BOND VALUATION AND RISK
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■ Bonds are debt obligations with long-term
maturities that are commonly issued by
governments or corporations to obtain long-term
funds.
■ The price of a bond is the present value of the cash
flows that will be generated by the bond, namely
periodic interest or coupon payments and the
principal at maturity.
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Current price of a bond (PV)
C
C
C  par
PV 

 ... 
1
2
n
1  k  1  k 
1  k 
Where:
C = coupon payment paid in each period
Par = par value
k = required rate of return
n = number of period to maturity
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Impact of the Discount Rate on Bond Valuation
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■
The appropriate discount rate for valuing any asset is the
yield that could be earned on alternative investments with
similar risk and maturities.
■
High risk securities have higher discount rates.
Impact of the Timing of Payments on Bond Valuation
■ Timing affects the market price of a bond
■
Funds received sooner can be reinvested to earn additional
returns
Valuation of Bonds with Semiannual Payments
■ First, divide the annual coupon by two
■ Second, divide the annual discount rate by two
■ Third, double the number of years
C/2
C/2
C / 2  par
PV of bond with 


...

1
2
2n
[
1

(
k
/
2
)]
[
1

(
k
/
2
)]
[
1

(
k
/
2
)]
Semiannual payments
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1. Discount bonds: Bonds Selling below Par
If coupon rate is below required rate, the price of the bond
is below par (PV < 1,000)
2. Par Bonds: Bonds Selling at Par
If coupon rate equals the required rate, the price of the
bond is equal to par value (PV = 1,000)
3. Premium Bonds: Bonds Selling above Par
If the coupon rate is above the required rate, the price of
the bond is above the par (PV > 1,000)
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Relationship between Required Return and Present
Value for a 10 Percent Coupon Bond with Various
Maturities
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1. Factors That Affect the Risk-Free Rate
R f  f (INF , ECON , MS , DEF )
a. Impact of Inflationary Expectations (INF)

If the level of inflation is expected to increase (decrease), there
will be upward (downward) pressure on interest rates and hence
on the required rate of return on bonds.

Inflationary expectations are partially dependent on oil prices
and exchange rate movements.
b. Impact of Economic Growth (ECON)- Strong economic growth
tends to generate upward pressure on interest rates, while
weak economic conditions put downward pressure on rates.
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1. Factors That Affect the Risk-Free Rate (Cont.)
c. Impact of Money Supply Growth (MS)
i.
The increased money supply may result in an increased supply of
loanable funds. If demand for loanable funds is not affected, the
increased money supply should place downward pressure on
interest rates, causing bond portfolio managers to expect an
increase in bond prices and thus to purchase bonds based on
such expectations.
ii. In a high-inflation environment, bond portfolio managers may
expect a large increase in the demand for loanable, which would
cause an increase in interest rates and lower bond prices.
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d. Impact of Budget Deficit (DEF)- An increase in the budget
deficit can put upward pressure on interest rates. An increase
in borrowing by the federal government can indirectly affect
the required rate of return on all types of bonds.
The value of any debt claim is equal to the sum of
the discounted future cash flows. The discount
[interest] for future cash flows is a function of the
level of riskiness for a particular bond and is termed
the Yield to Maturity (YTM).
Bond valuation model;
 Vb = Coupon * PVIFA + Face Value * PVIF
 Current Yield
 Coupon Yield
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Impact of Interest Rate Movements on Bond Prices
 Bond Prices move in the Opposite Direction to Interest
Rates.
 Bonds will sell at discounts or premiums or equal to Face
values
 Interest Rates move in the Same Direction as Inflationary
Expectations
 Interest Rates move in the Same Direction as Perceived
Riskiness
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
Determination of Bond Yields
 Yield to Maturity (YTM)
 Current Yield
 Tax treatment of gains and loses
 Premiums
 Discounts

Using Expectations to Manage Total Returns on
Bond Portfolios
 If you know were interest rates are going you know where
bond prices are going
 Riding the yield curve or betting on the movement of
interest rates
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
Factors Affecting Bond Price Interest Rate
Sensitivity
 The time remaining to maturity; direct relationship
 The size of the coupon payments
 The frequency of coupon payments; i.e., annual, Semi-,
quarterly, monthly

Measuring Sensitivity to Interest Rate Movements:
Duration
 As duration increases, the greater the sensitivity to
interest rate fluctuations
 Importance of determining the investment horizon
 Key strategy for immunizing the yield on a bond portfolio
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Bondholders know that the price of their bonds change when interest
rates change. But,

 How big is this change?
 How is this change in price estimated?
Macaulay Duration, or Duration, is the name of concept that helps
bondholders measure the sensitivity of a bond price to changes in bond
yields. That is:

Pct. Change in Bond Price  Duration 
Change in YTM
1  YTM
2


Two bonds with the same duration, but not necessarily the same maturity,
will have approximately the same price sensitivity to a (small) change in
bond yields.

Example: Suppose a bond has a Macaulay Duration of 11 years, and a
current yield to maturity of 8%.

If the yield to maturity increases to 8.50%, what is the resulting
percentage change in the price of the bond?

Pct. Change in Bond Price  - 11
0.085  0.08 
1 0.08 2
 -5.29%.
Some analysts prefer to use a variation of
Macaulay’s Duration, known as Modified Duration.

Modified Duration 
Macaulay Duration
YTM 

1 

2 

The relationship between percentage changes in
bond prices and changes in bond yields is
approximately:

Pct. Change in Bond Price  - Modified Duration  Change in YTM
Macaulay’s duration values are stated in years, and
are often described as a bond’s effective maturity.


For a zero-coupon bond, duration = maturity.
For a coupon bond, duration = a weighted average of
individual maturities of all the bond’s separate cash
flows, where the weights are proportionate to the
present values of each cash flow.

In general, for a bond paying constant semiannual
coupons, the formula for Macaulay’s Duration is:

Duration 
1  YTM  MC  YTM 
2
2
2M
YTM

YTM
YTM  C 1 
 1
2


1  YTM


In the formula, C is the annual coupon rate, M is the
bond maturity (in years), and YTM is the yield to maturity,
assuming semiannual coupons.

Calculating Duration
C=8% 2yr maturity Y=10% -> y = 5%
t
1
2
3
CF
40
40
40
PV$ @ y
0.9524
0.9070
0.8638
4
1040
0.8227
PV of CF
38.10
36.28
34.55
855.61
964.53
t x PV(CF)
38.10
75.56
103.66
3,422.44
3,636.744
Dmac = 3636.74/964.536 = 3.7704596
Dmod (1/2 yrs) = 3.7704596/1.05 = 3.590914 -> Dmac/(1+Y/2)
 All else the same, the longer a bond’s maturity, the
longer is its duration.
 All else the same, a bond’s duration increases at a
decreasing rate as maturity lengthens.
 All else the same, the higher a bond’s coupon, the
shorter is its duration.
 All else the same, a higher yield to maturity implies a
shorter duration, and a lower yield to maturity implies a
longer duration.

Use of Duration as an Immunization Strategy
"A portfolio of bonds is immunized from interest
rate risk if the duration of the portfolio equals the
desired investment horizon" Fisher and Weil
 Requires a known investment horizon; when do you need
the cash?
 Involves periodic adjustment in portfolio composition;
 Increase in YTM after the position is set results in a
decrease in duration and vice-versa (Reinvestment of cash
flows at higher rates than original YTM)
 Duration affected by calls, serial redemptions, and sinking
fund provisions
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Use of Derivative Securities as Hedges
 Interest rate futures, as well as options on futures
 May also involve currency hedges
 SWAP agreements may also be used
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

What cash flows are associated with bond investments?
What effects do interest rate increases (decreases) have
on;








Market values of bonds?
Current yields?
Yields to maturity?
Duration measures?
What does it mean when a bond sells at par, at a discount,
at a premium?
What does it mean to immunize a bond portfolio?
What information is necessary in order to make good
bond investments?
Q&A: 2, 5, 7, 8, 11, 14, 17 Interp: a, b, c
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