Tuckman Discount Factors

[jeff-1984]

Hi David,

In the 2011 PQ booklet for Tuckman chapter 1 Discount factors, exercise 11.4 which is about calculating the face amounts of a replicating portfolio related to an overvalued treasury note... Can you please explain to me how we're doing this ?

 

[jeff-1984]

that's the exercise and the solution in case anyone can explain it to me

11.4. A fixed income manager has determined that an eighteen-month (1.5 year maturity) Treasury note with a market price at $102 that pays a 4.0% semi-annual coupon is overvalued. She conducts an arbitrage by shorting the bond and buying the replicating portfolio, as it trades cheap, that consists of positions in the following three bonds:
1) Bond B(1): A six-month (0.5 year to maturity) bond with a market price of $99.80 that pays a 1.0% semi-annual coupon (i.e., 1.0% per annum coupon rate pays 0.5% every six months)
2) Bond B(2): A one-year (1.0 year to maturity) bond with a market price of $99.40 that pays a 2.0% semi-annual coupon
3) Bond B(3): A 1.5-year bond with a market price of $98.00 that pays a 3.0% semi-annual coupon
What are face amounts, respectively, of the replicating portfolio? (note: more tedious than a typical exam question)
a) B(1) = 0.00, B(2) = 0.10, B(3) = 103.54
b) B(1) = 0.49, B(2) = 0.49, B(3) = 100.49
c) B(1) = 1.60, B(2) = 2.85, B(3) = 4.67
d) B(1) = 1.43, B(2) = 99.65, B(3) = 103.66

11.4. B. B(1) = 0.4853, B(2) = 0.4877, B(3) = 100.49
We don't need the market prices to generate the replicating portfolio.
Starting with the final cash flow at 1.5, where F(x) = face amount:
As F(3) * (1+3%/2) = $102.00, F(3) = 102/1.015 = 100.4926
As F(2) * (1+2%/2) + F(3)*3%/2 = $2.00, F(2) = (2 - 100.4926* 3%/2)/(1+2%/2) = 0.4877;
As F(1) * (1+1%/2) + F(2)*(2%/2) + F(3)*3%/2 = $2.00, F(1) = (2 - 100.4926* 3%/2 - 0.4877*2%/2)/(1+1%/2) = 0.4853
The replicating portfolio must generate cash flows of: $2 at 0.5, $2 at 1.0, and $102 at 1.5 (same as the overvalued bond).
To confirm/check just the 0.5 year cash flows:
B(1): 0.4853*(1+1%/2) = $0.4877; i.e., coupon plus par
B(2): 0.4877*2%/2 = $0.00488
B(3): 100.49 * 3%/2 = $1.5074
And $0.4877 + $0.00488 + $1.507 = $2.00