Ec 173A – FINANCIAL MARKETS
LECTURE NOTES
Foster, UCSD
22Nov08
TOPIC 8 – BONDS & INTEREST RATES
A.
Bond Prices and Yields
[BKM, Ch. 14]
1.
Review:
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.
Examples:
•
Corporate bonds.
•
US Treasury Bonds and Notes.
•
State and local govt municipal bonds (“munis”).
•
Federal agency debt (bonds issued by Fannie Mae, Ginnie Mae, Freddie Mac, etc.)
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 interest rates rise from 4.0% to 4.5%,
they have increased by 50 basis points.
2.
Yield to Maturity:
1
a)
YTM  pure discount bonds.
1)
If B
0
grows to M in T years, r
y
is found by solving the following growth equation:
•
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  annual coupon interest.
1)
If the bond pays $I interest at the end of each year, a financial
calculator finds r
y
by solving the equation below (corresponding
to the bond value equation above) using an iterative algorithm:
•
B
0
= I
×
PVA
r,T
+ M
×
PV
r,T
2)
A formula for approximating YTM is available:
1
See the Appendix for
E
XCEL
spreadsheet formulas to compute YTM, accrued interest, and bond duration.
Bond Notation
B
0
current price of bond ($)
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
+
+
+
+
+
+
+
=
3
2
0
0
B
M
T
B
M
I
r
y
+

+
≈
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Ec 173A
BONDS & INTEREST RATES
p. 2
3)
Numerical example.
•
Given:
B
0
= $982, M = $1,000, I = $60/yr, T = 15 years
•
Using
E
XCEL
, r
y
= 6.188%.
•
Using the approximation, r
y
≈ 6.194%.
c)
YTM  semiannual coupon interest.
1)
If the bond pays $I/2 every 6 months, a financial calculator finds r
y
by solving the
following equation using an iterative algorithm:
•
B
0
= I/2
×
PVIFA
r/2,2T
+ M
×
PVIF
r/2,2T
2)
This corresponds to the bond value equation below.
The algorithm computes r/2,
then report r
y
= 2 (r/2), an annual percentage rate (APR).
3)
Numerical example.
•
Given:
B
0
= $982, M = $1,000, I = $30 per 6 months, T = 15 years
•
Using
E
XCEL
, r
y
= 6.186%.
3.
Interpretation of YTM:
a)
We interpret r
y
as the compound annual rate of return on the investment of $B
0
in the
bond if it is held to maturity
.
b)
This is easy to see with zerocoupon bonds.
1)
If B
0
= M/(1+r
y
)
T
, then B
0
(1+r
y
)
T
= M.
That is, B
0
grows to M in T years, so r
y
is the
annual rate of return (rate of growth of investment value).
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 Spring '09
 Foster
 Interest Rates, Yield Curve, Ry

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