Mathematical explanation of bond term
duration
"Duration, full name Macaulay Duration-Macaulay Duration, mathematical definition.
If the market interest rate is y, the duration of Macaulay cash flow (X 1, X2, ..., Xn) is defined as: d (y) = [1* x1(1+y)1+2.
That is, d = (1* pvx1+... n * pvxn)/pvx.
Among them, PVXi represents the present value of the first cash flow, and D represents the duration.
The definition of duration can be better understood through the following examples.
For example, suppose there is a bond whose cash flow in the next n years is (X 1, X2...Xn), where Xi represents the cash flow in the first stage. Assuming that the interest rate is Y0, the interest rate immediately becomes Y after investors hold cash flow. Q: How long should it be held to make its due value not less than the value?
The above questions can be quickly answered by the following theorem.
Theorem: PV (y0) * (1+y0) q
Q is the required 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=Y0. (simple)
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 2.0, the fund price will rise or fall by 2% for every change in interest rate of 1 percentage point; The portfolio duration of long-term bond funds is 12.0, so the price will rise or fall by 1 2% for every change of interest rate.
Development of bond maturity
Modified duration
As can be seen from the above discussion, for a given small change in yield to maturity, the relative change in bond price is directly 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 bond yield to maturity is very small. In order to describe the sensitivity of bond prices to yield to maturity changes more accurately, a modified duration model is introduced. The correction duration is defined as △ p/p =-(d *) dy+c [(dy) 2]/2.
From this formula, we can see that there is a strict proportional relationship between the relative change of bond price and the correction duration for a given small change of yield to maturity. Therefore, the modified duration is a modification of Macaulay duration on the basis of considering the yield term Y, and it is a more accurate measure of the sensitivity of bond prices to interest rate changes.
term of validity
There is an important assumption in the study of Macaulay duration model that the cash flow of bonds will not change with the fluctuation of interest rates. However, this assumption is difficult for financial instruments with implicit options, such as mortgages, callable (or saleable) bonds, etc. 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 change of bond price when the interest rate level changes specifically. It directly uses bond prices based on changes in different yields to calculate, and these prices reflect changes in the value of implied options. Its calculation formula is:
Duration (effective) =(V-? y-V+y)？ 2V0？ y[2]
These include:
Five-? The bond price when the Y interest rate drops by X basis points;
V+？ The bond price when the 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;
The initial price of V0 bond;
The effective duration does not need to consider the change of cash flow in each period, does not include the specific time when cash flow changes due to interest rate changes, and only considers the overall price situation under certain interest rate changes. 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 term of validity is equal to Macaulay term.
With the deepening of the study of duration model, some people put forward new duration models, such as direction duration, partial duration, key interest rate duration, approximate duration and risk adjustment duration, and added interest rate term structure, coupon rate change, credit risk and redemption clause to the model, and further developed the duration model.
Term of bond portfolio
Bond portfolio also has the 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.
Purpose of bond term
In bond analysis, duration has transcended the concept of time. The greater the bond price drop caused by interest rate rise, the greater the bond price rise caused by interest rate drop. It can be seen that under the same factors, bonds with small correction duration are more resistant to the risk of interest rate rise than bonds with large correction duration, but less resistant to 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 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.
It should be noted that the concept of duration is widely used not only in a single bond, but also in bond portfolio. 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 the duration of this type of bond. Therefore, investors can accurately judge the future interest rate trend when operating large funds, and then determine the duration of the bond portfolio. After the duration is determined, they can flexibly adjust the weight of various bonds, which will basically achieve the expected results.
Duration is a method to measure the average duration of bond cash flow. Because the sensitivity of bond price will increase with the increase of maturity time, duration can also be used to measure the sensitivity of bond to interest rate changes, and duration can be calculated according to the weighted average of each coupon interest or principal payment time of bond.
The duration is calculated just like 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 1% and the bond price changes by 3%, the duration is 3.