FRM Fun 1

[David Harper CFA FRM]

Today we start a series of fun (engagement) questions. Unlike the standard daily questions (or mock exam questions), these are not categorized according to exam topic. Instead, these are meant to give you a quick daily dose of brain candy. Also, we are going to award incentives to participants (yes, this is an experiment. We are simply trying to see if we can add some "fun" to the otherwise dreary task of studying!). Here is today's brain candy:

FRM Fun 1.

The most important formula for a bond's DV01 (dollar value of an '01) is DV01 = Price * Modified Duration / 10,000. If we assume continuous compounding, both durations (Mod and Mac) conveniently equal the maturity of a zero-coupon bond. In this way, under continuous, DV01 [zero-coupon bond] = Price * Maturity / 10,000. If the yield is 5.0%, the DV01 of the zero-coupon bond is HIGHEST at what maturity?

A. 20 years
B. 30 years
C. 40 years
D. Infinite

... is this too easy for you, smarty pants?
okay, then try something harder: prove the answer with calculus for any given yield = y.

Please answer below, and with any comments!

 

[LEO04]

Hi,

In case of a zero coupon bond , DV01 is a function of Price and maturity whereas price=fv of bond*e^-rt ,continuously compounded
Hence DV01 is highest at A)20 Years

 

[LEO04]

DV01 is a measure of how much a bonds price will increase in response to a one bp decline in bonds yield to maturity or vice versa

Using calculus percentage change in the bond’s price for a change in yield to maturity is (assuming year=20)

d p/p = dp =-20e^-20*ytm =-20
d(ytm) d(ytm)* p e^-20*ytm

so that's how the bonds negative term to maturity or duration is a function of DV 01

 

[David Harper CFA FRM]

Thanks akj, since you answered correctly, you are entered once with a "win" (gold star). Suzanne will be posting the rules shortly, etc (we just started this).

With respect to your proof, I don't quite agree (sorry, please correct me if i am wrong?).

I think another "win" is still available for whoever can show the general proof.

Let me re-phrase the calculus-based question:

• Given a zero-coupon bond with yield = y, and assuming continuous compounding, we want to solve for (find) the maturity (in years) with the highest DV01, as a function of the yield.
• As akj shows, since under continuous compounding P = F*exp(-yT), duration (D) = T, and DV01 = P*D/10,000, DV01[zero, CC] = F*exp(-yT)*T/10,000
• So we are looking for a "local maximum" with respect to maturity (T), not with respect to yield

 

[Mahesh Deep]

Hi David,

Please see my answer below.

Differentiating DV01[zero, CC] with respect to T using chain rule, and setting it equal to 0 to find the local maximum as follows

=> ((-T^2 )+1)* F*exp(-yT)/10000 = 0
above equation is equivalent to
=> ((T^2 )-1) =0

we get t=1 (we can discard t= -1).

So for t = 1 the DV01 is maximum.

P.S: it's tedious typing equations in a message, maybe I should have typed it in a document and uploaded the file

Thanks,
Mahesh

 

[David Harper CFA FRM]

Hi Mahesh,

Thanks but how did you get rid of the yield (y)? I think the correct answer is variant to the yield; ie, retains the yield.

I was thinking: DV01 = F*exp(-yT)*T/10,000, that just as you suggest we are looking for d[DV01]/dT, and since F is a constant, we can assume F = 100, so we have
DV01 = 100*exp(-yT)*T/10,000 = exp(-yT)*T/100 ... we might as well be looking for the max of = exp(-yT)*T.

So i think we have a both a product rule (http://en.wikipedia.org/wiki/Product_rule) and a chain rule. Setting this 1st derivative equal to zero, I think, gives us a solution in T which is a function of yield( (y); and which, btw, does return 20 years for y = 5%.

 

[aadityafrm]

Here is the solution, as a generic rule DV01 = - (1/10000) dP/dy,
[Since P.* Mod D /10000 = DV01 and Mod D = - (1/P) dP/dy, where P = F*exp(-yT)]
So local maxima for T can be obtained by putting d2P/dTdy = 0
Here, dP/dy = -F*T*exp(-yT)
So, d2P/dTdy = -F{exp(-yT) – y*T*exp (-yT)}
Equating above expression to zero, we get T= 1/y
In the give question, y=5%, which means T=20 yrs

 

[Mahesh Deep]

Sorry made a typo for the expression in parenthesis it should be
(-yt+1)
Instead of
(-t^2 +1)

Solving for t and setting it equal to 0 I get T= 1/y.


And you are correct I did have to use both product and chain rule.

Please let me know.

Thanks,
Mahesh

 

[David Harper CFA FRM]

I'm awarding the second star/trophy (our developers will be installing this feature functionality, so we don't quite have it yet! But we will be drawing at the end of this week) to aadityafrm for this excellent (IMO) explanation, bravo! Sorry Mahesh, this time, I think this answer better demonstrates the calculus:

Here is the solution, as a generic rule DV01 = - (1/10000) dP/dy,
[Since P.* Mod D /10000 = DV01 and Mod D = - (1/P) dP/dy, where P = F*exp(-yT)]
So local maxima for T can be obtained by putting d2P/dTdy = 0
Here, dP/dy = -F*T*exp(-yT)
So, d2P/dTdy = -F{exp(-yT) – y*T*exp (-yT)}
Equating above expression to zero, we get T= 1/y
In the give question, y=5%, which means T=20 yrs

Please note you can also get to the same place by maximizing DV01 w.r.t. maturity (T):

• Given DV01[zero, CC] = F*exp(-yT)*T/10,000, if F = 100
• d[DV01]/dT = d[exp(-yT)*T/100]/dT = [exp(-yT)*1 + exp(-yT)*(-y)*T]*1/100 = exp(-yT)*[1 - yT]*1/100, such that we are looking for:
• exp(-yT)*[1 - yT]*1/100 = 0 --> exp(-yT)*[1 - yT] = 0 --> 1-yT =0 --> Y = 1/y

The point of this is to highlight an interesting difference between duration and DV01 (Tuckman makes this point very well):

• While duration is an increasing function of maturity,
• In this case of a discounted (e.g., zero) bond, DV01 is not continuously increasing with maturity: since DV01 = D*P/10000, in the case of a deeply discounted bond, an increase in maturity does increase the (D) but tends to decrease the (P)