Question

Problem 3. Some bond calculus. (Hint: for this exercise, the formula for a geometric series might be useful: if q 1, then 1-q I will assume in the future, that this formula is well known. 1. Suppose a coupon bond has a face value F0 to be repaid after n years and promises to pay a coupon C equal to an interest rate of r percent of that face value for all years t = 1, . . . ,n (Note: I am using the notation x rather than i for the interest rate: that seems easier to read.). Currentlgy, ie in year t = 0, it is trading at price P

Answer 1

Part (a)

Face Value = F

Coupon, C = x% of F = xF

Period from t=1 to t=n

Hence, Price of the bond today, P = Present value of all future coupons + Present value of face value at the time of maturity. Hence,

$=C \times \sum_{k=1}^{n}(1+i)^{-k} + F\times (1+i)^{-n} = \frac{C}{i}\times [1-(1+i)^{-n}]+ F\times (1+i)^{-n} = \frac{xF}{i}\times [1-(1+i)^{-n}]+ F\times (1+i)^{-n}$

Hence

$\frac{P}{F}= \frac{x}{i}\times [1-(1+i)^{-n}]+ (1+i)^{-n}$

--------------------

Part (b)

If yield = coupon rate, this means i = x

In that case,

$\frac{P}{F}= \frac{i}{i}\times [1-(1+i)^{-n}]+ (1+i)^{-n} = 1$

Hence, P = F

Hence, when current price is equal to face value of the bond, in that case yield = coupon rate.

-----------------

Part (c)

If $n \to \infty$ then

$(1+i)^{-n} \to \0$

Hence, P / F = x / i

The relationship between P and i:

1. Inverse: When i increases, P decreases and vice versa
2. Non linear: The changes in P are disproportionate to changes in i

When F = 1000, x = 5% we can generate the following table

Price, P	Yield, i = xF / P
500	10.00%
750	6.67%
1,000	5.00%
1,250	4.00%
1,500	3.33%

Please see the plot below: