Transcript Time Value of Money - Long Island University

CHAPTER 4
BOND PRICES, BOND YIELDS, AND
INTEREST RATE RISK
Time Value of Money



A dollar today is worth more than a dollar
received at some future date.
Money may be spent on consumption or saved
by investing in real capital assets (machinery) or
by buying financial assets (deposits or stock).
Investing means giving up consumption.
Time Value of Money (concluded)


With a positive time preference for consumption,
investment means giving up consumption
(opportunity cost).
The opportunity cost of giving up consumption is
known as the time value of money. It is the
minimum rate of return required on a risk-free
investment.
Future Value or Compound
Value

The future value (FV) of a sum (PV) is
FV = PV (1+i)n.



(1+i)n is referred to as the Future Value Interest
Factor.
Multiply by the dollar amount involved to
calculate the FV of an investment.
Interest factor formulas are included in financial
calculators.
Present Value

The value today (at present) of a sum received at
a future date discounted at the required rate of
return.
PV = FV

1
(1 + i)
n
Given the time value of money, one is indifferent
between the present value today or the future
value received in the future.
Present Value (concluded)

With risk present, a premium return may be
added to the risk-free time value of money. The
higher the risk or higher the interest rate, the
lower the present value.
Valuing a Financial Asset

There are two necessary ingredients for valuing
financial assets.
– Estimates of future cash flows.
» The estimates include the timing and size of each
cash flow.
– An appropriate discount rate.
» The discount rate must reflect the risk of the asset.
The Mechanics of Bond Pricing



A fixed-rate bond is a contract detailing the par
value, the coupon rate, and maturity date.
The coupon rate is close to the market rate of
interest on similar bonds at the time of issuance.
In a fixed-rate bond, the interest income remains
fixed throughout the term (to maturity).
The Mechanics of Bond Pricing
(concluded)

The value of a bond is the present value of future
contractual cash flows discounted at the market
rate of interest.
C1
C2
CN + F N
PB =
+
+ ... 
1
2
N
(1 + i)
(1 + i)
(1 + i)
– Ci is the coupon payment and Fn is the face value
of the bond.
– Cash flows are assumed to flow at the end of the
period and are assumed to be reinvested at i.
Bonds typically pay interest semiannually.
– Increasing i decreases the price of the bond (PB).
Basic Bond Pricing Formula

The stream of coupon payments on a fixed rate
bond is an annuity which allows the pricing of a
bond with the following formula:
1

1



1  i n
PB  C 
i





F

 1  i n


Pricing Zero Coupon Bonds



Bonds that do not pay periodic interest payments
are called zero-coupon bonds.
Zero coupon bonds trade at a discount.
The value of the "zero" bond is
PB =

Fn
n
(1 + i)
There is no reinvestment of coupon payments
with zeros and thus, no reinvestment risk. The
yield to maturity, i, is the actual yield received if
held to maturity.
Bond Yields

Bond yields are related to several risks.
– Credit or default risk is the chance that some part
or all of the interest or principal payments will be
delayed or not paid.
– Reinvestment risk is the potential variability of
market interest rates affecting the reinvestment
rate of the periodic interest received resulting in an
actual, realized rate different from the expected
yield to maturity.
– Price risk relates to the potential variability of the
market price of the bond caused by a change in
market interest rates.
Bond Yields (continued)


Bond yields are market rates of return which
equate the market price of the bond with the
discounted expected cash flows of the bond.
A bond yield measure should reflect all three
cash flows from the bond and their timing:
– Coupon payments.
– Interest income from reinvestment of coupon
interest.
– Any capital gain or loss.
Bond Yields (continued)


The yield to maturity is the investor's expected or
promised yield if the bond is held to maturity and
the cash flows are reinvested at the yield to
maturity.
Bond yields-to-maturity vary inversely with bond
prices.
– If the market price of the bond increases, i, or the
yield to maturity declines.
Bond Yields (continued)
– If the market price of the bond decreases, the yield
to maturity increases.
– When the bond is selling at par, the coupon rate
approximates the market rate of interest.
– Bond prices above par are priced at a premium;
below par, at a discount.
Bond Yields (continued)

The realized yield is the ex-post, actual rate of
return, given the cash flows actually received and
their timing. Realized yields may differ from the
promised yield to maturity due to:
– A change in the amount and timing of the
promised cash flows.
– A change in market interest rates since the
purchase of the bond, thus affecting the
reinvestment rate of the coupons.
– The bond may be sold before maturity at a market
price varying from par.
Bond Yields (concluded)

The expected, ex-ante yield, assuming a realized
price and future interest rate levels, are
forecasted rates of return.
Bond Theorems



Bond yields vary inversely with changes in bond
prices.
Bond price volatility increases as maturity
increases.
Bond price volatility decreases as coupon rates
increase.
Bond Price Volatility

The percentage change in bond price for a given
change in yield is bond price volatility.
Pt  P t 1
%PB 
 100
Pt 1
– %PB = the percentage change in price.
– Pt = the new price in period t.
– Pt-1 = the price one period earlier.
Relationship Between Price,
Maturity, Market Yield, and Price
Volatility
(1)
Maturity
(years)
1
5
10
20
40
100
(2)
Bond Price
at 5 percent
Yield ($)
$1,000
1,000
1,000
1,000
1,000
1,000
PRICE CHANGE IF YIELD
CHANGES TO 6 PERCENT
(3)
(4)
(5)
Loss from
Price
Bond
Increase in
Volatility
Price ($)
Yield ($)
(percent)
$990.57
$ 9.43
-0.94%
957.88
42.12
-4.21
926.40
73.60
-7.36
885.30
114.70
-11.47
849.54
150.46
-15.05
833.82
166.18
-16.62
PRICE CHANGE IF YIELD
CHANGES TO 4 PERCENT
(6)
(7)
(8)
Gain from
Price
Bond
Decrease in Volatility
Price ($)
Yield ($)
(percent)
$1,009.62
$ 9.62
0.96%
1,044.52
44.52
4.45
1,081.11
81.11
8.11
1,135.90
135.90
13.59
1,197.93
197.93
19.79
1,245.05
245.05
24.50
Relationship Between Price,
Coupon Rate, Market Yield, and
Price Volatility
(1)
Coupon
Rate
(percent)
0%
5
10
(2)
Bond Price
at 5 percent
Yield ($)
$613.91
1,000.00
1,386.09
PRICE CHANGE IF YIELD
CHANGES TO 6 PERCENT
(3)
(4)
(5)
Loss from
Price
Bond
Increase in
Volatility
Price ($)
Yield ($)
(percent)
$ 558.39
$ 55.52
-9.04%
926.40
73.60
-7.36
1,294.40
91.69
-6.62
PRICE CHANGE IF YIELD
CHANGES TO 4 PERCENT
(6)
(7)
(8)
Gain from
Price
Bond
Decrease in Volatility
Price ($)
Yield ($)
(percent)
$ 675.56
$ 61.65
10.04%
1,081.11
81.11
8.11
1,486.65
100.56
7.25
Interest Rate Risk



Reinvestment risk--variability in realized yield
caused by changing market rates for coupon
reinvestment.
Price risk--variability in realized return caused by
capital gains/losses or that the price realized may
differ from par.
Price risk and reinvestment risk offset one
another, depending upon maturity and coupon
rates.
Duration



Duration is a measure of interest rate risk that
considers both coupon rate and term to maturity.
Duration is the ratio of the sum of the timeweighted discounted cash flows divided by the
current price of the bond.
Duration equals maturity for zero coupon
securities.
Duration Calculations
n
CFt (t )

t
t 1 (1  i )
D n
CFt

t
(
1

i
)
t 1
–
–
–
–
–
D = duration.
CFt = interest or principal at time t.
t = time period in which cash flow is received.
n = number of periods to maturity.
i = the yield to maturity (interest rate).
Duration Calculations (concluded)

Calculate duration of a bond with 3 years to
maturity, an 8 percent coupon rate paid annually,
and a yield to maturity of 10%.
$80(1) $80(2) $1,080(3)


1
2
(110
. )
(110
. )
(110
. )3
D
 2.78 years
$80
$80
$1,080


1
2
(110
. )
(110
. )
(110
. )3
Duration for Bonds Yielding 10%
(Annual Compounding)
MATURITY (YEARS)
1
2
3
4
5
ZERO COUPON
1.00
2.00
3.00
4.00
5.00
DURATION IN YEARS
4 PERCENT COUPON
1.00
1.96
2.88
3.75
4.57
8 PERCENT COUPON
1.00
1.92
2.78
3.56
4.28
Properties of Duration



The greater the duration, the greater is price
volatility.
Bonds with higher coupon rates have shorter
durations.
Generally, bonds with longer maturities have
longer durations.
Properties of Duration (concluded)


Except for bonds with a single payment, duration
is less than maturity. For bonds with a single
payment duration equals maturity.
The higher the yield to maturity, the shorter is
duration.
Using Duration to Estimate the
Percent Change in Bond Prices

The formula for estimating the percent change in
price for a given change in the market rate of
interest using duration is:
 i 
% PB   D 
x100

 1  i  
Convexity


The formula for estimating the percent change in
a bond’s price using duration works well for small
changes in interest rates, but not for large
changes in interest rates.
The formula can be modified to work well for
large interest changes and the modification is an
adjustment for convexity.
Calculating Convexity

The formula for convexity is:
 n t * (t  1) * C  n * ( n  1) * F

t  2   
n  2 
1  i 
 t 1 1  i 

convexity
Bond P rice

Using Duration and Convexity to
Estimate the Percent change in a
Bond’s Price

The formula for using duration and convexity to
estimate the percent change in a bond’s price is:
 

 i  1
2


%PB    D 

convexity

i
 2


1

i




 *100


Managing Interest Rate Risk with
Duration


Zero-coupon bonds have no reinvestment risk.
The duration of a zero equals its maturity. Buy a
zero with the desired holding period and lock in
the yield to maturity.
To assure that the promised yield to maturity is
realized, investors select bonds with durations
matching their desired holding periods.
(duration-matching approach).
Managing Interest Rate Risk with
Duration (concluded)

Selecting a bond maturity equal to the desired
holding period (maturity-matching approach)
eliminates the price risk, but not the reinvestment
risk.