Fin 221: Chapter 5
BOND PRICES AND
INTEREST RATE RISK
1
The Time Value of Money
• Investing—in financial assets or in real assets—means giving
up consumption until later.
• People have positive time preference for consumption,
indicating that money has different value in different time
periods, (time value of money)
• The time value of money indicate that a given amount of
money today (present value) is worth more than the same
amount received at some future time (future value).
• Positive time preference for consumption must be offset by
adequate return. Opportunity cost of deferring consumption
determines minimum rate of return required on a risk-free
investment
• Time value of money has nothing primarily to do with inflation.
– Inflation expectations affect discount rate, but
– Deferred consumption has opportunity cost by definition.
2
Future Value and Present Value
 The future value of an amount of money n periods
in the future (FV) at the present time (PV) is given
by (compounding):
FV = PV (1+i)n
where
i is the periodic interest rate and
n is the number of compounding periods.
 The Present Value, the value now of a sum
expected at a future time is given by (discounting):
PV = FV [1/ (1+i)n ]
3
Bond Pricing: What is a bond?
 A form of loan—a debt security obligating
a borrower to pay a lender principal and interest.
 Borrower (issuer) promises contractually to make periodic
payments (coupon payments) to lender (investor or
bondholder) over given number of years (maturity of the
bond)
 At maturity, holder receives principal, or the original amount
of the loan (face value or par value).
 Periodically before maturity, holder receives interest (coupon)
payments determined by coupon rate, original interest rate
promised as percentage of par on face of bond.
 For example, a $1,000 Par value, 5% Coupon Rate, maturing
30 years from today will generate payments of $50/year
interest for 30 years + $1,000 par at end of year 30.
4
Bond Pricing: bond cash flows
• Bondholder thus owns right to a stream or a series
of cash flows:
– Ordinary annuity of interest payments; plus
– Future lump sum in return of par value,
– Discountable to a present value at any time while bond
is outstanding.
• Therefore, the value (price) of a bond is the
present value of the future cash flows promised,
discounted at the market rate of interest (the
required rate of return on this risk class in today’s
market)
5
PV of bond cash flows
C1
C2
CN + F N
PB =
+
+
...
1
2
N
(1 + i) (1 + i)
(1 + i)
Where PB = price of bond or present value of promised payments;
Ct = coupon payment in period t, where t = 1, 2, 3,…, n;
Fn = par value (principal amount) due at maturity;
i = market interest rate (discount rate or market yield); and
n = number of periods to maturity.
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•
•
•
•
•
Bond pricing: principles
Cash flows are assumed to flow at end of the
period and to be reinvested at i. Bonds typically
pay interest semiannually.
Increasing i decreases price (PB); decreasing i
increases price; thus bond prices and interest
rates move inversely.
If market rate equals coupon rate, bond trades at
par (par bond).
If coupon rate exceeds market rate, the bond
trades above par, or at a premium ( premium
bond).
If market rate exceeds coupon rate, bond trades
below par or at a discount (discount bond).
7
Zero coupon bonds
• These are “pure discount” instruments.
• Have no periodic coupon payments, but only a single payment
at maturity.
• They eliminates reinvestment risk.
• PB is simply PV of FV represented by par value (the single
payment), discounted at market rate.
• The price is given by this formula:
PB zero coupon bond = Fmn/ [(1 +i/m)mn]
Where:
PB = price of the zero coupon bond
Fn = amount of cash payments at maturity.
i = interest rate (yield) for n periods
n = number of years until the payment is due
m = number of times interest is compounded each year
8
Bond yields: risks rewarded
 Cash flows from a bond depends on coupon
payments, income from reinvesting coupons, and
gains or losses resulting from changes in market
interest rates.
 Bond yield rewards investor for at least 3 risks:
• Credit or default risk: chance that issuer may be
unable or unwilling to pay as agreed.
• Reinvestment risk: potential effect of variability of
market interest rates on return at which payments
can be reinvested when received.
• Price risk: Inverse relationship between bond
prices and interest rates.
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Bond yields: set by market
• Discount rate at which bond price equals
discounted PV of expected payments.
• Measure of return ideally capturing impact of
- Coupon payments
- Income from reinvestment of coupons
- Any capital gain or loss
• Common yield measures are:
- Yield to maturity
- Realized yield
- Expected yield
10
Yield to maturity
• Investor's expected yield if bond is held to maturity
and all payments are reinvested at same yield.
• Normally determined by iteration—try different
discount rates until PB=present value of future
payments.
• The longer until maturity, the less valid the
reinvestment assumption.
• For example: an investor buys 5% percent coupon
(semiannual payments) bond for $951.90; bond
matures in 3 years. Solve the bond pricing equation
for the interest rate (i) such that price paid for the
bond equals PV of remaining payments due under
the bond.
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Yield to maturity …. Cont’d.
25
25
1,025
951.90 =
+
+ ...
1
2
6
(1 + (i / 2) ) (1 + (i / 2) )
(1 + (i / 2) )
Solving either by trial and error or with a financial
calculator results in yield to maturity of 3.4%
semiannually, or 6.8% annually.
12
Realized yield
• Investor’s ex post or actual rate of return,
given the cash flows actually received and
their timing. May differ from YTM due to—
- change in the amount or timing of
promised payments (e.g. default).
- change in market interest rates affecting
reinvestment rate.
- sale of bond before maturity at premium
or discount.
• It is the rate that equates the original price of
the bond to the discounted actual cash flows
from the bond.
13
Computing Realized Yield
Investor pays $1,000 for 10-year 8%
coupon bond; sells bond 3 years later for
$902.63.
Solve for i such that $1,000 (the original
investment) equals PV of 2 annual
payments of $80 followed by a 3rd annual
payment of $982.63 (the actual cash
flows this investor received).
14
Computing realized yield … cont’d
80
80
982.63
1000 =
+
+
...
1
2
3
(1 + i) (1 + i)
(1 + i)
Solving either by trial and error or with a financial
calculator results in a realized yield of 4.91%.
15
Expected yield
• Predicted yield for a given holding period, or the
expected yield if the bond is to be sold before
maturity.
• First, forecast the expected interest rate(s)
• Second, calculate bond price at end of holding
period, using the forecasted interest rate
• Third, plug forecast results into bond pricing
formula and calculate i which equates the original
price to the discounted expected cash flows from
the bond.
16
Bond price volatility (price risk)
Bond price and yield are inversely related. The response of PB to
changes in interest rates is measured by BP volatility. BP volatility
is the percentage change in price for given change in interest rates.
It is calculated using the formula:
%PB
Pt  Pt 1

 100
Pt 1
where %∆PB = percentage change in price
Pt = new price in period t
P t – 1 = bond’s price one period earlier
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19
Bond theorems
• Bond prices are inversely related to bond yields.
• BP volatility is affected by maturity and coupon
rate of the bond:
• The price volatility of a long-term bond is greater
than that of a short-term bond, holding the
coupon rate constant (volatility increases as
maturity increase).
• The price volatility of a low-coupon bond is
greater than that of a high-coupon, bond, holding
maturity constant (volatility decreases as coupon
rate increase).
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Interest rate risk and Duration
• Interest rate risk comprises price risk and reinvestment risk.
• Price risk is the variability in bond prices caused by their
inverse relationship with interest rates.
• Reinvestment risk is the variability in realized yield caused
by changing market rates at which coupons can be
reinvested.
• Price risk and reinvestment risk work against each other.
– As interest rates fall —
• Bond prices rise but
• Coupons are reinvested at lower return.
– As interest rates rise—
• Bond prices fall but
• Coupons are reinvested at higher return.
21
Interest rate risk and Duration
• Duration is a measure of interest rate risk that
considers both coupon rate and term to maturity.
It refers to the period necessary to offset price
risk and reinvestment risk, and thus eliminate
interest rate risk.
• It is measured as the ratio of the sum of the time
weighted discounted cash flows divided by the
current price of the bond. That is it is equal to
the PV of all cash flows weighted according to
length of time to receipt, divided by the price of
the bond.
22
Interest rate risk and Duration
• Duration is calculated using the formula:
CFt * t

t
t  1 (1  i)
D  n
CFt

t
t  1 (1  i)
n
where: D = duration of the bond
CFt = interest or principal payment at time t
t = time period in which payment is made
n = number of periods to maturity
i = the yield to maturity (interest rate)
23
24
Duration concepts (all else equal):
• Higher coupon rates mean shorter duration
and less price volatility.
• Duration equals term to maturity for zero
coupon securities.
• Longer maturities mean longer durations and
greater price volatility.
• The higher the market rate of interest, the
shorter the duration.
25
Duration can be calculated for an entire
portfolio
i 1
Portfolio Duration   wi Di
n
where: wi = proportion of bond i in portfolio and
Di = duration of bond i.
26
Duration and bond price volatility
Duration is used as a measure of price risk:
Using the 3-year, 4% coupon bond in Exhibit 5.6—
If yield increases to 12%:
 i 
%PB   D 
 100

 ( 1  i )
 0.02 
 2.88 

100


5
.
24
%

1
.
10


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Using Duration to Manage Interest Rate Risk
• Financial institutions use duration to manage
interest rate risk and actually achieve the desired
yield for the desired holding period.
 Zero-coupon approach: zero-coupon bonds
have no reinvestment risk. The duration of a “zero”
equals its term to maturity.
Buy a “zero” with
the desired holding period and lock in the YTM.
Must hold to maturity to evade price risk.
 Duration matching: To realize yield to maturity,
investors select bonds with durations matching
their desired holding periods.
 Maturity matching: Selecting a term to maturity
equal to the desired holding period eliminates
price risk, but not reinvestment risk.
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