# Bond prices and forward rate

#### td00553150

##### New Member
Hi guys,
I came across something on the textbook and found it very off.
The statement: 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 bond price rises.

Is this statement correct?

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#### David Harper CFA FRM

##### David Harper CFA FRM
Subscriber
Hi @td00553150 It's imprecise at best. We know what it's trying to say because it belongs to an ancient LO ("Assess the impact of maturity on the price of a bond and the returns generated by bonds"). What is true is the following:

If the forward rate is higher than the coupon rate, then the bond price decreases as maturity increases over the same period; e.g., if the forward rate, F(2.0, 2.5), is greater than the bond's coupon rate, then bond price decreases as maturity is extended from 2.0 years to 2.5 years. Please note that is equivalent to saying that the bond price increases as maturity is reduced from 2.5 years to 2.0 years, if the six-month forward rate beginning in two years exceeds the bond's coupon rate.

So the (first) problem with the statement is that the first clause ("In the case of an upward-sloping term structure, there will be a tendency") is unnecessary and not true (per the equivocating "tendency"): an upward-sloping term structure does not obligate a relationship between the forward rate and coupon rate. The (second) imprecision is that "bond prices rises" is ambiguous because it could refer to an extension of maturity (from 2.0 years to 2.5 years which is how Tuckman casts the example) or a reduction in maturity (from 2.5 years to 2.0 years which is probably natural for most of us because that's how time marches). I enjoy analyzing statements like this because it really forces us to understand and be careful with logic. I hope that's helpful

#### td00553150

##### New Member
Hi @td00553150 It's imprecise at best. We know what it's trying to say because it belongs to an ancient LO ("Assess the impact of maturity on the price of a bond and the returns generated by bonds"). What is true is the following:

If the forward rate is higher than the coupon rate, then the bond price decreases as maturity increases over the same period; e.g., if the forward rate, F(2.0, 2.5), is greater than the bond's coupon rate, then bond price decreases as maturity is extended from 2.0 years to 2.5 years. Please note that is equivalent to saying that the bond price increases as maturity is reduced from 2.5 years to 2.0 years, if the six-month forward rate beginning in two years exceeds the bond's coupon rate.

So the (first) problem with the statement is that the first clause ("In the case of an upward-sloping term structure, there will be a tendency") is unnecessary and not true (per the equivocating "tendency"): an upward-sloping term structure does not obligate a relationship between the forward rate and coupon rate. The (second) imprecision is that "bond prices rises" is ambiguous because it could refer to an extension of maturity (from 2.0 years to 2.5 years which is how Tuckman casts the example) or a reduction in maturity (from 2.5 years to 2.0 years which is probably natural for most of us because that's how time marches). I enjoy analyzing statements like this because it really forces us to understand and be careful with logic. I hope that's helpful

Thank you David, that clarifies and it helps a bunch.

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#### td00553150

##### New Member
Hi David,

I have another question related to this.
Question: The coupon rate on a five-year bond is higher than the forward rate between time 4.5 years and time five years. If forward rates do not change do you expect the bond price to increase or decrease during the next six months?
Answer: It will decrease. The decrease in value is the value of a lost forward rate agreement, which in this case is positive.

Is there anything wrong with this answer?
Thank you.

#### David Harper CFA FRM

##### David Harper CFA FRM
Subscriber
@td00553150 No, to me, nothing wrong, that's an accurate statement. Notice if i take my true statement above and switch the inequality, it matches your Q&A:
If the forward rate is lower than the coupon rate, then the bond price increases as maturity increases over the same period; e.g., if the forward rate, F(2.0, 2.5), is less than the bond's coupon rate, then bond price increases as maturity is extended from 2.0 years to 2.5 years. Please note that is equivalent to saying that the bond price decreases as maturity is reduced from 2.5 years to 2.0 years, if the six-month forward rate beginning in two years is less than the bond's coupon rate.

... although instead of the saying "The decrease in value is the value of a lost forward rate agreement, which in this case is positive," which might be clear to somebody else (and maybe I'm probably being dense but I'm not immediately clear on what is a lost forward rate), I'd probably explain with something like "because the coupon earned from 4.5 years to 5.0 years is greater than the forward rate, the 5.0 year bond earns an above-market return over those additional six months such that its price must be higher than the 4.5 year bond; i.e., the bond price must decrease from 5.0 to 4.5 years." Hope that helps,

#### td00553150

##### New Member
It is such a brain teaser to understand these things..
I think I'm almost getting to it, however there's one last thing I'm still confused with.
When it says "during the next six months", doesn't it mean from year 4.5 to year 5?
Also, based on the prevailing statement that bond price increases as maturity increases when coupon>forward.
How does the price of bond decrease during the next six months if given coupon > forward.

Thank you!

#### David Harper CFA FRM

##### David Harper CFA FRM
Subscriber
@td00553150 The relationship here is narrow, it asserts: if F(4.5, 5.0) < Coupon(%), then P(4.5) < P(5.0) under an assumption of unchanged term structure. That the 5-year bond has a higher price than the 4.5 year bond can be expressed either way: in the natural way time marches, as your Q&A implies, we can say the price decreases over the next six months, as it moves from current, higher P(5.0) to the future, lower P(4.5). Or, we can say the bond price is an increasing function of maturity, as in, its price increases as maturity is extended from 4.5 years to 5.0 years (although this is not how time marches). It is common to express dynamics as a function of increasing maturity, but of course on the other hand, as time marches, a bond's maturity naturally decreases. One of these expressions is likely to be counterintuitive if you prefer the other! These can be brain teasers but, they are less so when the question is carefully worded. This is the sort of question that begs to be carefully worded. Hope that helps,

#### td00553150

##### New Member
That perfectly explains, thank you David!

#### rajivbangalore25

##### New Member
@td00553150 The relationship here is narrow, it asserts: if F(4.5, 5.0) < Coupon(%), then P(4.5) < P(5.0) under an assumption of unchanged term structure. That the 5-year bond has a higher price than the 4.5 year bond can be expressed either way: in the natural way time marches, as your Q&A implies, we can say the price decreases over the next six months, as it moves from current, higher P(5.0) to the future, lower P(4.5). Or, we can say the bond price is an increasing function of maturity, as in, its price increases as maturity is extended from 4.5 years to 5.0 years (although this is not how time marches). It is common to express dynamics as a function of increasing maturity, but of course on the other hand, as time marches, a bond's maturity naturally decreases. One of these expressions is likely to be counterintuitive if you prefer the other! These can be brain teasers but, they are less so when the question is carefully worded. This is the sort of question that begs to be carefully worded. Hope that helps,

Hi David,

If the coupon rate is higher than the forward rate, wont more people buy this bond to take advantage of the higher coupon rate, which in turn will force the bond prices go upwards to a level where no arbitrage will be possible.

#### superturtle

##### New Member
Hi @David Harper CFA FRM , I see that you used 2 different approaches to get the semi-annual forward rate.

Firstly in P1.T3,Chp16,Pg18: you used the "longer method" ( the last equation in the page) to compute the semi-annual forward rate from C.C. zero rates.

Secondly in P1.T3,Chp20,Pg13: you calculated the semi-annual forward rate by converting from the C.C. forward rate. If I used this method for the case in Chp16, I get a different answer from the method used there.

The confusion is that I don't see the distinction in these two questions for me to decide which approach to use, which gives different answers.

Thanks
Ken

Subscriber

#### AUola2165

##### Member
Subscriber
Question:
The coupon rate on a five-year bond is higher than the forward rate between time 4.5 years and time five years. If forward rates do not change do you expect the bond price to increase or decrease during the next six months?

It will decrease. The decrease in value is the value of a lost forward rate agreement, which in this case is positive when long-maturity rates move up by less than short-maturity rates.

Could someone explain how this works? In my opinion, given that forward rates are expected to remain unchanged, the bond's coupon payments will continue to be higher than the implied market expectations for future interest rates and the price should increase.

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