# Q33

You have three portfolios, I, II and III, made up with bonds with terms being 2, 10, 16 and 30 years. All portfolio values are USD 100m. Portfolios II and III contain notably: - USD 25m of 10y bonds and - USD 25m and USD 50m of 30 y bonds respectively Rate curve is moving: 10y rate is moving down by [50]bps while 30y rate is moving up by 20bps. By how much the performance of portfolio II is going to exceed the one of portfolio III ? A) 1.45% B) 1.50% C) 2.95% D) 3.05% My answer: - Duration 30y bonds = 30 / 1.05 = 28.57 - Key rate 30 (ptf II) = 25/100 * 29 = 7.14 - Key rate 30 (ptf III) = 50/100 * 29 = 14.29 - Key rate 30 (ptf II - ptf III) = 7.14 - Impact = 7.14 * 0.20% = 1.43% Answer A) My issue is that to calculate the duration for the 30y bond, you need the 30y rate which was not given… You could assume that the rate is 0% - and then you obtain 1.50% - but that is stupid, no? I assumed 5%.

that is not stupid but right…these are ZERO coupon bonds their duration is their time to maturirty

I think that for a zero coupon, the duration equals the maturity divided by (1+rate)…

mhannebert Wrote: ------------------------------------------------------- > I think that for a zero coupon, the duration > equals the maturity divided by (1+rate)… Duration = time to maturity (for zeros)

For zero’s, the coupon is the duration. You just weight the maturities to get the key rates. In schweser they always start out key rates with an example of of zero coupon bonds.

Sorry guys, for zero coupons the duration is the maturity divided by (1+rate) - that is sure. Please, take any example if you are not convinced - with a change in rate of 0.01% or 0.001% only to limit the third order effects. BUT you are right, for the calculation of the key rates, you don’t use the duration but the maturity, as far as I remember in Schweser’s study notes. But I wonder whether that was just to simplify the calculation (i.e. for illustration purpose) or because this is absolutely right…

Straight out of the book again. There is a worked out example in the curriculum book. Don’t remember the exact answer, but we need to We need multiply the portfolio weight for each duration by the key rate. So, for example , if you have \$50 for 2 yr duration and \$50 for 30 yr. duration, then you go (50/100)*2= 1 and (50/100)*30= 15. Similarly calc. the values for the other port. based upon the market value weights. And then, based upon the basis point movements, calculate the change in value. So, if there was a minus 10 bp change, then the port. value will incr. by 0.10% for that key rate. Then calculate the difference. I think I got 1.45%, but I don’t remember the exact answer.

Btw, what was the answer for the other question, where they asked about which port. would have the highest change in value? I went with the port. that had the largest duration…

yes it was 16 or somethibg…B i think

mhannebert Wrote: ------------------------------------------------------- > Sorry guys, for zero coupons the duration is the > maturity divided by (1+rate) - that is sure. > Please, take any example if you are not convinced > - with a change in rate of 0.01% or 0.001% only to > limit the third order effects. Wrong. The duration of a bond can be calculated as sum( PV_cashflow * time_to_cashflow)/Value_bond. Value_ZCB = PV_cashflow, so duration = maturity. You are talking about calculating effective duration, which will still be pretty close to maturity after adjusting for convexity.

got the same result 1.45% but in a different way

You don’t have to divide by interest rate. Duration is just time to maturity. Question that remains is: Did the question ask for the return after a year (duration is 29 and answer is 1.45) or did it ask for the return shortly after now (duration is 30 and answer is 1.5)

There is no reason to compare the 2-yr movement I remember because these two porfolios have exactly the same weight on it so the movement will be the same. However, since the weights are 50 and 25 percent respectively, it will be: 30 x 0.5 (the weight) x 0.2 (10 basis point) - 30 x 0.25 x 0.2 = 3 - 1.5 = 1.5

But when did the 20 bps shift occur? Was that right away or after a year. If it was after a year the duration would be 29.

I didn’t read the question THAT close. So sorry, I can’t answer that question

Again, I’m sorry, but the duration of a z-bond is properly calculated when you divide the term by (1+r). But that is not the question they asked - my mistake. For the calculation of the impact on the two portfolios, the calculation suggested by chaucy seems right to me. The answer was B) 1.5. ;-( But then, how could you come to the other distractors, 2.95 and 3.05 ??? May be there was a mention that the rates will change in 1 year only…

mhannebert Wrote: ------------------------------------------------------- > Again, I’m sorry, but the duration of a z-bond is > properly calculated when you divide the term by > (1+r). It’s ok mate, you don’t have to be sorry. But it would be good if you tried to explain why that is. I can back up my assertion that you are wrong with many different sources. If you could back it up with just one source, I’d be very appreciative. I think you have forgotten that we divide by the price of the bond, not the par value.

if 1.5 + 1 for me

totally guessed 2.5

So was it 1.45 or 1.5? I had 1.5, and the calculation seemed obvious to me, which is why I’m wondering if it was wrong…