**Fixed Income II**

**Reading 46. Understanding Fixed-Income Risk and Return**

**Learning Outcome Statements**

b. define, calculate, and interpret Macaulay, modified, and effective durations;

c. explain why effective duration is the most appropriate measure of interest rate risk for bonds with embedded options;

d. define key rate duration and describe the use of key rate durations in measuring the sensitivity of bonds to changes in the shape of the benchmark yield curve;

*CFA Curriculum, 2020, Volume 5*

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### Subject 2. Macaulay, Modified and Effective Durations

**Yield duration**statistics measure the sensitivity of a bond's full price to the bond's own yield-to-maturity. They include the Macaulay duration, modified duration, money duration, and price value of a basis point.**Curve duration**statistics measure the sensitivity of a bond's full price to the benchmark yield curve, e.g., effective duration.

Duration is the weighted average time to receive the present value of each of the bond's coupon and principal payments. For example, a bond with a duration of three means that, on average, it takes three years to receive the present value of the bond's cash flows.

**Macaulay Duration**

Frederick Macaulay developed the concept of duration approximately 80 years ago. He demonstrated that a bond's duration was a more appropriate measure of time characteristics than the term to maturity of the bond, because duration incorporates both the repayment of capital at maturity, the size of the coupon and timing of the payments.

Macaulay duration is defined as the weighted average time to full recovery of principal and interest payments. The weights are the shares of the full price corresponding to each coupon and principal payment.

Alternatively, Macaulay duration can be calculated using a closed-form formula.

**Modified Duration**

Modified duration shows how bond prices move proportionally with small changes in yields. Specifically, modified duration estimates the percentage change in bond price with a change in yield.

-D

_{mod}= the modified duration for the bond

Di = yield change in basis points divided by 100

P = beginning price for the bond

Modified duration assumes that the price/yield relationship is a straight line. However, the price/yield relationship is convex, not linear. Suppose that the bond has an initial yield of Y

_{0}. A tangent line can be drawn to the price/yield relationship at Y

_{0}. The slope of the tangent line is related to the duration of the bond. If the yield falls to Y

_{1}, the price will rise to P

_{1}. Due to the linear assumption, the price change measured by duration is P

_{2}- P

_{0}.

To approximate modified duration:

V

_{-}= the price if yields declines

V

_{+}= the price if yield increases

V

_{0}= the initial price

For example, consider a 9% coupon 20-year option-free bond selling at 134.6722 to yield 6%. If the yield is decreased by 20 basis points from 6.0% to 5.8%, the price would increase to 137.5888. If the yield increases by 20 basis points, the price would decrease to 131.8439. Thus: ApproxModDur = (137.5888 - 131.8439)/(2 x 134.6722 x 0.002) = 10.66. This tells you that for a 1% change in the required yield, the bond price will change by approximately 10.66%.

Macaulay duration is mathematically related to modified duration.

A bond with a Macaulay duration of 10 years, a yield to maturity of 8% and semi-annual payments will have a modified duration of:

Dmod = 10/(1 + 0.08/2) = 9.62 years

**Effective Duration**

Effective duration measures interest rate risk in terms of a change in the

*benchmark yield curve*. It is very similar to approximate modified duration.

A pricing model can be used to estimate the price change resulting from a change in the benchmark yield curve instead of the bond's own yield-to-maturity. V- and V+ are adjusted to reflect any changes in the cash flows (due to embedded options) that result from the change in benchmark yield curve.

Effective duration should be used for bonds with embedded options.

**Key-Rate Durations**

It is important to distinguish interest rate risk from yield curve risk.

- The interest rate risk is the sensitivity of a bond to parallel shifts of the yield curve.
- The yield curve risk is a bond's sensitivity to changes in the shape of the yield curve.

Parallel shifts in the yield curve rarely occur. An analyst may want to measure the change in the bond's price by changing the spot rate for a particular key maturity and holding the spot rate for the other key maturities constant. The

**key rate duration**is the sensitivity of the value of a bond to changes in a single spot rate, holding all other spot rates constant. There is a key rate duration for every point on the spot rate curve so there is a vector of durations representing each maturity on the spot rate curve.

###
**User Contributed Comments**
22

You need to log in first to add your comment. ###### yodaddy

I am using both Schweser and analystnotes' notes. According to Schweser, you dont need anything besides Effective Dur. --> IS EFFECTIVE DUR. the only thing needed... can i listen to schweser and ignore modified.

###### tamagoochi

DO NOT ignore modified duration - pretty good chance it will be tested. Check the L.O.S's. In fact, modified duration is more important than effective when figuring out most questions.

###### jayson

Effective duration and modified duration essentially give the same information - the percentage change in price for a 1% change in yield. The only difference is in the way the duration is calculated - modified duration is based on the first derivative of price with respect to yield whereas effective duration is based on estimating a change in price for a given change in yield in either direction, through use of pricing models, for example. The implication of this is that effective duration is more flexible in its usage, since it can deal with, for example, callable bonds more easily.A question could give you a value for modified duration or effective duration, and you would use each identically.

###### yodaddy

PROBLEM here is Schweser has ignored modified and has called modified dur, effective dur. ONly diff it says... you can use effective for any situation and also you get the same answer. I wanna smack Mr. Schweser in the head if thats not true and hes sending me into the exam with such ludicrous consequences. POST ME A FORUMLA if you can for MODIFIED. FOR EFFECTIVE......it is....for example:v- - v+ -------- 2(vo) x change in yield in decimalv- price when yield fell v+ opposite vo = price of bond HELP ME PLEASE thanks.

###### kalps

Mr Swhweser is money making machine with a near monopoly in the CFA materials business - he is clever and richer than any of us will eve rbe - so he can afford to produce rubbish notes - he has the power to do whatever he likes

###### kevin

from schweser: For coupon paying bonds, duration is less than maturity. Duration is approximately equal to the point in years where the investor receives half of the present value of the bond's cash flows. Since zero-coupon bonds only have one cash flow at maturity, the duration is approximately equal to maturity. Any coupon amount will shorten duration because some cash flow is received prior to maturity.

I am confused. As far as I know, duration is a measure of the price sensitivity of a bond to a change in yield. But maturity is a time unit. How can we connect them together?

Any answer is greatly appreciated.

###### Hyper

Duration increases with square root of maturity!

###### lawrence

The longer the duration the more price sensitive the bond is. A change in interest rate affect distant cashflows more than present cashflows. There are also different measures of duration with different meaning.

###### Jimish

hey

see coupon paying bond are less sensitive because the cash flows are delated..

cuase when cash flows are delayed.. the appropriate discount rate also increases..
its the simple time... discount factor.. so as the zero coupon bond delays all the coupon till maturity... it has a greater duration..

as duration measures the interest sensitivity...... the most sensitive would be the one whose payment are deferred the most...

i hope u got it now

###### NillePet

Am i getting it right that mod. duration and effective duration are giving the same result? So it is the approx. change in price for a 100 Bps change in yield. But why is the result in the example of mod. duration stated in years then?

###### freyalam

Schweser doesn't ignore the modified duration. I'm doing the proQ test bank and questions about modified duration show up very often.

###### freyalam

kevin,

There is an advice that we ignore the unit (usually stated in years) and we need to interpret it as: A duration of x years = for 1% change in YTM, the bond's value will change approximately x %

"approximately" because we are assuming the effective duration as a linear function between the yield rate and bond's value while it is convex.

###### kondagadu

you cannot use modified duration for callable bonds .effective duration can be used for both callable and option-free bonds.

###### johntan1979

For those who are using Schweser and don't trust AnalystNotes, please direct your questions to Schweser, since you paid a hefty price for it. While AnalystNotes may not be perfect, many of us are here for the same reasons... value for money, and more comprehensive coverage of the CFA curriculum.

###### bidisha

I don't get how modified duration is different from normal duration?

###### bidisha

Never mind I get it. Normally used, the term duration IMPLIES modified duration. But technically, duration can be of 3 different kinds: modified, effective & Macaulay.

###### gill15

Its hilarious. All this time writing about or thinking about if you should know this....you do understand it is ONE page not even 75% of one page that you have to read in the CFA curriculum book...

###### jmorris

Jimish is correct. If we recall from the previous reading (was one of the basic bond valuation concepts), i sensitivity is greater on issues with long time horizons. These issues have an inherent discount built in to their PV.

###### robertdole

Also, with callable bonds, the YTM is not reliable (because the issue may be called at any given point in time in the future, if the issuer can refund the debt at a cheaper price); so EffDur provides a more accurate read at price sensitivity given moves in the yield curve.

###### robertdole

ModDur and EffDur are looking at two different catalysts for bond price movements.

###### robertdole

p.545 in the CFAI books, reading 56.

###### nvallabh

Can someone help me understand.....with MACAULAY duration, the indication/output is measured in YEARS. While MODIFIED is a % change in bond price? Confusing!