Dv01 garp 2010 practice question

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Hi david,

I have a doubt in

Q. 5. Sarah is a risk manager responsible for the fixed income portfolio of a large insurance company. The portfolio contains a 30-year zero coupon bond issued by the US Treasury (STRIPS) with a 5% yield. What is the bond’s DV01?

a. 0.0161
b. 0.0665
c. 0.0692
d. 0.0694

Answer: b
Explanation: The DV01 of a zero-coupon is DV01 = [{30 / (1 + y/2)^2T+1} *100 ]/10000 = 0.0665

Why they have taken formula to calculate modified duration as D/(1 + y/2)^2T+1 when it should be only D/(1 + y/2)?

Also this can be solved in other way , where we are discounting the price of bond?

DV01 = Price * Modified Duration / 10,000;
PV = 100 * EXP(-5%*30) = $22.31
Modified duration = 30/(1+5%/2) = 29.2683
DV01 = $22.73 * 29.26829 / 10,000 = 0.06652

And one other way using cont compounding . though my answer was not accurate
which is

p(y)= S * exp(-rt)
=100 exp(-30y)
and as DV01 is slope
so
p'(y)= -3000 exp(-30y)
=3000 exp(-30*.05)
=-669.39

and divide it by 10000 so we get .0669(not accurate though)

So please tell me, which approach is right? or if I am doing things wrong??

 

[David Harper CFA FRM]

Hi snigdha,

That's great. Both answers are correct: the question is at fault (imprecise) for not specifying semi-annual (the given answer) or continuous (your answer).

Please note, in their semi-annual answer, they used:
P = F/(1+y/2)^(2T) and duration(mod) = T/(1+y/2) only if bond is zero-coupon,
Such that DV01 = P*duration(mod)/10000 = F/(1+y/2)^(2T) * T/(1+y/2) /10000= (F*T)/(1+y/2)^(2T+1)/10000

Your continuous is great!
Minor: slope is dollar duration; -669.39 is dollar duration
i.e., for every 1 unit change in yield (where 1 unit = 100% = 100*100 = 10,000 basis points), bond price changes $670 dollars.
Such that DV01 = dollar duration/10000

David

 

[[email protected]]

Ok .. now i got it..

thank u so much david.. U rock!!!

 

[[email protected]]

Hi,

could you calculate the PV using your calculator?

where N=60, y=2.5, PMT=0, FV=100 (semi annual)

I get PV=22.7 (not totally accurate...)

Simon

 

[[email protected]]

Also, looking at the continuous compounding snigdha did, is dollar duration alway negative (negative relationship between yield and price)?

 

[David Harper CFA FRM]

Hi Simon: I get the same price (PV) under those semi-annual assumptions. Then you could separately get mod duration = 30/1.025 = 29.2683, such that DV01 = 22.728*29.2683/10000 = 0.0665 (as above, for semi-annual).

I didn't focus on the middle before, but it is inconsistent to use:
DV01 = Price * Modified Duration / 10,000;
PV = 100 * EXP(-5%*30) = $22.31 (assumes continuous)
Modified duration = 30/(1+5%/2) = 29.2683 (assumes semi-annual)
DV01 = $22.73 * 29.26829 / 10,000 = 0.06652
… the inconsistency here is that PV assume continuous (k = inf) while mod duration is semi-annual (k=2).

… so if we want to use continuous then this would be instead:
DV01 = Price * Modified Duration / 10,000;
PV = 100 * EXP(-5%*30) = $22.31
Modified duration = 30; the only case where Mod = Mac is continuous zero-coupon, as k = infinite.
DV01 = $22.31 * 30 / 10,000 = 0.6693

Re: Is dollar duration alway negative (negative relationship between yield and price)?
Yes, except notably for interest only (IO) strips. GARP likes to quiz them as the key exception.

David

 

[ankitagarwal]

Hi David,
Why does Interest-only (IO) strip exhibit a negative DV01?

 

[Nicole Seaman]

Hi David,
Why does Interest-only (IO) strip exhibit a negative DV01?

Hello @ankitagarwal

This question has also been discussed a great deal here in the forum, which may help to answer your question: https://forum.bionicturtle.com/threads/question-5-bonds-dv01-valuation.2098/.

Nicole

 

[ankitagarwal]

Thank you for replying.
But I am unable to access that page.

 

[Nicole Seaman]

Thank you for replying.
But I am unable to access that page.

@ankitagarwal

In that case, I'm sure David or another member can answer your question. I normally check to make sure a member is paid before referring them to a paid link, but didn't check this time. I'm just trying to help David by referencing links where questions have been answered because the forum is so busy the week of the exam, and it is difficult to keep up.

Nicole

 

[ankitagarwal]

Thank you Nicole.
@Davidharper please kindly answer the query.
While looking at the IO and PO curve it seems it should be PO which should have negative DV01 but the answer says otherwise. Is this true?

 

[David Harper CFA FRM]

Hi @ankitagarwal The IO strip (not the PO) is unique in that is has negative duration (and negative DV01) per https://forum.bionicturtle.com/threads/securities-with-negative-duration-garp16-p1-55.9969/ i.e.,

Hi @premseoul

Regular (positive) duration is an inverse relationship between rates and bond price; higher rates --> bond price. Mathematically, as the slope of the P/Y curve is negative, actual dollar duration is negative, but the negative sign (-) in the classic duration ensures that this typical relationship (higher rates --> lower price) is reflected in a positive duration: ΔP/P ≈ -D*Δ.

• The question wants the trade that exhibits negative duration such that an increase in rates implies an increase in value. To my thinking, we should immediately dismiss (a) and (d) because these are both calls on zero coupon bonds: the calls add leverage but the zero coupon bonds have duration equal to maturity, so these calls inherit the positive duration of the underyling zero coupon bonds.
• In regard to (b), the interest-only strip is a unique instrument which exhibits negative duration; i.e., for most yields, increase in rate implies increase in value. See Tuckman's explain below. A short position in the negative-duration instrument, therefore, will decrease in value with higher rates. So, directionally, it's a similar bet as calls on zeros.
• In regard to (c), the instrument is the same as (b), but the short position achieves the relationship wanted: higher rates --> higher position value.

S0 the "short maturity" is a red herring. I hope that helps!

Tuckman on the negative duration of IO mortgage strips:

"Figure 20.6 graphs the price of the same 5% 30-year MBS, labeled here as a pass-through, along with the prices of its associated IO and PO. When rates are very high and prepayments low, the PO is like a zero coupon bond, paying nothing until maturity. As rates fall and prepayments accelerate, the value of the PO rises dramatically. First, there is the usual effect that lower rates increase present values. Second, since the PO is like a zero coupon bond, it will be particularly sensitive to this effect. Third, as prepayments increase, some of the PO, which sells at a discount, is redeemed at par. Together, these three effects make PO prices particularly sensitive to interest rate changes.

The price-rate curve of the IO is, of course, the pass-through curve minus the PO curve, but it is instructive to describe the IO curve independently. When rates are very high and prepayments low, the IO is like a security with a fixed set of cash flows. As rates fall and mortgages begin to prepay, the cash flows of an IO vanish. Interest lives off principal. Whenever some principal is paid off there is less available from which to collect interest. But, unlike callable bonds or pass-throughs that receive such prepaid principal, when prepayments cause interest payments to stop or slow the IO gets nothing. Once again, its cash flows simply vanish. This effect swamps the discounting effect so that, when rates fall, IO values decrease dramatically. The negative DV01 or duration of IOs, an unusual feature among fixed income products, may be valued by traders and portfolio managers in combination with more regularly behaved fixed income securities." -- Tuckman, Bruce; Serrat, Angel. Fixed Income Securities: Tools for Today's Markets (Wiley Finance) (pp. 588-589). Wiley. Kindle Edition.

... and:

@juhsu I wrote a lot above but there are two dynamics, it's interesting. The first is just the plain old discounting. As you suggest, higher interest rate implies a higher discount rate which, given a fixed cash flow, reduces the present value (among the most fundamental ideas in finance!). But with respect to a floating rate note or an IO strip, if the rate that determines the floating coupon drops more than the discount rate (i.e., if they are not identical), then a lower rate will produce a lower PV because the lower cash flows will slightly/somewhat overwhelm the lower discount rate. This is a generalization on the idea that, when the floating coupon and discount rate are effectively the same, the floating rate note must price to par. So that's one effect ...

But the more relevant in the case of the MBS is that the lower rates imply greater prepayments, which itself impacts the price adversely, as above. Prepayments are a key valuation input in the mortgage/MBS. I hope that's helpful.