Impact of maturity on bond price

[afterworkguinness]

Hi David, I'm having trouble wrapping my head around this statement

"More generally, price increases with maturity whenever the coupon rate exceeds the forward
rate over the period of maturity extension."

Can you give me an idea of why this is ? Thanks !

 

[afterworkguinness]

Hi David,
I would be so grateful if you could find time to look at this by Friday night. I can imagine how busy you are this week; your help over the past few months has been invaluable. Thanks !

 

[David Harper CFA FRM]

Hi afterworkguinness sure thing, I'd love to have better text for this always-difficult idea. Here is a draft of what I've come up with just now (the relevant Tuckman, who is assigned, is copied below):

To illustrate, let's assume annual compounding (discounting). Start with a coupon-bearing bond of face value, F, that pays a coupon rate, c, with maturity of years, T, and yield, y.
For example, $100 par, 5-year bond, with 3.0% coupon and yield of 4.0%. The price = -PV(4%,5,100*3%,100) = $95.55; note: as c<y, it's good to mentally check that price < par.
Now split this bond into its coupons and principal (par, F), in PV terms:

• The single $100 principal cash flow is worth, in PV terms, $100/1.04^5 = $82.19; therefore
• The series of coupon cash flows must be worth $95.55 - $82.19 = $13.36; in a sense we've virtually created a P-STRIPS + 5 C-STRIPS

Now, let's ask the question: if we want the bond price to remain unchanged (e.g., $95.55) while we extend the maturity (e.g., from 5 years to 6 years), what will be required of the one-year forward rate, f(5,6)?

• The new 6-year bond has the same "first 5" coupon cash flows. The only cash flow difference is:
• The new 6-year bond has an additional coupon on the 6th year plus the par shifts from year 5 to year 6
• The bond price therefore will be unchanged if if PV(F, year 5) = PV(c*F, year 6) + PV(F, year 6)

Therefore:

• As only the forward rate is introduced, the bond price will be unchanged if FV(F, year 5) = Discounted to Year 5:[FV(c*F, year 6) + FV(F, year 6)]; i.e., because the 5 coupon cash flows[Years 1 to 5] are already identical, we only require that the (principal + coupon) in year 6, discounted to year 5, equals the principal in year 5
• Symbolically, as c*F[Years 1 to 5] already matches, price is unchanged if: F = (F + $coupon)/[1+f(5,6)] = F(1 + c)/[1+f(5,6)]
• F = F(1 + c)/[1+f(5,6)] --> (1+c) = [1+f(5,6)]. In words, if the coupon rate equals the forward rate, then discounting from (T+x) back to (T) ensures that the FV is par at (T) which corresponds to the price effect on on the original bond.
• Similarly, if c>f(5,6), then F(1 + c)/[1+f(5,6)] > F, which will increase the price and if c<f(5,6), then F(1 + c)/[1+f(5,6)] < F, which will decrease the price.

Here extracted Tuckman on same for the context:

"Extending maturity from six months to one year, the coupon rate earned over the additional six-month period is 4 7/8%, but the forward rate for six-month loans, six months forward, is only 4.851%. So by extending maturity from six months to one year investors earn an above-market return on that forward loan. This makes the one-year bond more desirable than the six-month bond and, equivalently, makes the one-year bond price of 99.947 greater than the six-month bond price of 99.935.
.... More generally, price increases with maturity whenever the coupon rate exceeds the forward rate over the period of maturity extension. Price decreases as maturity increases whenever the coupon rate is less than the relevant forward rate"

 

[surbhi.7310]

Hi @David Harper CFA FRM

I think there is a mistake here,

• The bond price therefore will be unchanged if if PV(F, year 5) = PV(c*F, year 5) + PV(F, year 6)

Shouldnt it be ?

• The bond price therefore will be unchanged if if PV(F, year 5) = PV(c*F, year 6) + PV(F, year 6)

Sorry about nitpicking, I just wanted to be clear!
Regards
Surbhi

 

[David Harper CFA FRM]

Hi @surbhi.7310 Yes, exactly! It's not nitpicking, it's helpful attention to detail. Fixed above. Thank you!!