Application of Yield Based Convexity to Table 12-6 pg 223 Tuckman

[Dr. Jayanthi Sankaran]

Hi David,

I have the following questions:

(1) Will the FRM require us to apply equation (12.49) on page 226 to data such as those on Table 12-6 pg 223 of Tuckman for yield based convexity?

(2) On time-weighting the Present Value in Table 12-6 by (t/2)*(t+1)/2, I get the convexity = 27.98 while Tuckman gets convexity = 25.29. I don't know if this is because he time weights it by t^2.

(3) On using equation (12.51) on page 226, for a five-year zero coupon bond yielding 2.092%, I get the same convexity as Tuckman = 26.93

(4) The convexity of the five year zero coupon bond = 26.93 > convexity of the five year coupon bond = 25.29 as in Tuckman. This is what one would expect.

(5) However, in my case the convexity of the five year coupon = 27.98 > convexity of the five year zero = 26.93 which is counterintuitive.

Is this because of rounding errors?

Thanks!
Jayanthi

 

[David Harper CFA FRM]

Hi @Jayanthi Sankaran

No way would the FRM require you to apply that convexity formula. Way too tedious for an exam. With respect to convexity, exam may query concepts and, quantitatively, might expect you to know that a 5.0 year bond has convexity in the rough neighborhood of 5^2. I input into my XLS, see here @ https://www.dropbox.com/s/tudfyvu3y54m166/09-17-convexity.xlsx?dl=0

1. First I just did it my way (Tab = David), by weighted the squares of time. Macaulay convexity is the weighted average of maturities-squared (to me this is intuitive and easy to remember). I get a Mac convexity of 23.435, such that modified convexity would be less (I get 22.95). I am not sure why this is lower than Tuckman's ... I mean, I see that he doesn't exactly weight the squared maturities, but I don't understand those adjustments
   
2. But in the second tab (see exhibit below), I replicated Tuckman's modified convexity (aka, convexity) of 25.2887. But I am not sure this is correct: a 5-year zero-coupon bond has (macaulay) convexity of 25.0 years-squared and like Mac duration, is yield invariant (!), I am pretty certain. Modified convexity would be less, under discrete compounding. And coupon-bearing would be less. So, I am not sure I agree with greater than 25. I hope this helps!

 

[Dr. Jayanthi Sankaran]

Hi David,

Thanks for taking the time to construct the above spreadsheet - I appreciate it

(1) For the five year coupon bond:
Like you, I get the Macaulay Duration = (1/100.16)*477.76 = 4.77.
However,
the Macaulay convexity = (1/100.16)*2586.07 = 25.82
and the Modified convexity = Macaulay convexity/(1+y/2)^2 = 25.82/(1 + 2.092%/2)^2 = 25.29 as in Tuckman

(2) For the five year zero coupon bond I get,
Macaulay convexity = T(T + 0.5) = 5*5.5 = 27.5 and the
Modified convexity = T(T + 0.5)/(1 + y/2)^2 = 27.5/(1 + 2.092%/2)^2 = 26.93 as in Tuckman.

i.e Modified convexity of the five-year zero coupon = 26.93 > Modified convexity of the five-year coupon bond = 25.29

This is intuitive since a coupon bond has some of its present value in earlier payments, and since the convexity contributions of those payments are less than that of the final payment at maturity, a coupon bond will have a lower convexity than a maturity- and yield-equivalent zero.

Hope that helps resolve the confusion on both sides
Thanks!
Jayanthi

 

[David Harper CFA FRM]

Hi @Jayanthi Sankaran Yes, totally agree, thank you!