Derivative Securities Pricing

Present Value and Bond Duration

This can be written as "X" and "Y". The term in square brackets is the ratio of the present value of the cash flow at time t to the bond price. The bond price is the present value of all payments. The duration is therefore a weighted average of the times when payments are made, with the weight applied to time being equal to the proportion of the bond's total present value provided by the cash flow at time t. The sum of the weights is 1.0. Note that, for the

Note that, for the duration of the bond, is defined as:

D, D, P

The duration of the bond, is defined as:

D, P

"ytn it c e "yt"ytnn i i iii¼1 t c et c e¼ ð4:12ÞD i ii iP i¼1i¼1¼ ð4:12Þ¼ ð4:12ÞDD B "ytn it c eBBi ii¼1

This can be written ¼ ð4:12ÞD

This can be written

This can be written "BX "yt "!

"!n ic eXX "yt"ytnn

This can be written i iic ec e¼D t ii "i¼¼D tD t BX "ytn ii ic e BBi¼1 i¼D ti¼1i¼1 i B

The term in square brackets is the ratio of the present value of the cash flow at time t to

The term in square brackets is the ratio of the present value of the cash flow at time to i¼1 t i i

The term in square brackets is the ratio of the present value of the cash flow at time to

The term in square brackets is the ratio of the present value of the

cash flow at time t_tot is the bond price. The bond price is the present value of all payments. The duration is the bond price. The bond price is the present value of all payments. The duration is the bond price. The bond price is the present value of all payments. The duration is the bond price. The bond price is the present value of all payments. The duration is

The term in square brackets is the ratio of the present value of the cash flow at time t_tot therefore a weighted average of the times when payments are made, with the weight therefore a weighted average of the times when payments are made, with the weight therefore a weighted average of the times when payments are made, with the weight the bond price. The bond price is the present value of all payments. The duration is therefore a weighted average of the times when payments are made, with the weight applied to time being equal to the proportion of the bond's total present value t_i applied to time applied to time being equal to the

proportion of the bond's total present value being equal to the proportion of the bond's total present value therefore a weighted average of the times when payments are made, with the weight provided by the cash flow at time . The sum of the weights is 1.0. Note that, for the applied to time t being equal to the proportion of the bond's total present value provided by the cash flow at time . The sum of the weights is 1.0. Note that, for the provided by the cash flow at time . The sum of the weights is 1.0. Note that, for the applied to time being equal to the proportion of the bond's total present value purposes of the definition of duration, all discounting is done at the bond yield rate of purposes of the definition of duration, all discounting is done at the bond yield rate of purposes of the definition of duration, all discounting is done at the bond yield rate of provided by the cash flow at time . The sum of the weights is 1.0. Note that, for

thetby the cash flow at time t . The sum of the weights is 1.0.iinterest, (We do not use a different zero rate for each cash flow in the way described iny. iinterest, (We do not use a different zero rate for each cash flow in the way described ininterest, (We do not use a different zero rate for each cash flow in the way described inpurposes of the definition of duration, all discounting is done at the bond yield rate ofy.y.Section 4.6.)When a small change y in the yield is considered, it is approximately true thatSection 4.6.)Section 4.6.)interest, (We do not use a different zero rate for each cash flow in the way described iny.When a small change !y in the yield is considered, it is approximately true thatWhen a small change !y in the yield is considered, it is approximately true thatWhen a small change !y in the yield is considered, it is approximately true thatSection 4.6.)When a small change !y in the yield is considered, it is approximately true thatdBdBdB¼

ð4:13Þ!B !y¼ ð4:13Þ¼ ð4:13Þ!B !y!B !ydydBdydy¼ ð4:13Þ!B !y

From equation (4.11), this becomes

From equation (4.11), this becomes

From equation (4.11), this becomes dy

From equation (4.11), this becomes XnXXnn "ytthis becomes (4.14)¼ "!y ð4:14Þ!B ic t e "yt"yti i¼ "!y ð4:14Þ¼ "!y ð4:14Þ!B!B iic t ec t eXn i iii¼1 "yt¼ "!y ð4:14Þ!B ic t ei¼1i¼1 i i

(Note that there is a negative relationship between and When bond yields increase,i¼1 B y.

(Note that there is a negative relationship between and When bond yields increase,

(Note that there is a negative relationship between and When bond yields increase,B y.B y.bond prices decrease. When bond yields decrease, bond prices increase.) From equa-bond prices decrease. When bond yields decrease, bond prices increase.) From equa-bond prices decrease. When bond

yields decrease, bond prices increase.) From equations (4.12) and (4.14), the key duration relationship is obtained:

(4.12) and (4.14), the key duration relationship is obtained:

(4.12) and (4.14), the key duration relationship is obtained:

bond prices decrease. When bond yields decrease, bond prices increase.) From equations (4.12) and (4.14), the key duration relationship is obtained:

¼ "BD ð4:15Þ!B !y

¼ "BD ð4:15Þ¼ "BD ð4:15Þ!B !y!B !y

This can be written ¼ "BD ð4:15Þ!B !y

This can be written

This can be written !B!B!B ¼ "D ð4:16Þ!y

This can be written that can be written ¼ "D ð4:16Þ¼ "D ð4:16Þ!y!yB!BBB ¼ "D ð4:16Þ!yB

Equation (4.16) is an approximate relationship

Between percentage changes in a bond price and changes in its yield, there is an approximate relationship described by Equation (4.16). This relationship, suggested by Frederick Macaulay in 1938, has become a popular measure known as duration. It is easy to use and provides valuable insights into the sensitivity of bond prices to changes in yields.

For example, consider a 3-year 10% coupon bond with a face value of $100.

Suppose that the yield on the bond is 12% per annum with continuous compounding. This means that

Consider a 3-year 10% coupon bond with a face value of $100. Suppose that the yield

suggested by Frederick Macaulay in 1938, has become such a popular measure.

changes in its yield. It is easy to use and is the reason why duration has become such a

on the bond is 12% per annum with continuous compounding. This means that

on the bond is 12% per annum with continuous compounding. This means that

on the bond is 12% per annum with continuous compounding. This means that

Consider a 3-year 10% coupon bond with a face value of $100. Suppose that the yield

¼ 0:12. Coupon payments of $5 are made every 6 months. Table 4.7 shows the

¼¼ 0:12. Coupon payments of $5 are made every 6 months. Table 4.7 shows the

0:12. Coupon payments of $5 are made every 6 months. Table 4.7 shows the

on the bond is 12% per annum with continuous compounding. This means that

Calculations necessary to determine the bond's duration. The present values of the calculations necessary to determine the bond's duration. The present values of the calculations necessary to determine the bond's duration. The present values of the¼ 0:12. Coupon payments of $5 are made every 6 months. Table 4.7 shows the

Consider a 3-year 10% coupon bond with a face value of $100. Suppose that the yield onybond's cash flows, using the yield as the discount rate, are shown in column 3 (e.g., the bond's cash flows, using the yield as the discount rate, are shown in column 3 (e.g., the bond's cash flows, using the yield as the discount rate, are shown in column 3 (e.g., the calculations necessary to determine bond's duration. The present values of the "0:12%0:5 the bond is 12% per annum with continuous compounding. This