Chapter 2: Term Structure of Interest Rate
1. Song: page 31, #1
Answer:
(1) If we assume that the coupon is paid annually,
For T-Bills: We have r n P n 12
1100＋=, so: 1-month: r 1211100653.99+=
, r = 4.178%; 3-month: r 12
31100866.98+=, r = 4.588% 6-month: r 12
61100766.97+=, r = 4.570%; 1-year: r +=1100666.95, r = 4.530% For T-Bonds and T-Notes :
We have n n n n n n n n n r i r i r i r i P )
1(100)1()1(111221+++++++++=-- , so 1)1(110011-+?-+=∑-=n n t t t n n n
n r i p i r ,
We have already known that r 1 = 4.530%, so we can calculate other zero rates by Excel: 2-year: r 2 = 4.522%;
3-year: r 3 = 4.532%;
4-year: r 4 = 4.551%;
5-year: r 5 = 4.562%;
6-year: r 6 = 4.624%;
7-year: r 7 = 4.571%;
8-year: r 8 = 4.632%;
Since there ’s no available information for the values from r 9 to r 19, we should use linear interpolation to determine them. The formula is )(n m n j r r n
m n j r r ---+
=, where n = 8, m = 20, r 8 = 4.632%, so: 9-year: r 9 =202012
1%2463.4%)632.4(82089%632.4r r +=---+; 10-year: r 10 = 2061%8603.3r +;
11-year: r 11 = 2041%4742.3r +
; 12-year:
r 12 = 2031%0882.3r +; 13-year:
r 13 = 20125%7022.2r +; 14-year:
r 14 = 2021%3162.2r +; 15-year:
r 15 = 20127%9301.1r +; 16-year:
r 16 = 2032%5441.1r +; 17-year:
r 17 = 2043%1581.1r +; 18-year:
r 18 = 2065%7721.0r +; 19-year: r 19 = 2012
11%386.0r +;。 Finally, we can get an high order equation about r 20. Solve this equation in Excel by trial-and-error.
Similarly, we can get the value of r 25and r 30 by solving another two equations.
(2) If we assume that the coupon is paid semiannually,
For T-Bills, the results are the same as in (1).
For T-Bonds and T-Notes, We have:
23
0.51 1.5222122221()()220.50.50.5(1/2)(1/2)(1/2)0.50.50.5*(1/2)(1/2)(1/2)n n n n n n n n n n
n n n i par i par i par P r r r i par i par i par par r r r ----??????=+++++?????+++++++
term
r term r 1
4.530% 4.5 4.568% 1.5
4.535% 5 4.573% 2
4.539%
5.5 4.611% 2.5
4.545% 6 4.650% 3
4.551% 6.5 4.619% 3.5
4.558% 7 4.588% 4
4.564% 7.5 4.618% 8 4.649%
Use linear interpolation to determine the values of zero rates with a maturity longer than 8 years.
2. Song: page 32, #2 Answer:
n
n n n n n n n n r i r i r i r i P )1(100)1()1(111221+++++++++=-- , Then 1)1(110011-+?-+=∑-=n n t t t n n n
n r i p i r
r 1 =3.465%
r 2 =3.762%
r 3 =3.815%
r 4 =3.949%
r 5 =4.143% Finally, 5
5511
1(1)1(1)t t t r i r =-
+=+∑=4.12%
3. The term structure is upward sloping. Put the following in order of magnitude:
a) The five-year zero rate.
b) The yield on a five-year coupon-bearing bond.
c) The forward rate corresponding to the period between 5 and 4
15years in the future. What is the answer to this question when the term structure is downward sloping?
Answer:
When term structure is upward sloping, c>a>b
When term structure is downward sloping, b>a>c
4. A $100 million interest rate swap has a remaining life of 10 months. Under the terms of the
swap, six-month LIBOR is exchanged for 12% per annum (compounded semiannually). The average of the bid-ask rate being exchanged for six-month LIBOR in swaps of all maturities is currently 10% per annum with continuous compounding. The six-month LIBOR rate was
9.6% per annum two months ago. What is the current value of the swap to the party paying floating? What is its value to the party paying fixed?
Answer:
Current value to the party paying floating:
-0.1*1/3-0.1*5/6-0.1*1/3
6e +106e 104.8e 103.328-101.364
=1.964million
-= value to the party paying fixed: -1.964million