associated with the duration of the bond and the second order part f'' x with the convexity of the bond.
There will also be a residual part because the Taylor series is only an approximation. He provides tables in which he measures how well each of these parts accounts for price changes for bonds of various maturities. Note that the duration measure that he uses is modified duration. The modified duration is However, you believe rates will move up over the next year.
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You don't want to sell the bonds - but you would like to protect yourself from this potential loss over the next year. This is where we would hedge. The goal of hedging is to provide a positive cash flow in the bad state. Suppose you plan to buy a house in the next two years. You know that mortgage rates are at 15 year lows and you want to lock in a rate. If you do nothing and rates move up, you will be faced with much more expensive financing.
You would like to protect yourself against this potentially higher cost. By hedging, we can lock the mortgage rate in today.
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In the second example, the hedge will deliver a payoff if mortgage rates go up. Even though you finance your mortgage at the higher rate, you are compensated by a positive payoff from the hedge. We usually use futures or options to execute our hedges. In our first example, there is a year bond contract trading on the CBOT. The bond has a duration of approximately 8.
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How did I get that number? Of course, if the rates move the other way, the gain on what we hold is wiped out by the loss in the futures. This is our first example of hedging. In actual practice, investors care about matching convexity and sometimes skewness and kurtosis. The techniques used to find the value of bonds are used throughout finance. We can use these same techniques to value an investment project or an acquisition being considered by a firm. For an investment project, it will likely have both costs and revenues.
Net present value is the value of the investment to you today -- net of costs, i. Consider the following scenario. You sit on the board of a firm that is considering an acquisition. The plan is to buy the firm, rationalize their operation and inject your firm's expertise in their production operations, and then sell after only two years.
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Ignore for now tax considerations. The net present value of the project is:. One point of fundamental importance is that any positive net present value project can be self- financed , i. Moreover, once carried out, the positive net present value is the gain measured in dollars today. To see it, we examine a strategy for the previous example. You purchase the firm in question. You pay the first year interest with the revenue from the project.
Remember, in the transactions we have executed, none of our capital has been touched. It should become clear that:. A forward interest rate is a marginal return for investing your money for an extra period, i. The annualized forward rate between period 1 and 2 can be written:. We can also calculate multiperiod annualized forward rates.
The forward rate between years 1 and 3 is:.
- Bond duration - Wikipedia.
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Finally, if the interest rates are compounded semiannually, then the annualized forward rate between years 3 and 4 is. There is an important alternative interpretation to the forward rate. Consider the following strategy. At the end of the second year, we realize the revenue from cashing in the two year bonds.