### CFA Practice Question

A zero-coupon bond has 12 years left until maturity. If the market yield is 5%, what is the modified duration of this bond? Assume annual discounting for simplicity.
A. 0.00
B. 11.43
C. 12.00
Explanation: There are two ways to calculate modified duration.

Assume that the yield changes by 15 bp, up and down.
Current price @ 5%, FV = 1,000; N = 12; I/Y = 5; CPT PV = 556.84
Price @ 5.15%, FV = 1,000; N = 12; I/Y = 5.15; CPT PV = 547.38
Price @ 4.85%, FV = 1,000; N = 12; I/Y = 4.85; CPT PV = 566.47
Average price change = (566.47 - 547.38) / 2 = 9.545, modified duration = (9.545 / 556.84) x (1 / 0.0015) = 11.43

Alternatively, we can use Macaulay's duration, which equals the maturity of the zero-coupon bond, and divide it by (1 + 0.05). Modified duration = 12 / 1.05 = 11.43

User Comment
dimos According to my notes, the duration of a zero-coupon is ALWAYS equal to its term to maturity. The correct answer is C? PLS HELP
llgoms I answered C also - zero cpn bond duration is always equal to term to mat. thought this was a give away.
nike If you are seeking the Macaulay duration of a zero-coupon bond, the duration would be equal to the bond's maturity, so there is no calculation required. However, this question asks for modified duration!
danlan Calculate as this:

PV=1000/(1+I)^n

I=YTM

so we should estimate

(1000/(1+I+d)^n-1000/(1+I-d)^n)/(2*1000*d/(1+I)^n)

when d gets close to 0, it becomes a limit, ...
steved333 Macaulay= Yield on 0 coupon bond. Modified= Macaulay/(1+ i/#compounds per year).

So you have 12/(1+ .05/1)= 12/1.05= 11.4286
panvino Using BAII: 2nd bond - to open worksheet.
sdt - settlement = purchase date - put in today's date - 5.2609 enter
cpn = coupon = 0
rdt = matuity date = 5.2621
rv = 1000
act = day count = 365
1/y coupon - t change from 2/y press 2nd enter
yld = 5
Then compute for price
scroll down and you have modified duration = 11.43
mekc thanks panvino!
boddunah learned a new thing about duration and macaulay duration .
ninad123 Effective Duration can be used for embedded bonds, modified is used for pure option free bonds
mrpman When i was practicing Schweser, everytime I had to calculate a zero coupon bond, I had to change it to semiannual. For example zero coupon bond (10 years till maturity, 5%YTM, 1000 FV) for this i would put (n=20, i/y= .025, pmt=0, fv= 1000, pv=0). Apparently this method is incorrect according to analyst notes, anyone else run into a problem like this??
localabel mrpman: The question states "Assume annual discounting.." Its true most bonds are on a semiannual basis, but on the test they don't force you to assume anything. On the test they will specify whether the bonds are semiannual or annual.
JLCQ Macaulay Duration=Bond's maturity (12=12)

Modified Duration=Macaulay Duration/1+YTM

Modified Duration= 12/1,05=11.43
dagibbo This was easy from a logical point of view. The modified duration is discounted, so it can't be the same as the maturity of 12, has to be less. Definitely not 0 either in this case, so it had to be B.
chesschh Thanks panvino, now Im sad I dont have the professional BA ii plus... I have the regular one