Question: 4. Your colleague Roger calculated a set of key rate 01s (KR01) for a bond and shared them with you, below, but he forgot to include the bond's specifications. He informs you that, although he selected only four key rates (two, five, 10 and 30 year), neighboring rates are shifted in a linear (interpolation) manner such that the entire term structure is characterized by the model. Here are the four key rates (KR01s), along with the bond's price of $94,000.00. Which is nearest to the bond's modified duration? a. 7.5 years b. 12.1 years c. 18.6 years d. 23.0 years Answer: 4. B. 12.1 years The sum of KRO1s will approximate the bond's DV01 (it will not be exact as a simultaneous shift of the key rates is not identical to a parallel shift--specifically, neighboring rates will shift less than one basis point--but it will be a good approximation). In this case: Estimate of bond's DV01 = 3.09 + 9.44 + 52.01 + 49.19 = $113.73; therefore, Since DV01 = Price * Modified Duration / 10,000, an estimate of bond's modified duration = DV01 * 10,000 / Price = $113.73 * 10,000 /$94,000 = 12.099 years ... which is also the sum of the individual key rate durations, where, for example key rate duration [2 years] = 3.09*10,000/$94,000 = 0.33 years.

I think there may be a typo in the formula and the subsequent calculations (P and 10,000 switched). Please correct me if my understanding is not correct. Estimate of bond's modified duration = DV01 * 10,000/ Price = $113.73 * $10,000/94,000 = 12.099 yrs 3.09*$10,000/94,000 = 0.33 years

Hi RK17, Yes, agreed: I transposed the interim formula (although the end result appears to be correctly computed: is why I use XLS for backup). Great catch! Fixed. Thank you!

Just curious, if the keyrate01's are not mentioned, how do you compute the bond price for the individual key rates

Hi laxsun, Our XLS replicates Tuckman's key rate shift (see http://www.bionicturtle.com/how-to/spreadsheet/2011.t5.b.3-key-rate-shift/ ). Re-pricing the bond for each key rate shift is the essence of the technique: in this case, the "before" is a flat par yield curve (ie, discounting all bond cash flows at the same yield), then we shift up the key rate (e.g., 5 year rate) and neighboring rates, so you have a mostly flat yield curve, but "yanked up" like a string at the key rate. Then, discount all cash flows, as normal, except b/c the key rates + neighbors are slightly higher, the bond price is slightly lower. This price output then determines the KR01. Thanks,