Online Finance Course – Pricing Interest Rate Swaps – Calculating the zero curve
Deriving the Zero Curve
We use the bootstrapping method for deriving the zero curve from the par term structure. This is an iterative process that allows use to derive a zero coupon yield curve from the rates/ prices of coupon bearing instruments. The step by step procedure employed in given below:
Step 4: Develop the cash flows matrix
Given the default par term structure above we calculate the cash flows for coupon bearing instruments for each tenor. The par value for each instrument is assumed to be 100. The instruments are assumed to be at par meaning that the coupon rate is equal to the par rate. An instrument with 1 year tenor, means a cash flow at maturity of the face value, 100 plus a coupon of par rate * face value= 12.15%*100 = 12.15 or a total of 112.15. An instrument with a tenor of 2 years will have a coupon payment at the end of year 1 and a payment of the face value + coupon at the end of year 2, and so on. Further details of the cash flows are given in the matrix below:
Coupon | 12.15% | 12.27% | 12.38% | 12.41% |
Tenor/ Duration | 1 | 2 | 3 | 4 |
1 | 112.15 | 12.27 | 12.38 | 12.41 |
2 | 112.27 | 12.38 | 12.41 | |
3 | 112.38 | 12.41 | ||
4 | 112.41 |
The columns pertain to a particular tenor; the rows pertain to the duration when a payment is due within the tenor.
Step 5: Developing the discounted value of cash flows matrix and the zero curve
As mentioned above, as the instruments are assumed to be priced at par the total present value of the future cash flows must equal the face value of the instrument, i.e. 100. In order to derive the discounted cash flows and the zero coupon rates we start with the shortest tenor and work our way to the larger tenors. This is illustrated for the first two tenors below using timelines to help in understanding the process more clearly.
As mentioned above the total present value of cash flows is equal to the face value. For the coupon bearing instrument with tenor 1 the cash flow are due at duration 1 amounting to 112.15. We know that the discount value of this total cash flow is 100. The zero coupon rate therefore would be the rate that discounts the cash flows to this value, i.e.
112.15 ÷ (1+ZC_{1}) = 100
Therefore ZC_{1} = [112.15 ÷ 100] -1 =12.15%
The cash flow at duration 1 will be discounted using the zero coupon rate determined earlier, i.e. ZC_{1}=12.15%. The cash flow at duration 2 will be discounted at ZC_{2} (annually compounded rate) which is an unknown at this point in time. Using the fact the total present value is equal to the face value, we determine ZC_{2} by solving the following equation:
100 = [12.27 ÷ (1+ZC_{1})] + [112.27 ÷ (1+ZC_{2})^{2}]
(1+ZC_{2})^{2} = 112.27 ÷ {100 – [12.27 ÷ (1+12.15%)]} = 1.2606021
ZC_{2} = (?1.2606021)-1 =12.277%
The same iterative and substitution process will be used to determine the zero coupon rates for the other tenors.
The matrix of discounted cash flows is given below:
Tenor/ Duration | 1 | 2 | 3 | 4 | |
1 | 100 | 10.9407 | 11.03879 | 11.06554 | |
2 | 89.0593 | 9.820558 | 9.844356 | ||
3 | 79.14065 | 8.739416 | |||
4 | 70.35069 | ||||
Total PV | 100 | 100 | 100 | 100 |
The zero curve is as follows:
t | ZC_{t} |
1 | 12.150% |
2 | 12.277% |
3 | 12.399% |
4 | 12.431% |