recently, there have been some fluctuations in the bond market, and investors have paid more attention to the bond market, which often appears in many analytical articles or investment suggestions on bonds? Duration? This word. So what does duration mean? Let's take a look at the rules of bond duration.
Mathematical explanation of bond duration
Duration
Duration, full name -Macaulayduration, mathematical definition
If the market interest rate is y, cash flow (X1,X2, ... The Macaulay duration of Xn) is defined as d (y) = [1 * x1/(1+y) 1+2 * x2/(1+y) 2+...+n * xn/(1+y) n]/[x+x1/(1+y).
The definition of duration can be better understood through the following examples.
example: suppose there is a bond whose cash flow in the next n years is (X1,X2,...Xn), where Xi represents the cash flow in the first phase. Assuming that the interest rate is Y, the interest rate immediately changes to Y soon after investors hold the cash flow. Q: How long should it be held to make its due value not lower than the value of?
the above question can be quickly answered by the following theorem.
theorem: PV (y) * (1+y) q <; The necessary condition for = PV (y) (1+y) q is q=D(Y). Here d (y) = (x1/(1+y)+2 * x2/(1+y) 2+...+n * xn/(1+y) n)/PV (y)
q is the desired time, that is, the duration.
the proof of the above theorem can be obtained by taking the reciprocal of the y derivative and making it take the local minimum at Y=Y. (Easy)
Simple explanation: Duration is the sensitivity of bond prices to normal changes in interest rates. If the portfolio duration of a short-term bond fund is 2., the price of the fund will rise or fall by 2% for every 1 percentage point change in interest rate; If the portfolio duration of a long-term bond fund is 12., the price will rise or fall by 12% for every 1 percentage point change in interest rate.
development of bond duration
correction duration
from the above discussion, it can be seen that for a given slight change in yield to maturity, the relative change of bond price is proportional to its Macaulay duration. Of course, this proportional relationship is only an approximate proportional relationship, and its establishment is based on the premise that the yield to maturity of bonds is very small. In order to describe the sensitivity of bond prices to yield to maturity changes more accurately, a ModifiedDurationModel is introduced. The correction duration is defined as △ p/p =-(d *) dy+c [(dy) 2]/2
From this formula, it can be seen that there is a strict proportional relationship between the relative change of bond price and the correction duration for a given small change in yield to maturity. Therefore, the modified duration is a modification of Macaulay duration on the basis of considering the yield term Y, which is a more accurate measure of the sensitivity of bond prices to interest rate changes.
Effective duration
There is an important assumption in the study of Macaulay duration model, that is, with the fluctuation of interest rate, the cash flow of bonds will not change. However, this assumption is difficult for financial instruments with implicit options, such as mortgage loans, callable (or sellable) bonds and so on. Therefore, Macaulay duration model should not be used to measure the interest rate risk of financial instruments whose cash flow is easily affected by interest rate changes. Aiming at the limitation of Macaulay duration model, FrankFabozzi put forward the idea of effective duration. The so-called effective duration refers to the percentage of bond price change when the interest rate level changes specifically. It directly uses the bond prices based on the change of different yields to calculate, and these prices reflect the change of implied option value. The formula is:
Duration(effective)=(V-? y-V+? y)? 2V? Y[2]
where:
V-? Bond price when y interest rate drops by x basis points;
V+? Bond price when y interest rate rises by x basis points;
-? Y initial rate of return plus x basis points;
+? Y initial rate of return minus x basis points;
initial price of p>V bond;
the effective duration does not need to consider the change of cash flow in each period, and does not include the specific time when the cash flow changes due to the change of interest rate, but only considers the overall price situation under a certain change of interest rate. Therefore, the effective duration can accurately measure the interest rate risk of financial instruments with the nature of implied options. For financial instruments without implied options, the effective duration is equal to Macaulay duration.
with the deepening of the research on duration model, some people have put forward some new duration models, such as direction duration, partial duration, key interest rate duration, approximate duration and risk adjustment duration, and added the term structure of interest rate, the change of coupon rate, credit risk and redemption terms to the model, which further developed the duration model.
bond portfolio duration
bond portfolio also has a corresponding concept of duration, and its duration is the weighted average of a single duration, which can be calculated by the following formula:
where is the weight of a single bond in the portfolio.
the purpose of bond duration
in bond analysis, duration has surpassed the concept of time, and the greater the decline of bond prices caused by rising interest rates, the greater the increase of bond prices caused by falling interest rates. It can be seen that, under the same factors, bonds with small correction duration have stronger ability to resist the risk of interest rate rise than bonds with large correction duration, but weaker ability to resist the risk of interest rate decline.
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 duration of bonds; 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.
It should be noted that the concept of duration is widely used not only in individual bonds, but also in bond portfolios. A long-term bond and a short-term bond can be combined into a medium-term bond portfolio, and increasing the investment ratio of a certain type of bond can tilt the duration of the portfolio to that of this type of bond. Therefore, when investors are operating large funds, they can accurately judge the future interest rate trend, and then determine the duration of the bond portfolio. When the duration is determined, they can flexibly adjust the weights of various bonds, which will basically achieve the expected results.
duration is a method to measure the average duration of cash flow in bonds. Because the sensitivity of bond price will increase with the maturity time, duration can also be used to measure the sensitivity of bond to interest rate changes, and the duration can be calculated according to the weighted average of each coupon interest or principal payment time of bond.
the calculation of duration is just like calculating the weighted average. The variable is time, the weight is the cash flow of each period, and the price is equivalent to the sum of the weights (because the price is calculated by discounted cash flow method). In this way, the calculation formula of duration is a weighted average formula, so it can be regarded as the average time to recover costs.
determining the duration, that is, affecting the sensitivity of bond prices to changes in market interest rates, includes three elements: maturity, coupon rate and yield to maturity.
different bond prices have different sensitivities to changes in market interest rates. Bond duration is the most important and main criterion to measure this sensitivity. Duration is equal to the price change caused by the change of interest rate by one unit. If the market interest rate changes by 1% and the bond price changes by 3%, the duration is 3.