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Term structure theories (Hull Chapter 4)

Thread starter #1
@David Harper CFA FRM First off- I want to make sure I am posting this question under the right thread.. :(:( if not, please point me to the right location...this was the closest thread I could find ... :(:(

So Hull Chapter 4 Practice Question - Hull 4.7
I hate to say I do not have a solid grasp on this one....and so am struggling with 4.7 . Also, unfortunately could not find the discussion thread for this topic - so hate to bug you again on this one @David Harper CFA FRM - you have been so kind and patient with me in my ignorant journey since my initial days on BT... Thanks to you I have come this far from where I was ...yet I have a long road to go( am sure)....

Also gratitude to any other member who might be kind enough to shed some light on this one... :)

I understand that if the Spot Rate Curve is upward sloping, then the Yield < Spot Rate @ Maturity...but am lost on where the Forward Rate would fall ....
hull 4.7-gargi.png

Edited by Nicole to post full Hull question so it shows up in search results (images do not show up in search results):

: The term structure of interest rates is upward-sloping. Put the following in order of magnitude:
  • The 5-year zero rate
  • The yield on a 5-year coupon-bearing bond
  • The forward rate corresponding to the period between 5 and 5.25 years in the future
What is the answer to this question when the term structure of interest rate is down-ward sloping?


When the term structure is upward sloping: c>a>b.
When the term structure is downward sloping: b>a>c.
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David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @gargi.adhikari I did move this, if you don't mind. Hull's question 4.7 seems like a good one (and classic finance-type question). With respect to the forward rate, from the Hull XLS I copied the 'forward rates' sheet below, but with two versions. The top panel is a steeply upward sloping zero rate curve (1%, 2%, 3%, ....). The implied forward curve is even steeper. The lower panel is the same except the year 5 zero is equal to the year 4 zero so that the zero rate curve is flat from year 4 to 5. Notice the corresponding one year forward rate, F(4,5), is then also 4.0%. For me the key is to go back to the no-arbitrage relationship that informs our solving for forward rates in the first place. In this case,

[1+s(4)]^4*[1+f(4,5)] = [1+s(5)]^5; i.e., the forward rate enforces, or is enforced by, the relationship between the spot rates.

So imagine our 4-year spot rate is 4.0%. If the f(4,5) forward rate is 4.0%, then I think it's easy to see why the 5-year spot rate must be also 4.0%. A forward that "matches" the 4-year spot rate "maintains" that spot rate over another year. But if the spot rate curve is upward sloping, then the 5-year spot rate is greater than 4.0%, in which case, we will require a forward rate greater than 4.0% to "pull up" the cumulative spot rate (as reflected on the right side of the equality). This, at least, is how I think about this intuitively. I hope that helps!

Thread starter #3
Hull Ch 4 Practice Question: Hull 4.12
For this I was trying to find the Yield by plugging the other variables into the calculator:
N = 3 yr --> 6 ( Semi-Annual)
PV = -$104 ( Price)
PMT = 8% Semi-Annual => $4
FV = $100 (Par Value)

Bond Yield 1/Y = 3.25%
So, Annualized Yield = 3.25% * 2 = 6.51 %

Why did this approach not give us the correct results in this problem....
Why did we have to do it by hand :-
4e -0.5y + 4e -1.0y + 4e -1.5y + 4e -2.0y + 4e -2.5y + 104e -3.0y = 100 --> This approach is adopted when are given the different Spot Rates for the different time periods and so we cannot plug them into the calculator and have to do it by hand this way ... but in this case, we need to find the Flat Yield, so why could we not plug the numbers into the calculator and have it spit out the Yield..?

I think I am making a mistake in assuming PV = -104...but that's the final Price as per the problem statement..what am I getting wrong here...:confused::confused::confused::eek::eek::eek:

P.S: Wish there was a discussion thread on these.. :-( so that I would not have to bug and invoke @David Harper CFA FRM ...my apologies....please let me know if there is any other platform that I might be missing to post questions on these Practice Questions... and many gratitude :):)


David Harper CFA FRM

David Harper CFA FRM
Staff member
@gargi.adhikari It's just compound frequency that is the difference. Of course, the by-hand calculation (where each cash flow is discounted; as in my exhibits above) is discounted spot rates such that it is perfectly flexible to the shape of the zero rate curve, while the calculator is returning the single-factor yield (yield to maturity, which is implicitly flat). However, if the zero rate curve is flat, then the yield is equal to each spot rate and they will return the same result except that the calculator is always assuming a discrete compound frequency (per your N = 6, semi-annual input). So, in your example, if you translate the calculator's semi-annual yield output of 6.5107% into its continuous equivalent--i.e., =LN(1+6.5107%/2)*2 = 6.40697%, and if you use this (y) is the long-hand, then you will get a price of exactly $104.00. I hope that helps!
Thread starter #5
Perfect ! Yea- my bad I had forgotten to convert the Discrete Rate of 6.51 % back to the Continuous of 6.41 % Thank you so much @David Harper CFA FRM once again for coming to the rescue ...
One last question though just to double check...
In Hull 4.12 Practice Question Footnote, you mention that :-
"I think solutions manual is mistaken. Should be:
4e -0.5y + 4e -1.0y + 4e -1.5y + 4e -2.0y + 4e -2.5y + 104e -3.0y = 100
With yield = 7.588%"

Should be disregard this because in problems so far, we have always equated all the periodic cash flows to the eventual price of the Bond which is given to be $104 in this case instead of the Par Value of $100...Could you please confirm that...or I might be missing a key conceptual thing somewhere in this :confused::confused::(:( ....so wanted to clarify ...


David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @gargi.adhikari My note there is incorrect, sorry :( You are correct that "Should be disregard this because in problems so far, we have always equated all the periodic cash flows to the eventual price of the Bond which is given to be $104 in this case instead of the Par Value of $100..." even my 7.588% appears to be an incorrect solution, yikes. We infer the yield from the traded or cash price (in this case, 104). We only substitute the par (aka, face) value in the rare case where we are looking for the par yield, because the par yield is (by definition) the yield that prices the bond to par. But in the above problem, where we don't have a zero rate curve, this would just lead us back (in circular fashion) to the coupon: if we take an 8.0% semi-annual coupon and solve for the (y) that discounts continuously, I get 7.84% continuous, and then translate that into semi-annual, I get =[exp(7.84%/2)-1]*2 = 8.0%. That is, a bond that prices to par will have a coupon rate equal to its yield. So my note that you've circled red really makes no sense because I will be solving for (y) that is the continuous version of the coupon rate. We definitely want to use the traded (cash) price of the bond to infer the yield. Sorry for the confusion, thank you!

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @sm@23 Yes, that is correct, please do not let the follow-on discussion about Hull 4.12 confuse you. My annotation on Hull 4.12 is mistaken. The issue here is between "yield" (which refers to yield to maturity) and "par yield." For practical purposes, we care much more about yield (YTM) and "par yield" is less commonly encountered, to be realistic.
  • Yield takes as an input the bond's observed (traded or market) price. It is the internal rate of return (IRR) assuming the price, so if the price changes, the yield changes. In Hull 4.12 above, the price is $104, such that the yield is N = 6, PV = -104, PMT = 4, FV = 100 and CPT I/Y = 3.26*2 = 6.511%. If the price increased to, say, $107 then the yield changes: N = 6, PV = -107, PMT = 4, FV = 100 and CPT I/Y = 3.72*2 = 5.44%
  • Par yield is the yield only under the special case (assumption) that the bond price is 100. I hope that helps!

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi sm@23 By traded price, I meant cash price. I usually write "traded" or "observed" to differentiate between "theoretical." In Hull 4.12, the cash price is used as an input to determine the yield; I probably should not have deviated from "cash price" but I'm used to the distinction between theoretical (aka, model price) and traded (observed) price. Traded might be additionally confusing because we also have the difference between a bond's cash price (aka, dirty price, full price) and the bond's quoted price (aka, clean price, flat price). When we solve for yield, we are assuming the cash (not quoted/clean/flat). So in this context, I do not mean to introduce any distinction between cash and traded. Hull's 4.12 correctly uses the bond's "observed or traded" cash price as an input into determining its corresponding yield (aka, yield to maturity). The '100' is simply the face (or par) value of the bond. Thanks!