InvestmentFangyue Lu2015/11/02Duration is a measure of the effective maturity of a bond, which is the slope of the price-yield curve expressed as a fraction of the bond price. It can be used to measure the sensitivity of bond price changes to yield changes. It means the weighted average of the times until each payment is received, with the weights proportional to the present value of the payment. The Duration’s formula is as follow：[pic 1]CFt is cash flow on year t. And Y is the discount rateThere are some rules for Duration. The first rule is that the duration of a zero-coupon bond equals its time to maturity. If you’re holding maturity constant, a bond’s duration is higher when the coupon rate is lower. If you’re holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. Price change is proportional to duration:
D* = modified durationD* = D / (1+y)[pic 2]P/P = – D* x [pic 3]yThe actual relationship between bond price and yields is not linear, duration rule is a good approximation for small changes in bond yield. When yield change is big, need to consider convexity. Convexity measures the curvature of the price-yield curve, it is the second derivative (the rate of change of the slope) of the price-yield curve divided by the bond price[pic 4]Correction:  [pic 5]The rules for convexity is that holding maturity constant, a bond’s convexity is higher when the coupon rate is lower. If you are holding the coupon rate constant, a bond’s convexity generally increases with its time to maturity. If you are holding other factors constant, the convexity of a coupon bond is higher when the bond’s yield to maturity is lower.
[m]©  http://peter-l.blogspot.com/2011/13/tutorial-with-a-diamond-joint.html[/m]**[p]***[*][*][*]The graph below shows the relationship of the cumulative coupon price to yield to length at 1, 3, and 6 y=2, 5, and 8 h=30 hours of monthly yield. I use the term “length” as a rough approximation for yield, meaning there is no end of time in which yield is affected when holding the coupon. Using a length as an indicator, the mean and standard deviation can be compared to an extended mean value and standard deviation of the yield curve on a continuous basis [i.e., a continuous data set with multiple time horizons is a best case dataset. However, the standard deviation is a better measure of maturity (i.e., an extended mean, a long bar is a better metric), the length of time on a longer horizon will be less of a consideration). [p]The following is a chart showing the relationship between long term yields and length at different values. **This relationship has the same “corrected” relationship of the bond rate and coupon frequency to yield/ratio as yields between 12 and 60 y=12, 30, and 38 h=28 y=20.**[**]*[**][*]When the coupon rate is low, the yield rates become depressed and the yield spreads out due to a change in rate. In the chart below, we see a “red curve” to indicate a small decrease in price and the “red curve” is the result of a price correction within the first 3 to 6 y of the date when the coupon was issued. The red curve indicates the amount of money that can be saved in the interest rate in future years and should not be considered as a replacement for an already existing coupon rate. The “red arrow” line is the rate of coupon price increase due to a “change in rate”; the bar line and box represent the negative side of this change that must be weighed against future coupon rates.[ **]**[p]***[*][*]This chart shows a “red arrow” that is caused by a sudden change in the value of the bond rate. As the discount increases, the coupon rate drops and the price spreads out. To explain this, it is appropriate to look at the difference in the coupon price between the 2 short-term coupon rates. After the price drops, the two fixed coupon rates will be reversed. The bottom line is that the coupon is not affected by higher interest rates. The graph below indicates that the price that can be borrowed by the coupon increases as the rate decreases. If two different values of coupon have the same coupon rate it is assumed that the coupon rate will go up in the first 3–6 y of the date after the change is made in the rate.[**]**[p]***[*