**3.13 EXERCISES**

1. What is the arbitrage-free price of a zero coupon bond (payable semi-annually) with two years left until maturity and has a face value equal to $110,000. Assume that the spot and forward interest rates for years 1, and 1-2 are 5%, and 5.5% respectively.

2. Suppose the spot rates of interest for maturities 1, 2, 3, 4, and 5 years are 3%, 4%, 5%, 5.5%, and 6% respectively, calculate the future value of $1,7500 at the end of each year for 5 years.

3. You want to have $250,000 in 5 year's time. As a one time investment how much would you have to invest now to reach this goal. Suppose the spot and forward rates of interest for years 1, 1-2, 2-3, 3-4, and 4-5 years are 6%, 7.5%, 8.0%, 8.5%, and 9% respectively.

4. Refer to exercise 3. Suppose at the end of Year 1 interest rates had declined so that the forward rates given in exercise 3 had all declined by 25 basis points. How much additional investment do you need to make at this time to guarantee your original target of $250,000?

5. What is the yield to maturity from investing in a newly issued 3 year zero coupon bond that has a face value equal to $100,000 and a coupon rate equal to 7.5% payable semi-annually. Suppose the current term structure of interest rates is downward sloping and the annualised spot rates equal:

0.5 year 8.5%, 1 year 8.25%, 1.5 years 8.0%, 2 years 7.85%, 2.5 years 7.7%, 3 years 7.6%.

6. Contrast a 3 year and 5 year coupon bond that both have a 5.5% coupon rate payable semi-annually. Suppose that over the first 3 years the term structure of spot interest rates are:

0.5 year 3.5%, 1 year 4.25%, 1.5 years 4.5%, 2 years 5.0%, 2.5 years 5.7%, 3 years 5.9%.

and then a flat 6.4% for years 4 and 5.

A. What is the market price of each bond if the face value of each bond is $10,000.

B. Compute the market price of each bond if spot interest rates increase by 10 basis points (0.1%) for years 1 and 2 and decrease by 10 basis points for years 3 to 5.

7. Consider a 5 year coupon bond with a 7% coupon rate payable semi annually and a face value equal to $10,000. Suppose that the spot and forward rates of interest equal:

0.5 year 5.5%, 1 year 5.6%, 1.5 years 6.1%, 2 years 6.7%, 2.5 years 6.9%, 3 years 7.3%, 3.5 years 7.7%, 4 years 8.1%, 4.5 years 8.3%, 5 years 8.3%.

A. At the time of issue what percentage of the market price arises from future coupon values and what percentage is due to the face value?

B. Immediately after the coupon payment in 2 1/2 years time what percentage of the market price is due to future coupon payments and what % is due to the eventual receipt of the face value?

C. Suppose immediately after the coupon payment in 5 1/2 years time interest rates had declined by 55 basis points (.55 of 1%). What percentage of the market price is due to future coupon payments and what percentage is due to the eventual receipt of the face value?

8. Suppose that the spot rates of interest as a function of maturity currently equal:

0.5 year 3.5%, 1 year 4.5%, 1.5 years 5.25%, 2 years 6.0%, 2.5 years 6.4%, 3 years 6.8%, 3.5 years 7.3%, 4 years 7.5%, 4.5 years 7.6%, 5 years 7.7%.

A. Under the unbiased expectations theory of the term structure construct the projected yield curve for 6 months from now?

B. Plot the yield curve from part A. and plot another yield curve that is consistent with the liquidity premium theory of the term structure of interest rates. Explain your graph.

C. Suppose in part A that the short end spot rates of interest increase as follows:

0.5 year 75 basis points, 1 year 50 basis points, 1.5 Year 35 basis points, 2 years 20 basis points and no shift thereafter.

Under the unbiased expectations theory of the term structure construct the projected yield curve for 6 months from now.

D. Using your answer to part C. calculate the projected value of a 2 year coupon bond in 6 months time. Suppose the coupon bond has a face value equal to $10,000 and a promised rate of 5% payable semi-annually.

9. Suppose that the spot and forward rates of interest currently equal:

0.5 year 4.5%, 0.5-1 year 4.75%, 1-1.5 years 5.25%, 1.5-2 years 5.75%, 2-2.5 years 6.0%, 2.5-3 years 6.4%, 3-3.5 years 6.8%, 3.5-4 years 7.2%, 4-4.5 years 7.4%, 4.5-5 years 7.6%.

Assume that present time is Time 0. Construct the forward contract prices for the following forward contracts:

A. Deliver a 2 year zero coupon bond (face value = $1,000) in 1 years time.

B. Deliver a 3 year zero coupon bond (face value = $1,000) in 1.5 years time.

C. Deliver a 4 year coupon bond in one years time. The face value is $10,000 and the coupon rate is 6.5% payable semi annually.

10. You can use the information provided in Table 1 to answer the six parts to this problem.

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 |

Table 1:

In table 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).

Complete the following six parts:

a) Complete Table 2.

Table 2

Time Period |
Spot Rate (A) October 1994 |
Forward Rate (A) |
Spot Rate (B) December 1994 |
Forward Rate (B) |

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

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

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

T,T+6 Month |
5.68 |
6.37 |
||

T,T+12 Month |
6.09 |
7.05 |
||

T,T+18 Month |
6.57 |
7.43 |
||

T,T+24 Month |
6.78 |
7.52 |
||

T,T+30 Month |
7.01 |
7.66 |
||

T,T+36 Month |
7.15 |
7.70 |
||

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

T,T+48 Month |
7.38 |
7.76 |
||

T,T+54 Month |
7.43 |
7.77 |
||

T,T+60 Month |
7.40 |
7.76 |

You should provide all working for two forward rates (T+6, T+12, and T+12, T+18). For remaining forward rates enter your answer without any support working.

Required for Parts b), c), and d).

Contrast the one month forward rate for December/January 1994/5 (T+2, T+3) computed from from the October 1994 yield curve, with the realized December/January spot rate (5.45%) and:

b) Apply the unbiased expectations theory of the term structure of interest rates, to evaluate these two rates.

c) Apply the liquidity premium theory of the term structure of interest rates, to evaluate these two rates.

d) Apply the market segmentation theory of the term structure of interest rates, to evaluate these two rates.

e) Under the unbiased expectations theory of the term structure of interest rates, use your answer to part a) to predict and interpret how the market’s interest rate expectations for the five years covered in part a), have shifted over the last two months.

f) How would consideration of the liquidity premium, and/or the segmented market theories of the term structure, modify your answer to part e)?