Bond Duration and Convexity

[LeeBrittain]

Hi David,

Out of all the topics covered in Part 1, this topic seems the most difficult for me to grasp for some reason. Which formulas are the most appropriate to use for duration and convexity and DV01? It seems like to me that the formulas used to calculate duration and convexity are different in the Financial Markets and Products notes vs. the Valuation and Risk Models notes.

Here are some of the issues i've come across....

BT Practice Problem 161.3) The estimated change in bond price is 2.75 given the 3-year, 4% annual coupon bond, FV 100, YTM 6% and a 100 bps drop in rates. I calculated duration with the (v- - v+)/2*Vo*(change in yield) and came up with -2.715. I calculated convexity as 5.29. I then try to calculate the change in the bond price based on -d* change in yield * price + price* convexity (change in yield) ^2. What is the price we are supposed to use in this calculation? Vo? V-? face value? The way I calculated it, I used Vo for price, so change in bond price= (94.65) * 2.715*.01 + 94.65*5.29*.0001, which gives me an incorrect answer of 2.62 for the change in the bond's price. Where did I go wrong?

Schwesser Exam 2 Problem 24. You are given a bond with: 1,000 par value, 4% semiannual coupon, YTM 5%. Calculate convexity given a 5 bp change in yield. The correct answer has convexity calculated with 1 in the denominator instead of 2 (i.e. (V- + V+ - 2 Vo) /( Vo * (change in yield)^2). Is there a standard formula that FRM prefers in calculations of bond convexity (i.e. if we are solely asked to calculate a bond's convexity instead of the convexity adjustment to determine the change in bond price, should we assume 1 or 2 in the denominator)?

Thanks,
Lee

 

[David Harper CFA FRM]

Hi Lee,

It's a great point. At the formula level, the FRM uses different durations, no way around it. But, with respect to convexity, you are much more likely to be asked to apply convexity in the adjustment, than calculate the measure, fwiw. Schweser shows it correctly, but as you will see, you won't be asked that question, really, because its double (2X) is also a fair answer.

Duration/convexity can be entirely arbitrated by the calculus. The calculus reveals but one idea, where variations are a matter of "how do we estimate" (or compound frequency, as usual!).

So, let me just copy the Taylor Series (Jorion's FRM handbook 6.9, tweaked notationally):
where DY= yield change, P(0) = bond price, and D = mod duration:
Bond Price Change est. $ = dP/dy * DY + 0.5 * d^2P/dy^2 * DY^2; i.e., Taylor Series
Bond Price Change est. $ = (slope) * DY + 0.5 * (rate of change of slope) * DY^2, and since slope = dollar duration = -D*Price(0):
Bond Price Change est. $ = - mod duration * P(0) * DY + 0.5 * CONVEXITY * P(0) * DY^2; this Taylor arbitrates

In regard to duration, the cause for differences arises:

• Macaulay duration (per my 161.3 that you cite) solves for the weighted (by PV cash flows) average maturity of bond. The Macaulay years can be (as in my solution) calculated exactly, given a compound frequency.
• Modified duration is often based from the Mac duration as Mod = Mac /(1+y/k). In this way, mod duration can inherit the precision of the Mac duration for a given price/yield
• Effective duration (your formula below) is just an way to approximate mod duration, if our best way is to simply re-price the bond (or if we NEED to re-price because the cash flows change with the yield, the "academic" distinction). Calculus-wise, effective duration looks for the slope secant of a line near to the tangent; it's an approximation, and unlike Mac duration, there are infinite proximate valid answers.

I go to the trouble to anchor it, just to show how i do the duration. (To be honest, I don't have the effective duration/convexity formulas memorized, as i rely entirely on the intuition, I don't bother to memorize them). For the duration, this is my sequence, if we just take my own 161.3 question as an example:

I konw want the slope of tangent line to P/Y curve and slope = rise/run.
Using a huge shock of 1% around 6%: get higher price = Price bond @ 5% = 96.94; get lower price @ 6% = 91.51.
That is all i need for slope which is rise/ run = (96.94 - 91.51) / (7% - 5%) = 271.51. This slope is dollar duration = D*Price(0),
So D = 271.51 / 94.18 = 2.88

I am using your formula, but we can resolve formula formatting issue with one insight: the slope is rise/run and we just need to be consistent about the rise/run. This non-profound realization renders us impervious to formatting variance:
D = rise/run * 1/P = [P(-DY) - P(+DY)]/(2*DY) * 1/P

And again, while this effective duration will approximate the modified duration, this is a brute-force calculation (we are picking two nearby points on the P/Y curve) so it's an approximation will an infinite number of valid answers (varies with each DY!).

In regard to your 161.3, I think the error is, it looks like you adjusted convexity in the wrong direction: for a 1% yield drop, your duration correctly implies a $2.71 increase in bond price. Please note that convexity must increase this CHANGE IN PRICE, so it can't go down to 2.62. I think if you add convexity, you are pretty close?

In regard to your EFFECTIVE convexity (measure): C = (V- + V+ - 2 Vo) /( Vo * (change in yield)^2)
Yes, that is probably the best default because it plugs into the Taylor Series above; i.e., if we calculate (C) per your formula then:
Convexity adjustment = 0.5 * C * P(0) * DY^2.
However, now you can see why (as Fabozzi says the convexity measure is "meaningless") it is equally fine to use:
C_alternative = (V- + V+ - 2 Vo) /[2 * ( Vo * (change in yield)^2)], if we use the following adjustment:
Convexity adjustment = C_alternative * P(0) * DY^2.

I hope that explains, David

 

[LeeBrittain]

Hi David,

It looks like I made my mistake on the way in which the bond prices are calculated. For example, I came up with a Vo of 94.65, but your answer is 94.18. If I input i=6, n=3, 100=Fv, 4=Pmt, I get 94.65, but with the continuous compounding rates this is an incorrect calculation I assume? I think this difference in prices lower my calculated change in bond price to the incorrect answer of 2.62.

Thanks,
Lee

 

[David Harper CFA FRM]

Hi Lee,

Yes, exactly. Under annual compounding, I get $94.65 versus $94.18 under continuous (and this will definitely impact the sensitivity calculations).

FWIW, I used continuous in T3.161 because these AIMs reference Hull Chapter 4 and Hull uses continuous, generally and here (which is the only case where you can use the term "duration" without the need for the modified/Macaulay qualifier).

Thanks, David