of 98
Patrick Wall is a new associate at a large international financial institution. His boss, C.D.
Johnson, is responsible for familiarizing Wall with the basics of fixed income investing.
Johnson asks Wall to evaluate the two otherwise identical bonds shown in Table 1. The
callable bond is callable at 100 and exercisable on the coupon dates only.
Wall is told to evaluate the bonds with respect to duration and convexity when interest rates
decline by 50 basis points at all maturities over the next six months.
Johnson supplies Wall with the requisite interest rate tree shown in Figure 1. Johnson
explains to Wall that the prices of the bonds in Table 1 were computed using the interest
rate lattice. Johnson instructs Wall to try and replicate the information in Table 1 and use his
analysis to derive an investment decision for his portfolio.
Table 1 Bond Descriptions
Non-callable Bond
Callable Bond
Price
$100.83
$98.79
Time to Maturity (years)
5
5
Time to First Call Date
--
0
Annual Coupon
$6.25
$6.25
Interest Payment
Semi-annual
Semi-annual
Yield to Maturity
6.0547%
6.5366%
Price Value per Basis Point
428.0360
--
Figure 1
15.44%
14.10%
12.69%
12.46%
11.85%
11.38%
9.75%
10.25%
10.05%
8.95%
9.57%
9.19%
7.91%
7.88%
8.28%
8.11%
7.35%
7.23%
7.74%
7.42%
6.62%
6.40%
6.37%
6.69%
6.54%
6.05%
5.95%
5.85%
6.25%
5.99%
5.36%
5.17%
5.15%
5.40%
5.28%
4.81%
4.73%
5.05%
4.83%
4.18%
4.16%
4.36%
4.26%
3.82%
4.08%
3.90%
3.37%
3.52%
3.44%
3.30%
3.15%
2.84%
2.77%
2.54%
2.24%
Years
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
.....................................................................................................................................................................................................
Given the following relevant part of the interest rate tree, the value of the callable bond at
node A is closest to:
3.44%
A
3.15%
2.77%
A) $101.53.
B) $103.56.
C) $100.00.
Explanation
The value of the callable bond at node A is obtained as follows:
Bond Value = the lesser of the Call Price or {0.5 × [Bond Valueup + Coupon/2] +
0.5 × [Bond Valuedown + Coupon/2]}/(1 + Interest Rate/2)]
So we have
Bond Value at node A = the lesser of either $100 or {0.5 × [$100.00 + $6.25/2] + 0.5 ×
[$100.00+ $6.25/2]}/(1+ 3.15%/2) = $101.52. Since the call price of $100 is less than the
computed value of $101.52 the bond price would be $100 because once the price of the
bond reached this value it would be called.
(Module 27.2, LOS 27.f)
Natalia Berg, CFA, has estimated the key rate durations for several maturities in three of her $25 million bond portfolios, as shown in
Exhibit 1.
Exhibit 1: Key Rate Durations for Three Fixed-Income Portfolios
Key Rate Maturity
Portfolio 1
Portfolio 2
Portfolio 3
2-year
2.45
0.35
1.26
5-year
0.20
0.40
1.27
10-year
0.15
4.00
1.23
20-year
2.20
0.25
1.24
Total
5.00
5.00
5.00
At a fixed-income conference in London, Berg hears a presentation by a university professor on the increasing use of the swap rate curve
as a benchmark instead of the government bond yield curve. When Berg returns from the conference, she realizes she has left her notes
from the presentation on the airplane. However, she is very interested in learning more about whether she should consider using the swap
rate curve in her work.
As she tries to reconstruct what was said at the conference, she writes down two advantages to using the swap rate curve:
Statement 1:
The swap rate curve typically has yield quotes at 11 maturities between 2 and 30 years.
The U.S. government bond yield curve, however, has fewer on-the-run issues trading at
maturities of at least two years.
Statement 2:
Swap curves across countries are more comparable than government bond curves
because they reflect similar levels of credit risk.
Berg also estimates the nominal spread, Z-spread, and option-adjusted spread (OAS) for the Steigers Corporation callable bonds in
Portfolio 2. The OAS is estimated from a binomial interest rate tree. The results are shown in Exhibit 2.
Exhibit 2: Spread Measures for Steigers Corporation Callable Bonds
Spread Measure
Benchmark
Nominal spread
25 basis points
Steigers Corp yield curve
Z-spread
35 basis points
Steigers Corp spot rate curve
OAS
−20 basis points
Steigers Corp spot rate curve
Nominal spread
120 basis points
Treasury yield curve
OAS
40 basis points
Treasury spot rate curve
Berg determines that to obtain an accurate estimate of the effective duration and effective convexity of a callable bond using a binomial
model, the specified change in yield (i.e., Δy) must be equal to the OAS.
Berg also observes that the current Treasury bond yield curve is upward sloping. Based on this observation, Berg forecasts that short-term
interest rates will increase.