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# P1.Ch16 Hull 4.22

#### thanhtam92

##### Active Member
Subscriber
Hull.4.22:
A 5-year bond with a yield of 11% (continuously compounded) pays an 8% coupon at the end of
each year.
What is the bond’s price?
What is bond’s duration?

Instead of setting the CF table as shows in the Answer, I am trying to use my calculator to calculate the price given 2% change in yield, and apply the following formula D =(-delta_B/b)/delta_y. However, I did not get the 4.256 as duration or even the bond price.

I am using HP12C and the setup is below
n = 5, i = 11%, PMT = 8, FV = 100, PV = 92.165
n = 5, i = 10.8%, PMT = 8, FV = 100, PV = 92.81

D = (-(92.165-92.81)/92.165)/0.2% = 3.49 years

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @thanhtam92 Your prices assume annual compound frequency, so to use your approach we'd want to translate the 11.0% (with CC) into exp(11%)-1 = 11.628% per annum with annual compound frequency. Note that we can confirm: n = 5, i = 11.628%, PMT = 8, FV = 100 and → CPT 86.01. Then, if we use a shock of 20 bps, Δy = 0.20%, the effective duration is given by:

($87.466 -$86.143)/(.2%*2) * 1/$86.801 = 3.8127%, is what I get. Okay but that is effective duration which approximates modified duration. But Hull's solution is for Mac duration (aka, weighted average maturity) such that 3.8127% (1 + 11.628%) = 4.256 years ... and, we have reconciled to his answer. It's instructive, we learn three things at least: • different compound frequencies can be resolved if we translate • effective duration is an approximation of modified duration (see other forum conversations: it retrieves a secant that approximates the tangent; the tangent's slope is the "true" dollar duration) • we can translation back/from from mod to Mac duration with mod D = Mac D/(1+y/k). I hope that's helpful, #### thanhtam92 ##### Active Member Subscriber Hi @thanhtam92 Your prices assume annual compound frequency, so to use your approach we'd want to translate the 11.0% (with CC) into exp(11%)-1 = 11.628% per annum with annual compound frequency. Note that we can confirm: n = 5, i = 11.628%, PMT = 8, FV = 100 and → CPT 86.01. Then, if we use a shock of 20 bps, Δy = 0.20%, the effective duration is given by: ($87.466 - $86.143)/(.2%*2) * 1/$86.801 = 3.8127%, is what I get.

Okay but that is effective duration which approximates modified duration. But Hull's solution is for Mac duration (aka, weighted average maturity) such that 3.8127% (1 + 11.628%) = 4.256 years ... and, we have reconciled to his answer. It's instructive, we learn three things at least:
• different compound frequencies can be resolved if we translate
• effective duration is an approximation of modified duration (see other forum conversations: it retrieves a secant that approximates the tangent; the tangent's slope is the "true" dollar duration)
• we can translation back/from from mod to Mac duration with mod D = Mac D/(1+y/k). I hope that's helpful,
thanks a lot @David Harper CFA FRM . I am not aware that we have to transfer to frequency compounding to use the calculator. And in the exam, would it clarify which duration we should calculate? Or we just assume it is Mac duration unless the security has an embedded option

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @thanhtam92 You don't necessarily need to translate between compound frequencies. Hull's problem 4.22 can be (easily) solved for the answer he seeks (\$86.80) but you can't use the TVM worksheet (N, I/Y, PV, PMT, and PV) because they presume discrete periods. I was just responding to your approach using the TVM. Re: duration: yes the exam should be specific although (i) often the context implies the duration wanted and (ii) often the difference is not material; we've really pushed a lot of feedback up on these points. GARP has gotten a lot better about choices (A, B, C, D) that are not proximate to each other such that a typical duration Q&A won't "break" based on the Mac/modified duration choice. Most of our applications (e.g., estimate price change given yield shock) want the modified duration where the effective duration is a reliable approximation (of the modified duration). In exam questions, the appearance of Mac duration (aka, weighted average maturity) is typically out of convenience when the questions wants the duration of a T-year zero-coupon bond. That's popular because we know the Mac duration is T.0 years and the modified duration is T/(1+y/k). But the question has the burden to be specific. Thanks,