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Question around Interest Rates

Roshan Ramdas

Active Member
Hi All,

Have 2 queries surrounding the below question from the document qr12.p1.t3.hull_134_185_v5 (page 103).

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Hull.04.03: The 6-month and 1-year zero rates are both 10% per annum. For a bond that has a life of 18 months and pays a coupon of 8% per annum (with semiannual payments and one having just been made), the yield is 10.4% per annum. What is the bond’s price? What is the 18month zero rate? All rates are quoted with semiannual compounding.

Answer: Suppose the bond has a face value of $100. Its price is obtained by discounting the cash flows at 10.4%. The price is 4/1.052 + 4/1.052^2 + 4/1.052^3 = 96.74 If the 18-month zero rate is R, we must have 4/1.05 + 4/1.05^2 + 104/ (1 + R/2)^3 = 96.74 which gives R = 10.42 ________________________________-_______ 1.We seem to have discounted cash flows using the discrete compounding method as opposed to continuous compounding (Cash Flow.e^-yt). Are there any ground rules surrounding the method of discounting to be used please ? 2.Also,...the question specifically highlights that 1 of the semi annual payments have been made I.e., 12 months to maturity. Yet the bond price being shown in the solution seems to be arrived at by discounting cash flows across all 18 months. Why is this the case please ? Lastly,...there seems to be a typo with the cash flow received in the last 6 months.....should be 104 as opposed to 4. Thank you, Roshan David Harper CFA FRM David Harper CFA FRM Staff member Subscriber Hi Roshan, Sorry for delay (I realize you asked this 1.5 weeks ago). This is a Hull question, btw (Hull.04.03 signifies that the first such question, in this case there is only one, is from Hull's textbook). 1. Re: compound frequency ground rules. This has been extensively discussed in the forum, and shared with GARP. The upshot: there are conventions associated with various instruments, and it is smart to be aware of them, however, there is no "ground rule" except the question must specify the assumption (and GARP will. Historically they specify an overall exam assumption; e.g., annual compounding, but they know to specify). And this question is good because it does specify semi-annual (discrete): "The 6-month and 1-year zero rates are both 10% per annum ... All rates are quoted with semiannual compounding." 2. Re: "the question specifically highlights that 1 of the semi annual payments have been made I.e., 12 months to maturity" I disagree but I see why you make that interpretation. First, the question says "For a bond that has a life of 18 months" and then, the potentially confusing clause is "one having just been made" which Hull inserted merely to confirm that the next coupon is not tomorrow but rather in 6 months. An exam question would not typically contain this clause, it would be assumed by the 18 month life; e.g., 2 years to maturity with semi-annual coupons always connotes next coupon in six months (not tomorrow). Re: Typo: I agree, the last cash flow clearly should be 104 (and is required to produce the 96.74). Tagged for non-urgent revision. Thanks, Jhoony New Member Subscriber Hi David I was searching for some specific forum chapters linked on following questions, but I couldn,t find them. I hope posting them here is ok. I have some additional comments about some questions in R20.P1.T4 Tuckman question set: 1. Question 26.2 (page 69 – 70) Is there a much simpler way to calculate the duration? I tried to formulate the PAR BOND such that the coupon equals YTM. After that I just changed the YTM in the BAII calculator (+/- 1bp) and got the PV(-1bp) and PV(+1bp) from which I computed the duration through the standard formula (PV- -PV+)/(2PV delta YTM). I got the result of 8,19 for MD and 8,35 for Mac. D., which is slightly different than the results in the answers set. 2. Question 28.2 (page 73) The YTM is probably 5% if we look at the answer and not 6% as assumed in the headline of the question. Why do you always discount the zero bond Mac. Duration with 1+r/2 and not only with 1+r? Why is the PV= ( 1/(1+5%/2)^30 )*100 if this is a zero coupon bond? Shouldn,t be just (1/(1+5%)^30 )*100? 3. Question 29.2 (page 76) Probably just a typo, but i think the answer under semiannual should be 98,97 and for annual 97,90 (just vice versa). And the last one J. I just want to check my understanding of duration and convexity. If the compounding frequency decreases (from continues compounding to semiannual or annual compounding) the convexity value of a bond also decreases? Is this reasoning OK? Thank you very much for you answers. David Harper CFA FRM David Harper CFA FRM Staff member Subscriber Hi @Jhoony Good points, all of them! Please note at the end of each answer there is a link to the source Q&A; e.g., "Discuss in forum here: http://www.bionicturtle.com/forum/threads/l2-t5-26-parperpetuity-duration.3450/" 1. Yes, yours is good and yours is how it should be done. The given answer, referencing a Tuckman formula, uses a formula because an analytical function happens to be available. The answer is the Q&A is therefore a precise "modified duration;" yours is an "effective duration" which approximates. We don't expect it to equal as you've computed the slope of a very nearby secant line as an approximation to the tangent line. I don't think GARP ever expected anybody to memorize the analytical duration of a par bond. Your approach is best. Source @ https://www.bionicturtle.com/forum/threads/l2-t5-26-par-perpetuity-duration.3450/ 2. Yes, it's my mistake and the PDF revision is overdue (sorry), see source at https://www.bionicturtle.com/forum/threads/l2-t5-28-dv01-duration-and-price-effects.3454/ With respect to price, I'm sourcing Tuckman and he always uses semi-annual compound (discount) frequency. Either way is okay, and you have an excellent point (implied): if there are no coupons, there is no natural reason to assume semi-annual frequency. I agree. The question needs to be specific. 3. Yes, my typo, see https://www.bionicturtle.com/forum/threads/l2-t5-29-convexity.3456/ Re: "If the compounding frequency decreases (from continues compounding to semiannual or annual compounding) the convexity value of a bond also decreases?" • No and yes. It's analogous to duration: "macaulay convexity" (similar to mac duration) will be unaffected by compound frequency, but "modified convexity" (similar to modified duration) will decrease as compound frequency (k periods per year) decreases. Although this detail (esoteric) has never reached the exam; i.e., I don't think the FRM has ever gone beyond the one definitional "flavor" of convexity in any specific way. I hope that's helpful, thanks, Lapoboss New Member Subscriber Hi @David Harper CFA FRM CIPM Sorry about Hull.04.03, how you get to 4/1.052 + 4/1.052^2 + 104/1.052^3 = 96.74 ? What's the formula? Specially how you arrived to 1.052. Thanks, Lapo David Harper CFA FRM David Harper CFA FRM Staff member Subscriber Hi @Lapoboss Source is here @ https://www.bionicturtle.com/forum/threads/hull-04-03.3743/ i.e., Question: The 6-month and 1-year zero rates are both 10% per annum. For a bond that has a life of 18 months and pays a coupon of 8% per annum (with semiannual payments and one having just been made), the yield is 10.4% per annum. What is the bond’s price? What is the 18-month zero rate? All rates are quoted with semiannual compounding. Answer: Suppose the bond has a face value of$100. Its price is obtained by discounting the cash flows at 10.4%. The price is
4/1.052 + 4/1.052^2 + 4/1.052^3 = 96.74

If the 18-month zero rate is R, we must have
4/1.05 + 4/1.05^2 + 104/ (1 + R/2)^3 = 96.74
which gives R = 10.42%.

1.052 = 1 + 10.4%/2
This is a good question from Hull. We need to use the yield to retrieve the price (the yield is rate which discounts all cash flows), and then the price is used to retrieve the 1.5 year zero rate. Thanks,