6.10 EXERCISES
1. Discuss the advantages and disadvantages associated with the three methods for hedging interest rate risk : cash matching, duration and minimum variance.
2. Consider the following yield curve:
Time |
Spot Rates |
Year 1 |
0.05 |
Year 2 |
0.055 |
Year 3 |
0.058 |
Year 4 |
0.060 |
Table 6.13
Using Bond Tutor construct an example that illustrates the following relationships:
A. Increase each spot rate by 25 basis points. Calculate the forward rates both before the shift and after the shift. How many basis points does each forward rate change?
B. For the forward rates implied from the spot rates in table 6.13 increase the year 1 spot rate and the years 2-4 forward rates by 25 basis points. Calculate the new implied spot rates from this shift. By how much does each spot rate change?
C. Using the results from part A describe the impact on the forward rate yield curve from a parallel shift in the spot rate yield curve. Similarly, using your results from part B describe the impact on the spot rate yield curve from a parallel shift in the forward rate yield curve.
D. Consider both a two and four year coupon bond each with a face value equal to $100 and each having a coupon rate equal to 5% payable annually.
i. Calculate the present value of each bond using Table 6.13.
ii. Calculate the duration of each bond by applying Macauley's definition of duration.
iii. Calculate the duration of each bond by applying Fisher-Weil's definition.
iv. Construct a portfolio of the two bonds that has zero Macauley duration.
E. Calculate the present value of each bond using the yield curve in part A. and part B. above separately.
F. Calculate the value of the portfolio constructed in part D. iv. using the yield curve in part A. and the yield curve in part B.
G. In view of your results, how useful is Macauley's duration for managing interest rate risk.
You can use the information provided in Table 6.14 to answer questions 3 and 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 |
Table 6.14:
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).
3.
Solve using Bond Tutor and show partial working for numerical answers. That is, indicate the steps that you took to calculate values supported by some intermediate numbers.
In this question consider the position of Treasury securities given in Table 6.15.
Table 6.15
Maturity (Time is denoted as the column 1 number in Table 1.1) |
Annualized % Promised Rate (payable semi-annually) |
Face Value |
Number of Contracts |
7 |
7 3/8 |
$10000 |
500 |
14 |
8 7/8 |
$10000 |
275 |
16 |
13 1/8 |
$10000 |
874 |
19 |
11 3/8 |
$10000 |
300 |
Assume that Treasury Strip contracts are available for each maturity period provided in Table 6.15. Let the face value of each Strip contract be $1000.
A. Using the principle of "Cash Matching" determine the portfolio of Treasury Strips that is the synthetic equivalent of the position in Table 6.15.
B. Referring to Table 6.14 compute the value of the position in Table 6.15 and the value and your Cash Matching synthetic position at two points in time: Time T (October 1994), and time T+ 2 (December 1994).
C. Compute Macauley’s Duration of the position given in Table 6.15.
D. By introducing two types of Treasury Strip contracts, namely maturity equal to 4 (i.e., Table 6.14 Column 1 (T+ 6)), and maturity equal to 22 (i.e., Table 6.14 Column 1 (T+22)), construct a position that has duration equal to zero at time T (October 1994). You can go short or long in each strip contract.
That is, treat the position given in Table 6.15 as fixed (i.e., cannot be traded), and use x units of the T+6 strip and y units of the T+22 strip to create a zero duration position as of October, 1994.
E. Compute the value of the zero duration position constructed in d) above at both time T (October 1994), and T+2 (December 1994).
4.
Evaluate and discuss using the numbers in parts b) and e) in Question 3, the two approaches (cash matching and duration) to managing the exposure of the value of the unhedged position in Table 6.15 to shifts in the yield curve.
5.
Minimum variance hedging is a statistical technique that is applicable to a wide range of risk managment problems. Consider a position that is exposed to some source of risk (interest rate risk or otherwise) and a hedging instrument that is exposed to the same source of risk. You can attempt to reengineer the exposure of your position by strategically introducing some amount of the hedging instrument.
For example, suppose you want to hedge a portfolio of property stocks (a property index fund) using a stock market index fund. The following table provides the total return for each year from 1980 to 1993 for the Australian All Ordinaries stock index and an index constructed from Property Trusts listed on the Australian Stock Exchange (ASX).
Year Ended |
All Ordinaries Return |
Property Trust Return |
1980 |
48.9% |
6.4% |
1981 |
-12.9% |
32.1% |
1982 |
-13.9% |
5.2% |
1983 |
66.8% |
50.2% |
1984 |
-2.3% |
10.1% |
1985 |
44.1% |
5.2% |
1986 |
52.2% |
35.4% |
1987 |
-7.9% |
5.7% |
1988 |
17.9% |
16.1% |
1989 |
17.4% |
2.3% |
1990 |
-17.5% |
8.7% |
1991 |
34.2% |
20.1% |
1992 |
-2.3% |
7.0% |
1993 |
45.4% |
30.1% |
Source: Property Investor August, 1994.
A. Assume that you invest $1 at the beginning of 1980 in each index. Construct for each year the end of year value for each $1 investment, assuming that no distributions or additions are made. That is at the end of 1981 the dollar invested in the all ordinaries index will grow to:
$1 x 1.489 x 0.871 = $1.297
Using a spreadsheet compute for each year and for each index the end of year value of this investment.
B. Define the geometric average (annualized) return, tRn, from an n year investment commencing at the present time t, as follows:
That is, if the $1 investment is compounded each year at the product of the one plus 1 year realized rates of return, then the geometric average return is the average rate of compounding over the investment horizon.
Using your spreadsheet from part a), compute the mean and variance of the return from investing $1 investment in each fund over the 14 years.
C. Suppose you have a position in the property index fund that you wish to hedge using the All Ordinaries index fund as the hedging instrument. Compute the minimum variance hedge ratio to hedge $10,000 invested in the Property Index.
D. For the sample period calculate the return and the variance of your hedged portfolio constructed in part C and evaluate your results.