Bootstrapping, Financial Management

In bootstrapping method, on-the-run 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 on-the-run 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 6-month and 1-year treasury securities are called treasury bills and they are issued as zero-coupon instruments. The annualized yield for the 6-month treasury securities and the 1-year treasury securities is equal to their respective spot rates. The value of 1.5-year Treasury rate is computed from the present value of the cash flows from the 1.5-year 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:

         108_bootstrapping.png

Where,

          r1  = one-half the annualized 6-month theoretical spot rate.

          r2  = one-half the 1-year theoretical spot rate.

          r3  = one-half the 1.5-year theoretical spot rate.

We know that the 6-month spot rate is 4.00% and the 1-year spot rate is 4.02%, therefore:

         r1 = 0.020 and r2= 0.021

Present value of the 1.5-year coupon Treasury security can be calculated as follows:

         708_bootstrapping12.png

Equating the price of 1.5-year coupon Treasury security to the par value of the security, we get:

         1434_bootstrapping13.png

Solving the above equation we get,

         2.2549 + 2.2064   1054_bootstrapping3.png      = 100

          1946_bootstrapping4.png     = 95.5387

               1950_bootstrapping5.png

Bond-equivalent yield which is the theoretical 1.5-year 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

          1453_bootstrapping6.png   

We can compute the theoretical 2-year spot rate with the help of the given theoretical 1.5-year 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:

         354_bootstrapping7.png

Where,

          r4 = one-half the 2-year theoretical spot rate.

         620_bootstrapping8.png

 

Substituting the values of r1, r2 and r3 in the above equation, we get:

         95_bootstrapping9.png

 

Equating the price of the 2-year coupon Treasury security to the par value of the security, we get:

         778_bootstrapping10.png

 

The theoretical 2-year spot rate is then:

         2121_bootstrapping11.png

Posted Date: 9/10/2012 2:37:01 AM | Location : United States







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