Bond maturity

The concept of 1. duration

Duration refers to the fluctuation range of bond portfolio value when the market yield changes by one percentage point. It is used to measure the sensitivity of bond portfolio to interest rate changes.

The concept of duration was first put forward by Macaulay in 1938, so it is also called Macaulay duration (abbreviated as D). Macaulay duration is to calculate the average maturity of bonds in the form of weighted average. It is the weighted average of the time when the bond will generate cash flow in the future, and its weight is the proportion of cash value in each period in the bond price.

The specific calculation is: divide the present value of each bond cash flow by the bond price to get the weight of each cash payment, then multiply the time of each cash flow by the corresponding weight, and finally sum up to get the duration of the whole bond. Therefore, the unit of Macaulay duration is years.

Macaulay duration formula:

MacD indicates Macaulay duration;

Ti represents the time of I cash flow;

V stands for bond value;

CFi represents the amount of the first cash flow;

Y stands for rate of return.

The calculation formula of duration is a weighted average formula, so it can be regarded as the average time to recover the cost.

Case 1

Macaulay duration calculation example.

There are the following keys (as shown in table 1 and table 2):

Table 1 Summary of Bond Information

Table 2 Bond Cash Flow

Macaulay duration =

(3.86 1084 × 0.96438+0506849+3.3838+09563 × 1.9439+93+05 × 2.59889.99989899995

Modify duration

In the actual bond analysis, duration has surpassed the concept of time, and investors use it more to measure the sensitivity of bond price changes to interest rate changes, that is, to measure how much bond price changes when the rate of return changes to a certain extent, which can accurately reflect the impact of quantitative interest rate changes on bond prices after certain corrections. From this, we have the concept of correction duration.

Modified duration formula:

K is the number of interest payments in a year;

V is the bond price;

Y is the output.

Calculation of correction duration:

In the case of 1, what is the correction duration of the bond?

Modified duration = Macaulay duration /( 1+ yield/interest frequency) =2.7 148.

The modified duration is the first derivative of bond price to yield, and its economic meaning can be understood as that if the modified duration is 5, the bond price will rise by 5% when the yield drops 1%.

The longer the correction lasts, the more sensitive the bond price is to the change of yield, the greater the decline of bond price caused by the increase of yield, and the greater the increase of bond price caused by the decrease of interest rate. It can be seen that under the same factors, bonds with short correction duration are more resistant to the risk of rising yield than bonds with long correction duration, but when the yield drops, the profitability is weak.

It is the above characteristics of duration that provide reference for our bond investment. When we judge that the current interest rate level is likely to rise, we can focus on investing in short-term varieties and shorten the bond duration; When we judge that the current interest rate level is likely to decline, we should lengthen the duration of bonds and increase the investment in long-term bonds, which can help us get a higher premium in the rise of the bond market.

Duration can bring the theoretical basis of an extremely important strategy in bond investment management-"immune strategy". According to this strategy, when the bond portfolio duration of the trading subject is equal to the bond holding period, the trading subject will achieve the goal of "immunity" in the short term, that is, the total wealth in the short term will not be affected by interest rate fluctuations. The principle is to adjust the bond portfolio duration to 0, so that the change of yield will not affect the value of bonds.

02

Convexity of bonds

We introduce index duration to measure the sensitivity of bond price to the change of yield, and we can really approximate how much the bond price changes when the yield changes by using duration. However, when the yield changes greatly, there will be some errors in calculating the change of bond price by duration, such as the bonds mentioned in Table 3:

Table 3 Summary of Bond Information

We know that the modified duration is 2.7 148, and the full price is 100.8 143 yuan. If the yield rises by 10 basis point, that is, from 3.8% to 3.9%, the recalculated bond price is 100.54438+02 yuan. The price adjusted according to the duration is

100.8143-0.1%× 2.7148 ×100.8143 =100.5406 Yuan, two yuan. If the yield rises by 100 basis point, that is, from 3.8% to 4.8%, the recalculated bond price is 98. 1278 yuan. The price adjusted according to the duration is

100.8143-1%× 2.7148×100.8143 = 98.0774 yuan, with a difference of 5 points.

As can be seen from the above example, when the yield changes in a large range, there will be some errors in calculating the change of bond price by duration, because duration itself will change with the change of yield, so in order to measure the influence of bond yield change on bond price more accurately, we need to introduce a new concept, convexity.

The convexity of bond is a concept put forward by StanleyDiller in 1984, which is a measure of the bending degree of bond price curve. Strictly speaking, convexity refers to the degree of bond price change caused by the change of bond yield to maturity. Convexity is the second derivative of bond price to yield. The greater the convexity, the greater the curvature of the bond price curve, and the greater the error caused by measuring the interest rate risk of bonds with the revised duration. Convexity is also an index to measure the sensitivity of bond maturity to interest rate. When the price yield changes greatly, their fluctuation range is nonlinear. The prediction made by the duration will deviate. Convexity is a correction to this deviation.

The convexity of bonds has the following characteristics:

(1) convexity increases with the increase of duration. If the yield and duration remain unchanged, the greater the coupon rate, the greater the convexity.

(2) For bonds without implied options, the convexity is always greater than 0, that is, the interest rate drops and the bond price will accelerate; When interest rates rise, bond prices will slow down and fall.

(3) The convexity of bonds with implied options is generally negative, that is, the price decreases with the decrease of interest rate, or the effective duration of bonds decreases with the decrease of interest rate and extends with the increase of interest rate. Because when the interest rate falls, the possibility of buying options increases.

Case 2

If the duration of a bond is 5 and the convexity is 30, if the yield rises by 100 basis points, then the change of bond price is:

-5× (1%)+1/2× 30×1%×1%=-4.85%, and the price drop is lower than that calculated only by duration (-5×1%) =-5.

If the yield drops by 65,438+000 basis points, then 5× (65,438+0%)+65,438+0/2× 30× 65,438+0% = 5.65,438+05%, and the price increase ratio is only calculated according to the duration (5× 65

Case 2 can verify that positive convexity will lead to the increase of bond price when the yield decreases, which is greater than when the yield increases at the same rate.