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# Miller EOC Question 3-11

#### jshi

##### New Member
Hi @David Harper CFA FRM, there is also this Miller end of chapter practice question I had a question on. Please move if not in the right place.

Question 11 A $100 notional, zero coupon bond has one year to expiry. The probability of default is 10%. In the event of default, assume that the recovery rate is 40%. The continuously compounded discount rate is 5%. What is the present value of this bond? Answer for PV: e^-0.05 * 94 = 89.42 1) I understand that future value is expected value etc, but I don't recall the basic PV formula consisting of the exponential. Is this an exception when looking at bond returns? 2) How much should we pay attend to Miller's end of chapter practice questions? They are much simpler in nature to BT's but seem to be concept driven (e.g. proofs) and calculation heavy (calculating means and variances with >4 inputs). Slightly related to this, I was reading another thread that knowledge of calculus is not expected, however a lot of the Miller and BT questions result in calculation of integral of pdf? Thank you! #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @jshi 1. This is the continuous equivalent (or analog) to the discrete form that you are expecting. When the rate, r, is discrete, PV*(1+r/k)^(k*T); e.g., if the discount rate is 5.0% per annum with semi-annual compound frequency, then because$89.47055*(1 + 0.050/2)^(2*1) = $94.00, it's therefore true that the PV of$94.00 received in one year discounted at 5.0% with semi-annual compound frequency is given by PV = \$94*(1+5%/2)^(-2*1.0). As k increases (i.e., as the compound frequency or periods per year) in FV= PV*(1+r/k)^(k*T), the solution converges on FV = PV*exp(rT) such that PV = FV*exp(-rT) under continuous compounding.
2. It's hard for me to say, I wish GARP would opine on this. My sense is that Miller's EOC are not much representative of the exam, and instead helpful directly to reinforce the chapter's material. These may seem like identical ideas, but I don't think they are. I agree with you that they tend to "concept driven (e.g. proofs) and calculation heavy (calculating means and variances with >4 inputs)" and therefore not highly predictive of GARP's exam questions. Re calculus: actually, we may have several calculus questions, but in the scheme of things, we don't really have a lot of them. We have them where the LOs point to such material in Miller. I presume you say the response I got from Bill May (of GARP) on this question at https://www.bionicturtle.com/forum/...o-know-calculus-for-the-frm.13435/#post-57562 i.e., where Bill replied to my question:
"2) Specifically regarding calculus, the FRM Exam does not explicitly require an understanding of calculus and, as a perusal of the learning objectives will indicate, does not test directly on differentiation, integration, or other aspects of calculus. That being said, an understanding of calculus, not to mention matrix manipulation that is typically associated with linear algebra, would likely prove beneficial to candidates as they prepare for the Exam. Many of tools and techniques related to probability, statistics, modelling, and estimation draw on these concepts and the better candidates understand these topics the better prepared they are likely to be. I hope that helps." -- Bill
Thanks!