P1.T3 Hull Chapter 4 reading practice question 2

[desaiha]

David,

The question provides the spot curve with continuous compounded spot rates. and asks for a 2 year par yield. Wouldn't the solution be 2 year rate (.i.e 5%) why is the solution treating the spot curve as semi-annual compounding and calculating and equivalent continuous par yield. Don't quite follow this reasoning?

Thanks

 

[Antibiotix]

Hi, unfortunately I am not replying with a solution. In fact I am perplexed by the same problem above. Any help clarifying would be appreciated. Or perhaps point me to a solution somewhere else in this forum ? Thanks.

 

[David Harper CFA FRM]

Hi @Antibiotix The source question is here @ 158.3 https://forum.bionicturtle.com/threads/l1-t3-158-bond-price-using-spot-rates.4481/
... the question is that way simply because it intends to follow Hull's Par Yield in example 4.4. In Hull's example, the spot rate curve is characteristically given with continuous compounding, yet a par yield (coupon rate) is semi-annual (matching the intervals of the curve). Although, the question should specify semi-annual coupons (sorry). Please note that some conversion is necessary: the spot (zero) rates are expressed as continuously compounded; however the coupon rate must be (or at least will be) discrete, such that the par yield will necessarily be solved with a discrete compound frequency. I hope that explains. Thanks,

 

[Milan]

Hi.

I am having a problem with this question also.
You say in the solution "A 5.0% semiannual coupon rate is the solution that prices that bond exactly at par, given this theoretical spot rate curve."
Could you please elaborate on this - why 5%?
Also, as I do not have quantitative background, I was not aware what is "par yield" and needed to search on the internet about that. Do we need to complete some prerequisites before starting with Bionic Turtle readings, or is BT sufficient to finish FRM?

Thanks.

 

[David Harper CFA FRM]

Hi @Milan

No you don't need to complete some prerequisites, our material is based on the assignments. In the case of par yield (or par rate), it is one of those topics in the FRM with a flexible history. It's in Hull Chapter 4 but really assigned in Tuckman Chapter 2 (not that its specific locations matter, the definition is the same). Rather than being a basic prerequisite, I would characterize "par yield" as a difficult (if not highly difficult concept) that also tends to elude the exam. I am actually pretty happy with this old youtube video I recorded which attempts to explain how the "par yield" can be viewed as a special case of the yield to maturity ("yield") which is the far more important concept for the FRM. (see below)

You ask why 5%. Here is how I look at is. Well, we are given a spot rate curve; the spot rate curve (equivalent to a set of discount factors) is the "truth" in the marketplace. The drawback of the spot rate curve is that it is several rates, one for each maturity. In the question, the spot rate curve is 2.0% at 0.5 years; 3.0% at 1.0 year; 4.0% at 1.5 years; and 5.0% at 2.0 years. True, but not very compact. The advantage of yield to maturity (aka, yield) is that, by definition, a yield is only one rate! It summarizes all of the information in the spot rate curve, or at least summarizes all of the rates up to a given maturity. However, the one "problem" with yield is that it varies by the bond's cash flows, it is not independent of the bond. So, given a spot rate curve up to a maturity of T.0 years, we cannot say there is a single corresponding yield associated with maturity of T.0 years as there are different yields for different bonds (with different coupon rates). However, we can say there is a single par rate at maturity T.0 years. So, it's the best of both worlds! It is yield which impounds all the information of the spot rate, and among different yields, it is the yield which matches a coupon rate which prices the bond to par; i.e., at each maturity it is the coupon rate for a bond that would price. (All of that said, due to I think conceptual difficulty, it remains of low testability). That is how I think about it, and I think Aaron Brown here makes a most excellent summary statement: "The confusing thing is that "yield to maturity" is a property of a bond at a specific market price. "Par yield" is a property of a yield curve at a specific maturity." http://www.wilmott.com/messageview.cfm?catid=8&threadid=5582

Here is the relevant, assigned Tuckman: "Par Rate: Consider 100 face or notional amount of a fixed-rate asset that makes regular semiannual coupon or fixed-rate payments of and a terminal payment at year T of that 100. The T-year, semiannual par rate is the rate such that the present value of this asset equals par or 100. But that is exactly the definition of swap rates given earlier in this chapter. Hence, swap rates in Table 2.1 are, in fact, par rates."

And my youtube video: