As in previous homework, assume you work for a company that has to pay an obligation of USD 1 mln in 1.5 years from today. There are two bonds on the market - one is a 3%-coupon bond, has one year to maturity and is traded at price 101.7854. Another has 2 years to maturity, has 4% annual coupon rate and is traded at price 104.9214. Assume both principals are 100.
The spot rate curve is given as
6 months 1 year 18 months 2 years
1% 1.2% 1.4% 1.5%
Assume that spot rates follow semi-annual compounding convention, and that coupons are also paid semi-annually.
You want to use the two bonds to construct a portfolio, that would pay off the 1 mln obligation at 1.5 years from today, and would be immunized against parallel shifts in the interest-rate curve (in the sense of quasi-modified duration). Please amounts invested in each bond.
To construct a portfolio that is immunized against parallel shifts in the interest-rate curve, we need to calculate the durations of the two bonds and match them with the duration of the liability.
First, let's calculate the durations of the two bonds:
For the 1-year bond:
Coupon rate = 3% (annual), which is 1.5% semi-annually
Time to maturity = 1 year
Price = 101.7854% (or 101.7854)
Using the semi-annual compounding convention, we can calculate the cash flows from this bond:
Year 1: (1.5% x 100) + 100 = 102.5
The duration of this bond will be calculated as (P x T) / Price, where P is the present value of each cash flow weighted by the time to receive it, T is the time (in years) to receive each cash flow, and Price is the current price of the bond.
[PV of Year 1 cash flow x T] / Price = [(102.5/1.015) x 1] / 101.7854 = 1.0059 years
For the 2-year bond:
Coupon rate = 4% (annual), which is 2% semi-annually
Time to maturity = 2 years
Price = 104.9214% (or 104.9214)
Using the semi-annual compounding convention, we can calculate the cash flows from this bond:
Year 1: (2% x 100) + 100 = 102
Year 2: (2% x 100) + (2% x 100) + 100 = 104
The duration of this bond will be calculated as (P x T) / Price, similar to the previous bond:
[ (PV of Year 1 cash flow x T) + (PV of Year 2 cash flow x T)] / Price = [(102/1.015) x 1 + (104/1.015^2) x 2] / 104.9214 = 1.9927 years
Now, let's calculate the duration of the liability:
Time to maturity of the liability = 1.5 years
Once we have the durations of the bonds and the liability, we can calculate the amounts invested in each bond:
For Bond 1:
Amount invested = (Duration of Bond 2 - Duration of Liability) / (Duration of Bond 2 - Duration of Bond 1) x Liability
= (1.9927 - 1.5) / (1.9927 - 1.0059) x 1,000,000
= 25,009.42 USD
For Bond 2:
Amount invested = Liability - Amount invested in Bond 1
= 1,000,000 - 25,009.42
= 974,990.58 USD
Therefore, the amounts to be invested in each bond to construct the immunized portfolio are approximately 25,009.42 USD in the 1-year bond and 974,990.58 USD in the 2-year bond.