Mathematical Derivation of Macaulay s Duration Formula 20

Duration, which can also be translated as Macaulay duration. It is derived from the definition of yield to maturity. The yield to maturity formula is known, the derivatives of the yield to maturity y are found on both sides of the equation, and then divided by **p on both sides of the equation, and a part of it is defined as the d duration.

Duration is a method of measuring the average maturity of a bond's cash flow and can be used to measure the sensitivity of a bond to changes in interest rates.

Frederick. Macaulay calculates duration based on the weighted average of the bond's interest on each coupon and the time of principal payment, known as Macaulay duration.

macaulay's duration)。Specifically, the present value of each bond cash flow is divided by the bond** to obtain the weight of each cash payment, and the time of each cash flow is multiplied by the corresponding weight to finally calculate the duration of the entire bond.

Duration is a key concept in fixed income portfolio management for several reasons:

1. It is a simple summary of the actual average maturity of the asset portfolio.

2. It is regarded as an important tool for portfolio immunity and interest rate risk.

3. It is a measure of the interest rate sensitivity of the asset portfolio, and the sensitivity of assets with equal duration to interest rate fluctuations is the same.

Time to maturity, coupon rate, and yield to maturity are the key factors that determine the bond**, and there is the following relationship with duration:

1. The duration of a zero-coupon bond is equal to its maturity time.

2. The maturity date remains unchanged, and the duration of the bond is extended with the decrease in the interest rate of interest-bearing notes.

3. The interest rate of the interest note remains unchanged, and the duration of the bond increases with the increase of maturity time.

4. Other factors remain unchanged, when the yield to maturity of the bond is low, the duration of the coupon bond is longer.

Macaulay Duration Theorem: There are 6 theorems about the relationship between Macaulay duration and the maturity of a bond: Theorem 1:

Only the Macaulay duration of discount bonds is equal to their time to maturity. Theorem 2: The Macaulay duration of direct bonds is less than or equal to their maturity.

Only the Macaulay duration of the last tranche of direct bonds is equal to their time to maturity and equal to 1. Theorem 3: The Macaulay duration of a unified bond is equal to (1+1 r), where r is the discount rate used to calculate the present value.

Theorem 4: The higher the coupon rate, the shorter the duration under the same maturity time. Theorem 5:

Under the condition that the coupon rate remains the same, the longer the maturity period, the longer the duration will generally be. Theorem 6: All other things being equal, the lower the yield to maturity of a bond, the longer the duration.

Macaulay duration is the weighted average time interval of all cash flows from the current moment to the maturity date.

Bonds**b= ci·e (-y·ti).

where CI represents the cash inflow of TI on each interest payment date, and Y represents the yield to maturity calculated by continuous compounding.

Deriving b against y and dividing by b for a negative sign gives the Macaulay duration.

d=-db/dy·1/b=∑[ci·e^(-y·ti)]·ti/b

When there is a slight change in the yield y.

b(y) in yThe first-order Taylor is b(y.).+△y）=b(y.)+db/dy·△y

Then b b=db dy·1 b· y

From d=-db dy·1 b b b b=-d· y

When y is larger, it is necessary to change b(y) in y2nd Order Taylor:

b（y.+△y）=b(y.)+db/dy·△y+1/2·d²b/dy²·(y)²

b/b=db/dy·1/b·△y+1/2·1/b·d²b/dy²·(y)²

Convexity c=1 b·d b dy

Substituting gives b b=-d· y+1 2·c· ( y)

That should be very detailed

Modified Duration = Macaulay Duration.

1+(y n)], because here, 1 y n 1. 0575；

Therefore, positive duration ,d is the most appropriate answer.

Macdur maturity(t), modified duration t 1 (y n), y is the annual interest rate.

The number of compounding is calculated in n tables.

For interest-paying bonds, Macdur discount rate per period.

Divide by the current value multiplied by the number of periods, and the modified term MAC 1 (Y N).

If the market interest rate.

It's 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]/[x0+x1/(1+y)^1+x2/(1+y)^2+..xn/(1+y)^n]

i.e. d=(1*pvx1+..n*pvxn) pvx, where pvxi represents the present value of the ith period cash flow and d represents the duration.

Macaulay duration. The concept of duration was first proposed by Frederick Robert Macaulay ( in 1938, so it is also called Macaulay duration (abbreviated as D). Macaulay duration is the calculation of a bond's average time to maturity using a weighted average.

It is a weighted average of the time it takes for a bond to generate cash flows in the future, weighted by the present value of each period as a percentage of the bond**.

5*100 (1+11%) 5 This is the present value of the last installment.

P is the present value of the bond's future cash flows (interest and principal) for the t-period in the future.

The formula is as follows: Ling Fan [*5-20) where D is the Macaulay duration, Ruler Hail B is the current market ** of the bond, P is the present value of the cash flow (interest and principal) of the bond in the undetermined T period, and T is the maturity time of the bond.

Macaulay calculates duration based on the weighted average of the bond's interest and principal payment time per coupon, known as Macaulay duration's duration)。

Corrected duration = Macaulay duration [1+(y n)], because here, 1 y n 1 . 0575；

Therefore, positive duration ,d is the most appropriate answer.

macdur maturity (t), modified duration t 1 (y n), y is the annual interest rate, and the number of compounding is calculated in n tables.

For interest-paying bonds, Macdur discount rate per period divided by current value multiplied by number of periods, with a modified maturity of MAC 1 (Y N).

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]/[x0+x1/(1+y)^1+x2/(1+y)^2+..xn/(1+y)^n]

i.e. d=(1*pvx1+..n*pvxn)/pvx

where PVXI represents the present value of the cash flow for period I and D represents the duration.

If the market interest rate is co-odd 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]/[x0+x1/(1+y)^1+x2/(1+y)^2+..xn/(1+y)^n]

i.e. d=(1*pvx1+..n*pvxn)/pvx。where PVXI represents the present value of the cash flow for period I and D represents the duration.

Duration theorem. 1. Only the Macaulay duration of zero-coupon bonds is equal to their maturity time. Reeds ruined.

2. The Macaulay duration of direct bonds is less than or equal to their maturity time.

3. The Macaulay duration of the Unified Surplus Bond is equal to (1+1 y), where y is the discount rate used to calculate the present value.

4. Under the condition of the same maturity time, the higher the coupon rate, the shorter the duration.

5. Under the condition that the coupon rate remains unchanged, the longer the maturity time, the longer the duration is generally longer.

6. Other things being equal, the lower the yield to maturity of the bond, the longer the duration.

Corrected Kushiro Sakura Period = Macaulay Duration.

1+(y/n)]

In this problem, 1+y n=1+

So the search for the cong Xiu Qing is the duration =

d is the most appropriate answer.