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Chapter 10
I.
Not on exam
II. Bond Pricing
A. Because a bond’s coupon and principal repayments all occur months or years in the future, the price an investor would
be willing to pay for a claim to those payments depends on the value of dollars to be received in the future compared
to dollars in hand today
1.
This “present value” calculation depends in turn on market interest rates
B. The nominal riskfree interest rate equals the sum of (1) a real riskfree rate of return and (2) a premium above the real
rate to compensate for expected inflation
C. The discount rate will embody an additional premium that reflects bondspecific characteristics such as default risk,
liquidity, tax attributes, call risk, and so on
D. In practice, there may be different discount rates for cash flows accruing in different periods
E.
To value a security, we discount its expected cash flows by the appropriate discount rate
F.
The cash flows from a bond consist of coupon payments until the maturity date plus the final payment of par value
1.
Therefore,
bond value = present value of coupons + present value of par value
2.
If we call the maturity date
T
and call the discount rate
r
, the bond value can be written as:
3.
Bond value = Σ[coupon/(1+r)
t
] + [par value/(1+r)
T
]
4.
Each coupon is discounted based on the time until it will be paid
5.
The first term is the present value of an annuity
6.
The second term is the present value of a single amount, the final payment of the bond’s par value
G. The present value of a $1 annuity that lasts for
T
periods when the interest rate equals
r
is
(1/r)[1 – {1/(1+r)
T
}].
1.
Call this the Tperiod
annuity factor
for an interest rate of
r
2.
We call
1/(1+r)
T
the
PV factor
, i.e., the present value f a single payment of $1 to be received in
T
periods
H. We write the price of a bond as:
1.
Price = Coupon x (1/r)[1{1/(1+r)
T
}] + par value x {1/(1+r)
T
}
2.
Or
Price = Coupon x Annuity factor(r, T) + Par value x PV factor(r, T)
I.
At a higher interest rate, the present value of the payments to be received by the bondholder is lower
1.
Therefore, the bond price will fall as market interest rates rise
2.
The
general rule in bond valuation
: when interest rates rise, bond prices must fall because the present value of the
bond’s payments is obtained by discounting at a higher interest rate
J.
The bond price curve
1.
The negative slope illustrates the inverse relationship between prices and yields
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Note that the shape of the curve implies that an increase in the interest rate results in a price decline that is smaller
than the price gain resulting from a decrease of equal magnitude in the interest rate
3.
This property of bond prices is called
convexity
because of the convex shape of the bond price curve
4.
This curvature reflects the fact that progressive increases in the interest rate result in progressively smaller reductions
in the bond price
5.
Therefore, the price curve becomes flatter at higher interest rates
K. Corporate bonds typically are issued at par value
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 Fall '08
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