# effective duration calculation

Why did they decide to use 5% duration for this problem?? Does it matter what change in yield you use since one is not given, shouldn’t the duration be the same no matter what change you use? A non-callable bond with 18 years remaining maturity has an annual coupon of 7% and a \$1,000 par value. The current yield to maturity on the bond is 8%. Which of the following is closest to the effective duration of the bond? A) 11.89. B) 8.24. C) 12.72. D) 9.63. The correct answer was D) 9.63. First, compute the current price of the bond as: FV = \$1,000; PMT = \$70; N = 18; I/Y = 8%; CPT → PV = –\$906.28. Then compute the price of the bond if rates rise by 50 basis points to 8.5% as: FV = \$1,000; PMT = \$70; N = 18; I/Y = 8.5%; CPT → PV = –\$864.17. Then compute the price of the bond if rates fall by 50 basis points to 7.5% as: FV = \$1,000; PMT = \$70; N = 18; I/Y = 7.5%; CPT → PV = –\$951.47. The formula for effective duration is: (V- – V+) / (2V0Δy). Therefore, effective duration is: (\$951.47 – \$864.17) / (2 × \$906.28 × 0.005) = 9.63.

I don’t think it matters. I got the same answer when using a change of 100 bp. Remember that effective duration lies along a straight line so the price change p/bp change should be constant no matter what variable you use. If you were to calculate the convexity adjusted duration then you would need to know how many bp to use as a variable. Best, TheChad

I think it matters a bit - the only reason that you get the same answers is that the two secant lines you get are nearly parallel because we don’t have optionality or anything else other than plain vanilla convexity going on here. Try this near an inflection point and it matters a lot what increment you use.

I am a bit rusty on my calc., but since the slope of an effective duration curve is constant, does it even have an inflection point? I agree 100% however, if one were trying to calculate the true duration (see “convexity adjusted duration” above) then the actual bp change would be very relevant do to the convexity of the “true” yield to price curve. Please let me know if I am thinking about this wrong so I can adjust my understanding of duration if I need to. Thanks, TheChad

The curve I am talking about is the usual price vs yield curve. The effective duration is just getting the slope of a secant line on that curve. The secant is usually close to parallel with the tangent so it’s a good estimate of duration. Here we have two secants, one drawn at ±50 bp and one at ± 100 bp. The price vs yield curve can have an inflection point if it has negative convexity.

OK we are on the same page then…I just wanted to make sure I understood it correctly. Thanks for the clarification. Best, TheChad