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# Bond price vs Forward Rate

#### nag_san

##### New Member
Hi, In Part 1 Valuation and Risk Models under 10.4 GARP material states the following. Does this apply only to Forward Rates, as usually Interest Rate (Presumably Spot Rates) and Bond prices are negatively correlated (ie if Interest rate > Coupon rate of the bond, then Bond price falls). The below suggests that if Forward Rate > Coupon, then Bond Price increases. If there is an existing thread on this topic, please advise as I could not locate any.

In the case of an upward-sloping term structure, there will be a tendency for the forward rate to be higher than the coupon so that the bond price rises.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nag_san This is GARP's new re-write of Tuckman's Chapter 2 (Maturity and Price or Present Value) and is trying to explain the first part of this LO ("Assess the impact of maturity on the price of a bond and the returns generated by bonds"). Specifically, this refers to
"Assess the impact of maturity on the price of a bond ..."

I will need to re-read the chapter later because, at the moment, I think I disagree with the emphasized portion of "In the case of an upward-sloping term structure, there will be a tendency for the forward rate to be higher than the coupon so that the bond price rises." But it might be just because it's only in the context and the explanation in this section is weak. I do know what it's trying to say, however, because it's an old LO and the source is effectively Tuckman.

The concept is illustrated here in question #1 https://www.bionicturtle.com/forum/threads/weekly-trivia-4-28-14-duration-dv01-and-convexity.7776/

But here is a simpler example. I will use continuous compounding to make things easier. Assume a upward-sloping term structure, super convenient: 1.0% at 1.0 year, 2.0% at 2.0 years, and 3.0% at 3.0 years (i.e., spot rates with continuous compounding). Annual only, because my bond(s) will be annual payers. The implied forward rate term structure is also convenent: F(1,2) = 3.0% and F(2, 3) = 5.0%. That's our upward-sloping term structure.

Now price two 3-year annual pay bonds:
• 4.0% annual pay coupon bond price (3-year, continuous zero rates) = $102.85; i.e.,$4*exp(-1%*1) + $4*exp(-2%*2) + 4*exp(-3%*3) = 102.85 • 6.0% annual pay coupon bond price (3-year, continuous zero rates) =$108.52
Now let one year expire (forward in time ...) with unchanged term structure:
• 4.0% bond (now 2 years to maturity) price increases to $103.88. The idea is this bond's price rises because the forward rate, F(2,3) = 5% is greater than the 4.0% coupon; but • 6.0% bond price decreases to$107.78 because the 6.0% coupon is higher than the 5% forward rate (the same forward rate over which the bond "elapses" from 3 to 2 years).
The sentence you quoted is confusing because an upward-sloping term structure per se does not imply the forward rate is higher than the coupon rate. I haven't sufficiently analyzed the entire section to validate my perspective, but if were to edit into a logically correct standalone sentence then I would write (eg) the following complicated sentence in order to be precise:
"When the term structure is upward-sloping and our scenario is unchanged term structure, over the next period of time the bond price increases if the final forward rate (e.g., the six-month forward rate beginning in 1.5 years as an initial two-year semi-annual pay bond ages six months; the one-year forward rate beginning in 3.0 years as an initial four-year annual-pay bond ages one year) is greater than the bond's coupon rate as the bond ages (i.e., as its maturity reduces) by one period."

BTW, here is a question to provoke thought: why isn't my 4.0% bond pulling to par?

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#### nag_san

##### New Member
@David Harper CFA FRM I know you said you'd review later but this is already incredibly helpful. The sentence unchanged term structure after a year or so is what was missing in the content, I think. Even then I may have still needed some examples to understand better. With your examples and the now clear "Unchanged Term structure" sentence, I believe I got it. I may sound greedy but will wait for your further response whenever you get a chance with anything you may wish to add additionally. Thanks a bunch for your assistance as always.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nag_san thank you! The first sentence of the section (Maturity Effects) reads "What happens to the price of a bond if the term structure remains unchanged over a six-month period?" so that assumption is there, but I will tell you that I could not follow the rest of the new text, on its own, as it is written. It's obviously re-written from Tuckman's Chapter 2 (Maturity and Price or Present Value; Tuckman does give an illustrated example) but I don't see how anybody could get the idea without an illustrated example (this is my view of many concepts in the new material. Some people have observed that it's shorter, as an advantage, but I've argued that it's shorter mostly because many illustrated examples have been omitted; for me, that's a big deduction because I need illustrated examples). I wrote my summary sentence of the dynamic, as best I could, above, but still its dense even when edit-trimmed.... stay tuned!

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