An amortizing bond is a bond which pays the principal not at its maturity, but prior to its maturity, according to some schedule, typically (but not necessarily) in equal amounts.
In particular, consider a floating-rate amortizing bond, which pays 25% of its principal amount each year until maturity (in 4 years from today). In addition, the bond pays a yearly coupon, where the coupon rate for each period is defined as the short rate for this period. Note that the coupon payment is calculated proportional to the principal outstanding, so for a bond of principal amount USD 100, the cash stream would look like
1 year 2 years 3 years 4 years
25+100*r1 25+75*r2 25+50*r3 25+25*r4
where r1, r2, r3, r4 are the corresponding short rates. Assume the spot rate curve given as in the previous problem:
1 year 2 years 3 years 4 years
0.5% 1% 1.4% 2.1%
Assuming also the expectation dynamics, find the price of this bond.
To find the price of this bond, we need to calculate the present value of all the cash flows. Here's how you can do it step by step:
1. Determine the cash flows for each period:
Year 1: Coupon payment = 25% of principal + coupon rate for Year 1
Year 2: Coupon payment = 25% of principal + coupon rate for Year 2
Year 3: Coupon payment = 25% of principal + coupon rate for Year 3
Year 4 (Maturity): Coupon payment = 25% of principal + coupon rate for Year 4 + principal amount
2. Calculate the present value of each cash flow using the spot rates given in the problem.
For example, the present value of Year 1 cash flow would be:
PV1 = (Coupon payment for Year 1) / (1 + spot rate for Year 1)
Similarly, the present value of Year 2 cash flow would be:
PV2 = (Coupon payment for Year 2) / ((1 + spot rate for Year 1) * (1 + spot rate for Year 2))
Continue this process for each cash flow.
3. Sum up all the present values to get the price of the bond:
Bond price = PV1 + PV2 + PV3 + PV4
Now let's calculate the bond price using the given spot rates:
Year 1:
Coupon payment = 25 + 100 * 0.5% = 25.5
PV1 = 25.5 / (1 + 0.5%) = 25.5 / 1.005 = 25.37
Year 2:
Coupon payment = 25 + 75 * 1% = 25.75
PV2 = 25.75 / ((1 + 0.5%) * (1 + 1%)) = 25.75 / (1.005 * 1.01) = 25.42
Year 3:
Coupon payment = 25 + 50 * 1.4% = 25.7
PV3 = 25.7 / ((1 + 0.5%) * (1 + 1%) * (1 + 1.4%)) = 25.7 / (1.005 * 1.01 * 1.014) = 25.42
Year 4 (Maturity):
Coupon payment = 25 + 25 * 2.1% = 25.525
Principal amount = 25
PV4 = (25.525 + 25) / ((1 + 0.5%) * (1 + 1%) * (1 + 1.4%) * (1 + 2.1%)) = 50.525 / (1.005 * 1.01 * 1.014 * 1.021) = 46.82
Bond price = PV1 + PV2 + PV3 + PV4 = 25.37 + 25.42 + 25.42 + 46.82 = 122.03
Therefore, the price of this bond is approximately USD 122.03.