 # Duration : A unit of time ?

In Reading 28, duration is often used as a unit time to match the horizon of the liability in an immunization. In Level 1 & 2, we learned that duration is a sensitivity of price change given a small change. Why the duration can be regarded as a unit of time ? On the other hand, duration will change when only the yield changes at any point of time (i.e., even time is not passed over), then how can duration be regarded as a unit time without time passed over ?

Correction to my previous message. In Reading 28, duration is often used as a unit time to match the horizon of the liability in an immunization. In Level 1 & 2, we learned that duration is a sensitivity of price change given a small change IN YIELD. Why the duration can be regarded as a unit of time ? On the other hand, duration will change when only the yield changes at any point of time (i.e., even time is not passed over), then how can duration be regarded as a unit time without time passed over ?

The fixed income section of L3 is quite hopeless. Learn because you want to pass the exam and not because you want to gain knowledge.

believe the macauley duration is expressd in years (weighed av of cash flows) modified and effective as %

duration = change in price(pricereturn) / change in yield yield = pricereturn/unit time so the dimensional math shows it is some unit of time

alimesoda Wrote: ------------------------------------------------------- > believe the macauley duration is expressd in years (weighed av of cash flows) > modified and effective as % My understanding is that while Macauley Duration is a measurement of time, Modified/Effective Duration are not. Moreover, matching the duration of a bond (asset) with a liability’s horizon date means that the cash flows of the bond can meet the payment to the liability ?

I always thought time was a foolish and confusing way to express duration. My understanding is that duration is the sensitivity of price to changes in yield, such that the duration times 100 basis points will roughly equal the price change for a 100-basis-point shift in yield. The reason it’s expressed in years is that for a zero-coupon bonds, the duration will be the same as the maturity in years. So a five-year zero-coupon bond will have a duration of 5.

Why the duration for a zero is the same as the maturity in years? I have never figured it out since L2 although I memorise it just for the sake of passing the exam. Here is a simple calculation on a zero coupon of maturity 10 years for \$100 par. The discount rate, yield to maturity is the same. The various present values for the respective discount rates are: 4.50% 64.3927682 5.00% 61.39132535 5.50% 58.54305794 Duration = (V- - V+)/(2 * Vo * (Delta Y) ) Where V- = 64.3927682 V+ = 58.54305794 Vo = 61.39132535 Delta Y = 0.005 Thus, duration = 9.528561611

Wrong formula. Duration is = - % change in price / % change in yield Thus, duration has no dimension. janakisri Wrote: ------------------------------------------------------- > duration = change in price(pricereturn) / change > in yield > > yield = pricereturn/unit time > > > so the dimensional math shows it is some unit of > time

seemorr Wrote: ------------------------------------------------------- > The reason it’s expressed in years is that for a zero-coupon bonds, the duration will be the > same as the maturity in years. So a five-year zero-coupon bond will have a duration of 5. It is true that the durations are same as the maturities in years for zero-coupon bonds and the cash flow of a zero-coupon bond with duration of 5 surely can match a liability with a horizon of 5 years. Since bonds used in immunization are not limited to zero-coupon bonds, then why the (effective/modified) duration of a “coupon bond” can be a unit of time ? How the cash flows of a “coupon bond” with duration of 5 can match the payment of a liability with a horizon of 5 years ?

Let us go back to the basics. Duration is the sensitivity of a bond to a change in interest rates, right? If you go along the yield curve the sensitivities of each maturity is different from the next one. I assume we are together up to this point. If you have a liability of a certain point along the yield curve and you need to match it with an asset, what will you do? I believe the best way to do it is to create an asset that will have the same sensitivity to interest rates (read: the same duration) with the liability. By doing this, you are free from the vagaries of interest rate movements. Most liabilities are a one time cash outflow but it is difficult to get zero coupon bonds to match the cash flow. The next best thing to do is that get a family of bonds and adjust the duration to match the liability.

What does the “TIME HORIZON” of a liability mean ? A liability with a time horizon of 5 years means the liabilty has a duration (value / price change given a small change in yield / interest rate) of 5 ?

alta168 Wrote: ------------------------------------------------------- > What does the “TIME HORIZON” of a liability mean ? > A liability with a time horizon of 5 years means > the liabilty has a duration (value / price change > given a small change in yield / interest rate) of > 5 ? The maturity? If you have a 30 year mortage, the time horizon is 30 years at origination. I think you’re getting way too deep into this.

alta168 Wrote: ------------------------------------------------------- > What does the “TIME HORIZON” of a liability mean ? > A liability with a time horizon of 5 years means > the liabilty has a duration (value / price change > given a small change in yield / interest rate) of > 5 ? According to CFA Level 3 Fixed Income Readings… Yes