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 FixedIncome Securities:
a)
Fixedincome securities  medium and longterm 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
semiannually ($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  semiannual 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
+
+
+
+
+
+
+
=
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INTEREST RATE RISK
p. 2 of
21
2)
It is only true with coupon bonds if the bondholder reinvests
each future coupon
payment at today’s r
y
.
3.
Bond Prices:
a)
Initial issue bond pricing and convergence to M.
1)
Before 2008, there were 1012 new corporate bond issues every day.
Issuers pick
coupon rate r
c
so that the bonds sell close to par in primary market (
i.e.
, so that B
0
=
M).
2)
If coupon rate on a given bond equals YTM on bonds of other companies with same
credit rating at the time, then r
c
= r
y
= r, I = rM and B
0
= M.
Proof:
3)
As t approaches the date of bond maturity:
•
T
→
0
•
PVA
r,T
→
0
•
1/(1+r)
T
→
1
•
Bond price B
t
→
M (net of accrued interest
b)
Bond price, accrued interest, and daycount conventions:
1)
Bonds that pay coupon interest usually make payments every 6 months.
The date
of these payments can be deduced from the date of maturity (the date of the final
semiannual payment).
If it matures April 15, interest is paid April 15 and October
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 Fall '08
 Foster,C
 Interest Rates, Yield Curve, Zerocoupon bond, rate risk

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