Econ 174 – FINANCIAL RISK MANAGEMENT
LECTURE NOTES
Foster, UCSD
October 16, 2011
INTEREST RATE RISK
A. Bond Yields, Risk, and Duration
[Review]
1.
Bonds and Fixed-Income Securities:
a)
Fixed-income securities -- medium and long-term bonds or debt instruments involving a
promise to pay a certain fixed amount (principal) to the holder at a certain future date
(maturity), often with fixed “coupon” interest payments along the way.
b)
Basic bond value equation and definitions.
1)
Annual interest I = r
c
M.
Most bonds pay
semi-annually ($I/2 every 6 months).
2)
Yield to maturity r
y
is the discount rate that
equates DCF with current price B
0
.
It is like
the annual rate of return on the bond.
3)
Basis point
-- 1/100 of a percentage point; if the interest rate rises from 4.0% to
4.5%, it has increased by 50 basis points.
2.
Yield to Maturity:
1
a)
YTM -- pure discount bonds (“zeros”).
1)
If B
0
will grow to M in T years, YTM is found by solving the following for r
y
:
•
M = B
0
(1+r
y
)
T
⇒
Ln(1+r
y
) = Ln(M/B
0
)/T = γ, 1+r
y
= e
γ
2)
Numerical example.
•
B
0
= $887, M = $1,000, T = 34 months (2.83 years)
•
γ = Ln(1000/887)/2.83 = 0.0423, e
0.0423
= 1.0432, r
y
= 4.32%
b)
YTM -- semi-annual coupon interest.
1)
If the bond pays $I/2 every 6 months, a financial calculator finds r/2 solving the
following equation using an iterative algorithm, then reports r
y
= 2 (r/2):
•
B
0
= I
×
PVA
r/2,2T
+ M/(1+r/2)
2T
2)
Numerical example.
•
Given:
B
0
= $982, M = $1,000, I = $30 per 6 months, T = 15 years
•
Using E
XCEL
, r
y
= 6.186%.
c)
r
y
= compound annual rate of return on investment of $B
0
if bond held to maturity
.
1)
This is easy to see with zero coupon bonds.
1
See the Appendix for E
XCEL
spreadsheet formulas to compute YTM, accrued interest, and bond duration.
Bond Notation
B
0
current price of bond ($)
M
par or face value (usually
$1,000)
T
term to maturity (years)
r
c
coupon interest rate (%/yr)
I
annual coupon interest ($)
r
y
yield to maturity (YTM, %/yr)
T
y
y
y
r
M
I
r
I
r
I
B
)
1
(
)
1
(
)
1
(
2
0
+
+
+
+
+
+
+
=