In bootstrapping method, ontherun treasury issues are used as they are fairly priced, and there is no credit risk or liquidity risk involved. In practice observed yield is rarely used for ontherun treasury coupon issues. Instead, the coupon rate is adjusted in a way that the price of an issue would be equal to par value.
Using the treasury par yield curve, let us see the calculation of the spot rates. The treasury par yield curve and the spot rates obtained using them are shown in table 8. In this table, the par yield curve shown is for 24 treasury securities and the longest maturity is 12 years.
Table: Hypothetical Treasury Par Yield Curve
Period

Years

Annual Yield to Maturity (in %)

Price

Spot Rate (in %)

1

0.5

4.00



4.0000

2

1.0

4.20



4.2000

3

1.5

4.60

100.00

4.6109

4

2.0

4.95

100.00

4.9721

5

2.5

5.30

100.00

5.3382

6

3.0

5.60

100.00

5.6558

7

3.5

5.90

100.00

5.9790

8

4.0

6.10

100.00

6.1949

9

4.5

6.15

100.00

6.2449

10

5.0

6.25

100.00

6.3498

11

5.5

6.35

100.00

6.4603

12

6.0

6.45

100.00

6.5732

13

6.5

6.55

100.00

6.6887

14

7.0

6.65

100.00

6.8068

15

7.5

6.75

100.00

6.9275

16

8.0

6.80

100.00

6.9835

17

8.5

6.88

100.00

7.0828

Period

Years

Annual Yield to Maturity (in %)

Price

Spot Rate (in %)

18

9.0

6.95

100.00

7.1702

19

9.5

7.00

100.00

7.2309

20

10.0

7.09

100.00

7.3531

21

10.5

7.18

100.00

7.4785

22

11.0

7.25

100.00

7.5756

23

11.5

7.35

100.00

7.7248

24

12.0

7.50

100.00

7.9647

The 6month and 1year treasury securities are called treasury bills and they are issued as zerocoupon instruments. The annualized yield for the 6month treasury securities and the 1year treasury securities is equal to their respective spot rates. The value of 1.5year Treasury rate is computed from the present value of the cash flows from the 1.5year coupon treasury security. The spot rate at the time of receipt is used as the discounting factor. Since all coupon bonds are selling at par i.e., $100, the coupon rate would be the yield to maturity for each bond.
0.5 year

 0.046 ´ $100 ´ 0.5

=

$2.3

1.0 year

 0.046 ´ $100 ´ 0.5

=

$2.3

1.5 years

 0.046 ´ $100 ´ 0.5 + 100

=

$102.3

The present value of the cash flows is then:
Where,
r_{1} = onehalf the annualized 6month theoretical spot rate.
r_{2} = onehalf the 1year theoretical spot rate.
r_{3} = onehalf the 1.5year theoretical spot rate.
We know that the 6month spot rate is 4.00% and the 1year spot rate is 4.02%, therefore:
r_{1} = 0.020 and r_{2}= 0.021
Present value of the 1.5year coupon Treasury security can be calculated as follows:
Equating the price of 1.5year coupon Treasury security to the par value of the security, we get:
Solving the above equation we get,
2.2549 + 2.2064 = 100
= 95.5387
Bondequivalent yield which is the theoretical 1.5year spot rate is equal to 4.6109% (2 x 2.3054%). This is the rate to be used to value all the treasury cash flows that are to be received 1.5 years from now
We can compute the theoretical 2year spot rate with the help of the given theoretical 1.5year spot rate as follows:
0.5 year 

0.0495 ´ $100 ´ 0.5

=

$2.475

1.0 year 

0.0495 ´ $100 ´ 0.5

=

$2.475

1.5 years 

0.0495 ´ $100 ´ 0.5

=

$2.475

2.0 years 

0.0495 ´ $100 ´ 0.5 + 100

=

$102.475

The present value of cash flows is:
Where,
r_{4 }= onehalf the 2year theoretical spot rate.
Substituting the values of r_{1}, r_{2} and r_{3 }in the above equation, we get:
Equating the price of the 2year coupon Treasury security to the par value of the security, we get:
The theoretical 2year spot rate is then: