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In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received.When an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.The dual use of the word "duration", as both the weighted average time until repayment and as the percentage change in price, often causes confusion. Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield. Both measures are termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them.When yields are continuously compounded Macaulay duration and modified duration will be numerically equal. When yields are periodically compounded Macaulay and modified duration will differ slightly, and there is a simple relation between the two. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond.Modified duration, on the other hand, is a derivative (rate of change) or price sensitivity and measures the percentage rate of change of price with respect to yield. (Price sensitivity with respect to yields can also be measured in absolute (dollar) terms, and the absolute sensitivity is often referred to as dollar duration, DV01, BPV, or delta (δ or Δ) risk). The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration. Modified duration is used more than Macaulay duration.For every-day use, the equality (or near-equality) of the values for Macaulay and modified duration can be a useful aid to intuition. For example a standard ten-year coupon bond will have Macaulay duration somewhat but not dramatically less than 10 years and from this we can infer that the modified duration (price sensitivity) will also be somewhat but not dramatically less than 10%. Similarly, a two-year coupon bond will have Macaulay duration somewhat below 2 years, and modified duration somewhat below 2%. (For example a ten-year 5% par bond has a modified duration of 7.8% while a two-year 5% par bond has a modified duration of 1.9%.)==Macaulay duration==Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:: V = \sum_^PV_i The Macaulay duration is defined as::(1) MacD = \frac} = \sum_^t_i \frac where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all future cash payments from the asset.In the second expression the fractional term is the ratio of the cash flow PV_i to the total PV. These terms add to 1.0 and serve as weights for a weighted average. Thus the overall expression is a weighted average of time until cash flow payments, with weight \frac being the proportion of the asset's present value due to cash flow i.For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely t_1 (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if cash flows are, in order, (t_1, ..., t_n), then::t_1 \leq MacD \leq t_n,with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond.Macaulay duration has the diagrammatic interpretation shown in figure 1. This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the PV(i)::(2) V = \sum_^PV_i = \sum_^CF_i \cdot e^ :(3) MacD = \sum_^t_i\frac} Where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* CF_i is the cash flow of the ith payment from an asset,* y is the yield to maturity (continuously compounded) for an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all cash payments from the asset until maturity.Macaulay gave two alternative measures:* Expression (1) is Fisher–Weil duration which uses zero-coupon bond prices as discount factors, and* Expression (3) which uses the bond's yield to maturity to calculate discount factors.The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield y, not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.Duration versus Weighted Average LifeSimilarities in both values and definitions of Macaulay Duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5 year fixed-rate interest only bond would have a Weighted Average Life of 5, and a Macaulay Duration that should be very close. Mortgages behave similarly. The differences between the two are as follows:
In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received.
When an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.
The dual use of the word "duration", as both the weighted average time until repayment and as the percentage change in price, often causes confusion. Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield.
Both measures are termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them.〔When yields are continuously compounded Macaulay duration and modified duration will be numerically equal. When yields are periodically compounded Macaulay and modified duration will differ slightly, and there is a simple relation between the two. 〕 Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond.
Modified duration, on the other hand, is a derivative (rate of change) or price sensitivity and measures the percentage rate of change of price with respect to yield. (Price sensitivity with respect to yields can also be measured in absolute (dollar) terms, and the absolute sensitivity is often referred to as dollar duration, DV01, BPV, or delta (δ or Δ) risk). The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration. Modified duration is used more than Macaulay duration.
For every-day use, the equality (or near-equality) of the values for Macaulay and modified duration can be a useful aid to intuition. For example a standard ten-year coupon bond will have Macaulay duration somewhat but not dramatically less than 10 years and from this we can infer that the modified duration (price sensitivity) will also be somewhat but not dramatically less than 10%. Similarly, a two-year coupon bond will have Macaulay duration somewhat below 2 years, and modified duration somewhat below 2%. (For example a ten-year 5% par bond has a modified duration of 7.8% while a two-year 5% par bond has a modified duration of 1.9%.)
==Macaulay duration==
Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:
:
The Macaulay duration is defined as:〔〔〔
:(1)
where:
* indexes the cash flows,
* is the present value of the th cash payment from an asset,
* is the time in years until the th payment will be received,
* is the present value of all future cash payments from the asset.
In the second expression the fractional term is the ratio of the cash flow to the total PV. These terms add to 1.0 and serve as weights for a weighted average. Thus the overall expression is a weighted average of time until cash flow payments, with weight being the proportion of the asset's present value due to cash flow .
For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if cash flows are, in order, , then:
:
with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond.
Macaulay duration has the diagrammatic interpretation shown in figure 1.
This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.
For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the :
:(2)
:(3)
Where:
* indexes the cash flows,
* is the present value of the th cash payment from an asset,
* is the cash flow of the th payment from an asset,
* is the yield to maturity (continuously compounded) for an asset,
* is the time in years until the th payment will be received,
* is the present value of all cash payments from the asset until maturity.
Macaulay gave two alternative measures:
* Expression (1) is Fisher–Weil duration which uses zero-coupon bond prices as discount factors, and
* Expression (3) which uses the bond's yield to maturity to calculate discount factors.
The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield , not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.
Duration versus Weighted Average Life
Similarities in both values and definitions of Macaulay Duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5 year fixed-rate interest only bond would have a Weighted Average Life of 5, and a Macaulay Duration that should be very close. Mortgages behave similarly. The differences between the two are as follows:
# Macaulay Duration only measures fixed period cash flows, Weighted Average Life factors in all principal cash flows whether they be in fixed or floating. Thus for Fixed Period Hybrid ARM mortgages, for modeling purposes, the entire fixed period ends on the date of the last fixed payment or the month prior to reset.
# Macaulay Duration discounts all cash flows at the corresponding cost of capital. Weighted Average Life does not discount.
# Macaulay Duration uses both principal and interest when weighting cash flows. Weighted Average Life only uses principal.
抄文引用元・出典: フリー百科事典『 Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:: V = \sum_^PV_i The Macaulay duration is defined as::(1) MacD = \frac} = \sum_^t_i \frac where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all future cash payments from the asset.In the second expression the fractional term is the ratio of the cash flow PV_i to the total PV. These terms add to 1.0 and serve as weights for a weighted average. Thus the overall expression is a weighted average of time until cash flow payments, with weight \frac being the proportion of the asset's present value due to cash flow i.For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely t_1 (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if cash flows are, in order, (t_1, ..., t_n), then::t_1 \leq MacD \leq t_n,with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond.Macaulay duration has the diagrammatic interpretation shown in figure 1. This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the PV(i)::(2) V = \sum_^PV_i = \sum_^CF_i \cdot e^ :(3) MacD = \sum_^t_i\frac} Where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* CF_i is the cash flow of the ith payment from an asset,* y is the yield to maturity (continuously compounded) for an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all cash payments from the asset until maturity.Macaulay gave two alternative measures:* Expression (1) is Fisher–Weil duration which uses zero-coupon bond prices as discount factors, and* Expression (3) which uses the bond's yield to maturity to calculate discount factors.The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield y, not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.Duration versus Weighted Average LifeSimilarities in both values and definitions of Macaulay Duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5 year fixed-rate interest only bond would have a Weighted Average Life of 5, and a Macaulay Duration that should be very close. Mortgages behave similarly. The differences between the two are as follows: ">ウィキペディア（Wikipedia）』
■Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:: V = \sum_^PV_i The Macaulay duration is defined as::(1) MacD = \frac} = \sum_^t_i \frac where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all future cash payments from the asset.In the second expression the fractional term is the ratio of the cash flow PV_i to the total PV. These terms add to 1.0 and serve as weights for a weighted average. Thus the overall expression is a weighted average of time until cash flow payments, with weight \frac being the proportion of the asset's present value due to cash flow i.For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely t_1 (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if cash flows are, in order, (t_1, ..., t_n), then::t_1 \leq MacD \leq t_n,with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond.Macaulay duration has the diagrammatic interpretation shown in figure 1. This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the PV(i)::(2) V = \sum_^PV_i = \sum_^CF_i \cdot e^ :(3) MacD = \sum_^t_i\frac} Where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* CF_i is the cash flow of the ith payment from an asset,* y is the yield to maturity (continuously compounded) for an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all cash payments from the asset until maturity.Macaulay gave two alternative measures:* Expression (1) is Fisher–Weil duration which uses zero-coupon bond prices as discount factors, and* Expression (3) which uses the bond's yield to maturity to calculate discount factors.The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield y, not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.Duration versus Weighted Average LifeSimilarities in both values and definitions of Macaulay Duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5 year fixed-rate interest only bond would have a Weighted Average Life of 5, and a Macaulay Duration that should be very close. Mortgages behave similarly. The differences between the two are as follows: ">ウィキペディアで「In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received.When an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.The dual use of the word "duration", as both the weighted average time until repayment and as the percentage change in price, often causes confusion. Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield. Both measures are termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them.When yields are continuously compounded Macaulay duration and modified duration will be numerically equal. When yields are periodically compounded Macaulay and modified duration will differ slightly, and there is a simple relation between the two. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond.Modified duration, on the other hand, is a derivative (rate of change) or price sensitivity and measures the percentage rate of change of price with respect to yield. (Price sensitivity with respect to yields can also be measured in absolute (dollar) terms, and the absolute sensitivity is often referred to as dollar duration, DV01, BPV, or delta (δ or Δ) risk). The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration. Modified duration is used more than Macaulay duration.For every-day use, the equality (or near-equality) of the values for Macaulay and modified duration can be a useful aid to intuition. For example a standard ten-year coupon bond will have Macaulay duration somewhat but not dramatically less than 10 years and from this we can infer that the modified duration (price sensitivity) will also be somewhat but not dramatically less than 10%. Similarly, a two-year coupon bond will have Macaulay duration somewhat below 2 years, and modified duration somewhat below 2%. (For example a ten-year 5% par bond has a modified duration of 7.8% while a two-year 5% par bond has a modified duration of 1.9%.)==Macaulay duration==Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:: V = \sum_^PV_i The Macaulay duration is defined as::(1) MacD = \frac} = \sum_^t_i \frac where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all future cash payments from the asset.In the second expression the fractional term is the ratio of the cash flow PV_i to the total PV. These terms add to 1.0 and serve as weights for a weighted average. Thus the overall expression is a weighted average of time until cash flow payments, with weight \frac being the proportion of the asset's present value due to cash flow i.For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely t_1 (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if cash flows are, in order, (t_1, ..., t_n), then::t_1 \leq MacD \leq t_n,with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond.Macaulay duration has the diagrammatic interpretation shown in figure 1. This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the PV(i)::(2) V = \sum_^PV_i = \sum_^CF_i \cdot e^ :(3) MacD = \sum_^t_i\frac} Where:* i indexes the cash flows,* PV_i is the present value of the ith cash payment from an asset,* CF_i is the cash flow of the ith payment from an asset,* y is the yield to maturity (continuously compounded) for an asset,* t_i is the time in years until the ith payment will be received,* V is the present value of all cash payments from the asset until maturity.Macaulay gave two alternative measures:* Expression (1) is Fisher–Weil duration which uses zero-coupon bond prices as discount factors, and* Expression (3) which uses the bond's yield to maturity to calculate discount factors.The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield y, not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.Duration versus Weighted Average LifeSimilarities in both values and definitions of Macaulay Duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5 year fixed-rate interest only bond would have a Weighted Average Life of 5, and a Macaulay Duration that should be very close. Mortgages behave similarly. The differences between the two are as follows: 」の詳細全文を読む
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