Rate duration in finance
Key rate duration measures how the value of a security or portfolio changes at a specific maturity point along the entirety of the yield curve. Key rate duration is an important concept in estimating the expected changes in value for a bond or portfolio of bonds because it does so when the yield curve shifts in a manner that is not perfectly parallel, which occurs often. This is why the key rate duration is such a valuable metric. Key rate duration and effective duration are related.
Key Rate Duration Definition
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 the price of 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. 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 mathematical derivative rate of change of price 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 or euro , etc. 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 often than Macaulay duration in modern finance. For every-day use, the equality or near-equality of the values for Macaulay and modified duration can be a useful aid to intuition. 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] [2] [3] [5]. These terms add to 1. 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 ,.
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. 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.
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. 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:. In contrast to Macaulay duration, modified duration sometimes abbreviated MD is a price sensitivity measure, defined as the percentage derivative of price with respect to yield the logarithmic derivative of bond price with respect to yield.
Modified duration applies when a bond or other asset is considered as a function of yield. In this case one can measure the logarithmic derivative with respect to yield:. When the yield is expressed continuously compounded, Macaulay duration and modified duration are numerically equal.
In financial markets, yields are usually expressed periodically compounded say annually or semi-annually instead of continuously compounded. Then expression 2 becomes:. This gives the well-known relation between Macaulay duration and modified duration quoted above.
It should be remembered that, even though Macaulay duration and modified duration are closely related, they are conceptually distinct. Macaulay duration is a weighted average time until repayment measured in units of time such as years while modified duration is a price sensitivity measure when the price is treated as a function of yield, the percentage change in price with respect to yield.
This will give modified duration a numerical value close to the Macaulay duration and equal when rates are continuously compounded. Formally, modified duration is a semi- elasticity , the percent change in price for a unit change in yield, rather than an elasticity , which is a percentage change in output for a percentage change in input.
Modified duration is a rate of change, the percent change in price per change in yield. Modified duration can be extended to instruments with non-fixed cash flows, while Macaulay duration applies only to fixed cash flow instruments. Modified duration is defined as the logarithmic derivative of price with respect to yield, and such a definition will apply to instruments that depend on yields, whether or not the cash flows are fixed. Modified duration is defined above as a derivative as the term relates to calculus and so is based on infinitesimal changes.
Modified duration is also useful as a measure of the sensitivity of a bond's market price to finite interest rate i. Thus modified duration is approximately equal to the percentage change in price for a given finite change in yield. Fisher—Weil duration calculates the present values of the relevant cashflows more strictly by using the zero coupon yield for each respective maturity.
Key rate durations also called partial DV01s or partial durations are a natural extension of the total modified duration to measuring sensitivity to shifts of different parts of the yield curve. Thomas Ho [9] introduced the term key rate duration. Reitano covered multifactor yield curve models as early as [10] and has revisited the topic in a recent review. Key rate durations require that we value an instrument off a yield curve and requires building a yield curve.
Ho's original methodology was based on valuing instruments off a zero or spot yield curve and used linear interpolation between "key rates", but the idea is applicable to yield curves based on forward rates, par rates, and so forth. Many technical issues arise for key rate durations partial DV01s that do not arise for the standard total modified duration because of the dependence of the key rate durations on the specific type of the yield curve used to value the instruments see Coleman, [3].
For a standard bond with fixed, semi-annual payments the bond duration closed-form formula is: [ citation needed ]. The total PV will be:.
The modified duration, measured as percentage change in price per one percentage point change in yield, is:. The steps to compute duration are the following:. Discounting to present value at 6. The detail is the following:. The DV01 is analogous to the delta in derivative pricing The Greeks — it is the ratio of a price change in output dollars to unit change in input a basis point of yield.
Dollar duration or DV01 is the change in price in dollars, not in percentage. It gives the dollar variation in a bond's value per unit change in the yield. It is often measured per 1 basis point - DV01 is short for "dollar value of an 01" or 1 basis point.
PV01 present value of an 01 is sometimes used, although PV01 more accurately refers to the value of a one dollar or one basis point annuity.
For a par bond and a flat yield curve the DV01, derivative of price w. This formula can be used to calculate the VaR of the portfolio by ignoring higher order terms.
Typically cubic or higher terms are truncated. Quadratic terms, when included, can be expressed in terms of multi-variate bond convexity. One can make assumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, in some special cases e. The formula can also be used to calculate the DV01 of the portfolio cf. The primary use of duration modified duration is to measure interest rate sensitivity or exposure.
Thinking of risk in terms of interest rates or yields is very useful because it helps to normalize across otherwise disparate instruments.
Consider, for example, the following four instruments, each with year final maturity:. All four have a year maturity but the sensitivity to interest rates, and thus the risk, will be different: the zero-coupon has the highest sensitivity and the annuity the lowest.
Modified duration is a useful measure to compare interest rate sensitivity across the three. The zero-coupon bond will have the highest sensitivity, changing at a rate of 9. The annuity has the lowest sensitivity, roughly half that of the zero-coupon bond, with a modified duration of 4. The BPV will make sense for the interest rate swap for which modified duration is not defined as well as the three bonds.
Modified duration measures the size of the interest rate sensitivity. Sometimes we can be misled into thinking that it measures which part of the yield curve the instrument is sensitive to.
For example, the annuity above has Macaulay duration of 4. But it has a cash flows out to 10 years and thus will be sensitive to year yields. If we want to measure sensitivity to parts of the yield curve we need to consider key rate durations. The yield-price relationship is inverse, and the modified duration provides a very useful measure of the price sensitivity to yields. As a first derivative it provides a linear approximation.
For large yield changes, convexity can be added to provide a quadratic or second-order approximation. Alternatively, and often more usefully, convexity can be used to measure how the modified duration changes as yields change. Similar risk measures first and second order used in the options markets are the delta and gamma.
Modified duration and DV01 as measures of interest rate sensitivity are also useful because they can be applied to instruments and securities with varying or contingent cash flows, such as options. For bonds that have embedded options , such as putable and callable bonds, modified duration will not correctly approximate the price move for a change in yield to maturity.
Consider a bond with an embedded put option. This bond's price sensitivity to interest rate changes is different from a non-puttable bond with otherwise identical cashflows. To price such bonds, one must use option pricing to determine the value of the bond, and then one can compute its delta and hence its lambda , which is the duration.
The effective duration is a discrete approximation to this latter, and will require an option pricing model. These values are typically calculated using a tree-based model, built for the entire yield curve as opposed to a single yield to maturity , and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see Lattice model finance Interest rate derivatives. Thus the index, or underlying yield curve, remains unchanged.
Floating rate assets that are benchmarked to an index such as 1-month or 3-month LIBOR and reset periodically will have an effective duration near zero but a spread duration comparable to an otherwise identical fixed rate bond.
The sensitivity of a portfolio of bonds such as a bond mutual fund to changes in interest rates can also be important. The average duration of the bonds in the portfolio is often reported. The duration of a portfolio equals the weighted average maturity of all of the cash flows in the portfolio.
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. Key rate duration is a measure of the sensitivity of a security or the value of a portfolio to a 1% change in yield for a given maturity.
Unlike the maturity date, which tells you when the issuer has promised to repay your principal, duration, which takes the bond's interest payments into account, helps you to evaluate how volatile the bond's price will be over time. Basically, the longer the duration -- expressed in years -- the more volatile the price. Related to Duration: Macaulay Duration , pregnancy duration. Duration A common gauge of the price sensitivity of a fixed income asset or portfolio to a change in interest rates.
The maturity of a fixed income investment, such as a bond, is simply how long it is until the investment is finally repaid.
Bond prices change inversely with interest rates, and, hence, there is interest rate risk with bonds. One method of measuring interest rate risk due to changes in market interest rates is by the full valuation approach , which simply calculates what bond prices will be if the interest rate changed by specific amounts. The full valuation approach is based on the fact that the price of a bond is equal to the sum of the present value of each coupon payment plus the present value of the principal payment.
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As a fiduciary to investors and a leading provider of financial technology, our clients turn to us for the solutions they need when planning for their most important goals. Matt Tucker takes a look at two key fixed income concepts and explains how each one behaves in a rising interest rate environment. Two terms that often get confused are duration and maturity. When investors believe interest rates are going to increase, they generally shift to a lower duration strategy to reduce the interest rate risk in their portfolios. Calculating duration rather involved, taking into account yields, bond coupons and that final maturity payment.
Key Rate Duration
Why Zacks? Learn to Be a Better Investor. Forgot Password. If you have a bond portfolio, you should understand the differences between duration and maturity. Duration and maturity are key concepts that apply to bond investments. Effective duration and average maturity apply if you have a portfolio consisting of several bonds. While maturity refers to when a bond expires, or matures, duration is a measure of the bond's price sensitivity to changes in interest rates. While the two concepts are related, they also differ significantly. When investing in bonds, this distinction is critical to grasp. Unlike stocks, bonds come with expiration dates -- they mature.
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.
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Duration: Understanding the relationship between bond prices and interest rates
Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates. A bond's duration is easily confused with its term or time to maturity because they are both measured in years. However, a bond's term is a linear measure of the years until repayment of principal is due; it does not change with the interest rate environment. Duration, on the other hand is non-linear and accelerates as time to maturity lessens. At the same time, duration is a measure of sensitivity of a bond's or fixed income portfolio's price to changes in interest rates. In general, the higher the duration, the more a bond's price will drop as interest rates rise and the greater the interest rate risk. The duration of a bond in practice can refer to two different things. The Macaulay duration is the weighted average time until all the bond's cash flows are paid. By accounting for the present value of future bond payments, the Macaulay duration helps an investor evaluate and compare bonds independent of their term or time to maturity. The second type of duration is called " modified duration " and, unlike Macaulay duration, is not measured in years.
