**4.9 CHAPTER 4: EXERCISES**

Table 4.9.1: **Information for questions 1-4**

Num |
Time Period |
Spot Rate (A) October 1994 |
Spot Rate (B) December 1994 |
Time Period |
Spot Rate (B) Aligned with (A) |

1 |
T,T+1 Month |
4.70 |
5.45 |
||

2 |
T,T+2 Month |
4.93 |
5.65 |
||

3 |
T,T+3 Month |
5.22 |
5.82 |
T+2,T+3 |
5.45 |

4 |
T,T+6 Month |
5.68 |
6.37 |
T+2,T+6 |
6.20 |

5 |
T,T+12 Month |
6.09 |
7.05 |
T+2,T+12 |
6.98 |

6 |
T,T+18 Month |
6.57 |
7.43 |
T+2,T+18 |
7.39 |

7 |
T,T+24 Month |
6.78 |
7.52 |
T+2,T+24 |
7.47 |

8 |
T,T+30 Month |
7.01 |
7.66 |
T+2,T+30 |
7.62 |

9 |
T,T+36 Month |
7.15 |
7.70 |
T+2,T+36 |
7.68 |

10 |
T,T+42 Month |
7.28 |
7.73 |
T+2,T+42 |
7.73 |

11 |
T,T+48 Month |
7.38 |
7.76 |
T+2,T+48 |
7.74 |

12 |
T,T+54 Month |
7.43 |
7.77 |
T+2,T+54 |
7.75 |

13 |
T,T+60 Month |
7.40 |
7.76 |
T+2,T+60 |
7.65 |

In table 4.9.1 columns 3 and 4 provide the yield curve at two points in time: October 1994 for column 3, (Spot Rate (A)), and December 1994, for column 3, (Spot Rate (B)). The rows for columns 3 and 4 give the 1 month, 2 month, 3 month, 6 month, 12 month etc., spot rates of interest for zero coupon Treasury securities.

Calender time is not aligned by row for columns 3 and 4. The 1 month spot rate for column 3 covers a one month period of time from T = October to T+1 = November 1994, whereas column 4 covers a one month period of time from T = December to T+1 = January 1995.

Column 6 provides the current *spot rates* time aligned with column 3. For example, in column 3 the 3-month Spot Rate (A), (T,T+3), is the 3-month period of time commencing October, 1994 to January 1995. In column 6 the corresponding spot rate labelled (T+2,T+3), covers the one month period of time starting December 1994 (i.e., T+2 = October plus 2 months) , to January 1995 (i.e., T+3 = October plus 3 months = January). That is, each row for Columns 3 and 6 is *aligned in end of period calender time* and thus can be used to compute the present value of cash flows at two points in time (October 1994, and December 1994).

In addition consider Treasury contracts listed in Table 4.9.2.

Table 4.9.2

Maturity (Time |
Annualized % Promised Rate (payable semi-annually) |
Face Value |

October 1996 (T + 24) |
7 3/8 |
$10000 |

October 1998 (T+48) |
8 7/8 |
$10000 |

Assumptions for questions 1-4:

For the following parts, let the current time be period T (i.e., T = October 1994), and the next coupon payment date be six months from October 1994 (period T+6).

For calculating the present value of cash inflows let the __compounding period be monthly__. For example, if the spot 4 month rate equals 12% per annum then apply this rate at 12%/12 = 1% per month for four months. This assumption allows you to answer each question in a consistent manner.

1. i. Using the information provided in Tables 4.9.1 compute the present value of the 7 3/8 Treasury contract in Table 2 at time T (T = October 1994). (Show all working for each contract assuming that a unit time period is 1 month).

ii. Using your answer to part a, compute the yield to maturity for the 7 3/8 contract in Table 4.9.2. Show working to verify your answer.

iii. Using the information provided in Table 4.9.1 column 6, compute the present value of the 7 3/8 contract in Table 4.9.2 as of *December 1994* (T+2). Show all working.

2. i. Using the information provided in Tables 4.9.1 compute the implied *forward rates* that cover the period of time up to October 1998. Using these forward rates compute the present value of the 13 1/8 Treasury contract in Table 2 at time T (T = October 1994). (Show all working and assume that a unit time period is 1 month).

ii. Using your answer to part a, compute the yield to maturity for the 13 1/8 contract in Table 4.9.2. Show working to verify your answer.

iii. Using the information provided in Table 1 column 6, compute the present value of the 13 1/8 contract in Table 4.9.2 as of *December 1994* (T+2). Show all working.

3. i. Using the information provided in Tables 4.9.1 compute the present value of both the 7 3/8 and 13 1/8 Treasury contracts in Table 2 at time T (T = October 1994) under the following conditions:

Spot rate - 10 basis points (- 0.001), Spot rate, Spot rate plus 10 basis points.

ii. Contrast your results in part a. to determine how each security responds to a constant shift in the spot rates of interest.

4. i. Using the information provided in Tables 4.9.1 compute the implied *forward rates* that cover the period of time up to October 1998. Using these forward rates compute an exposure table of bond values at time T (T = October 1994) constructed as follows:

Forward rate - 10 basis points (-0.001) (i.e., shift forward rates only by -10 basis points), Spot and Forward rates, Forward rates plus 10 basis points.

ii. Contrast your results in part a. to determine how each security responds to a constant shift in the forward rates of interest.

5. Suppose the quoted discount is 5.73% for a 13-week (91-day) *T-*bill with a face value of $1million. What is the price, in terms of dollars, that you would have to pay for this *T-*bill?

6. Suppose that at the close of trading on April 28, 1995 the following Bid prices were recorded.

Type (Promised rate) |
Maturity |
Bid |

T-Bill |
7/27/95 |
5.71 |

T-Bill |
10/26/95 |
5.83 |

T-Bill |
1/11/96 |
5.87 |

T-Bill |
4/04/96 |
5.92 |

Coupon* 8 1/2 |
5/97 |
103.16 |

Coupon* 9 |
5/98 |
106.20 |

Coupon* 8 7/8 |
5/00** |
108.16 |

Coupon* 10 |
5/05** |
119.5 |

Coupon* 8 1/8 |
2/21 |
107.7 |

* Assume that the maturity date is the 15th of the month and ignore any accrued interest.

** Callable bonds: The earliest call date provided.

i. Compute the yields for each security ignoring accrued interest and plot the yield curve.

ii. For the Coupon 10 5/05 calculate what the accrued interest is if the bond matured on the 15th of April and paid interest on a semi-annual basis.

7. Suppose that at the close of trading on April 28, 1995 the following Bid prices were recorded.

Type (Promised rate) |
Maturity |
Bid |

T-Bill |
7/27/95 |
5.71 |

T-Bill |
10/26/95 |
5.83 |

T-Bill |
1/11/96 |
5.87 |

T-Bill |
4/04/96 |
5.92 |

Strips* |
5/97 |
87.18 |

Strips* |
5/98 |
81.20 |

Strips* |
5/00 |
71.4 |

Strips* |
5/05 |
48.26 |

Strips* |
5/21 |
14.1 |

* Assume that the maturity date is the 15th of the month.

i. Compute the yields for each strip and plot the yield curve.

8. Compare and critically evaluate your answers to part i. in questions 6 and 7.

9. (Bond Tutor/Spreadsheet exercise) Select a Treasury Bond/Note from the financial press that has been stripped. Check whether there is any difference between the buying/selling price of the underlying issue and the sum of its stripped components.