CHAPTER 5 BOND PRICES AND RISKS. Copyright© 2003 John Wiley and Sons, Inc. Time Value of Money A...

CHAPTER 5

BOND PRICES AND RISKS

Copyright© 2003 John Wiley and Sons, Inc.

Time Value of Money

A dollar today is worth more than a dollar in the future.

Current income may be spent on current consumption or saved by investing in real assets (machines and equipments) or by investing in financial assets (stocks and bonds).


Time Value of Money

With a positive time preference for consumption, investment (real or financial) means giving up consumption (opportunity cost).

The opportunity cost of giving up consumption is known as the time value of money. It is the required return on a risk-free investment (the interest rate).


Future Value The future value (FV) of a present value (PV) is FVn = (1+i)n PV0

(1+i)n is referred to as the Future Value Interest Factor (FVIF).

Multiply the FVIF by the PV to calculate the FV.


Present ValueThe present value of a future value is PV0 = FVn/(1+i)n

1/(1+i)n is referred to as the Present Value Interest Factor (PVIF).

Multiply the PVIF by the FV to calculate the PV.

Given the time value of money, one is indifferent between the present value or the future value.


Valuing a Financial Asset

There are two components for valuing financial assets:

Estimates of future cash flows including the timing and size of each cash flow.

An appropriate discount rate must reflect the risk of the asset.


Pricing of Coupon Bonds

A fixed-rate bond is a contract detailing the face/par value, the coupon rate, and maturity date.

The coupon rate equals the market interest rate on similar bonds when the bond is issued.

In a fixed-rate bond, the interest income (coupon payment) remains fixed throughout the term to maturity.


Pricing of Coupon Bonds

The value of a bond is the present value of future cash flows discounted at the market interest rate, i

Ci is the coupon payment and Fn is the face or par 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 = C(1 + i)

+ C(1 + i)

+ ... C + F(1 + i)

1

1

2

2

N N

N


Pricing of Zero-Coupon Bonds

Bonds that pay no interest payments are called zero-coupon bonds.

The value of the "zero" bond is P = Fn/(1 + i)n.

Zero-coupon bonds sells at a discount.

There is no coupon payments with zeros and thus, no reinvestment risk. The yield-to-maturity is the realized yield if the bond is held to maturity.


Bond Yields

Bond yield is the interest rate which equates the price of the bond with the discounted expected cash flows of the bond.

A measure of the bond yield should reflect all three cash flows from the bond and their timing:

Coupon payments.

Interest income from reinvestment of coupon payments.

Any capital gain or loss incurred if the bond is sold before maturity.


Bond Yields

The yield-to-maturity is the expected yield if the bond is held to maturity and the coupon payments are reinvested at the yield-to-maturity.The yield-to-maturity varies inversely with bond price.When the bond is selling at par, the coupon rate approximates the market interest rate.


Bond Yields

If the coupon rate is less than the market interest rate on similar bonds, the bond will sell below its face value – at a discount.

If the coupon rate is higher than the market interest rate on similar bonds, the bond will sell above its face value – at a premium.


Bond YieldsThe realized yield is the actual yield, given the cash flows actually received. It may differ from the yield-to-maturity due to:

A change in market interest rates since the purchase of the bond, thus affecting the reinvestment rate of the coupons – reinvestment risk.The bond may be sold before maturity at a price different from its par value – price risk.


Bond RisksCredit or Default Risk is the chance that some or all of the interest payments or face value will be delayed or not paidReinvestment Risk: Variability in the realized yield caused by changing interest rates when coupons are reinvestment.Price Risk: Variability in the realized yield caused by changing interest rates when bonds are sold before maturity.Reinvestment risk and interest rate risk offset one another, depending upon maturity and coupon rates.


Bond Price Volatility

The percentage change in bond price for a given change in yield is bond price volatility.

%P = the percentage change in price.

Pt = the new price in period t.

Pt-1 = the price one period earlier.

%PBP P

Pt t

t

1

1

100


Bond Facts

Bond yields vary inversely with bond prices.

Bond volatility increases as maturity increases.

Bond volatility decreases as coupon rates increase.


Relationship Between Price, Maturity, Yield, and Price Volatility

PRICE CHANGE IF YIELD

CHANGES TO 6 PERCENT



(1)

Maturity(years)

(2)Bond Priceat 5 percent

Yield ($)

(3)

BondPrice ($)

(4)Loss fromIncrease in

Yield ($)

(5)Price

Volatility(percent)

(6)

BondPrice ($)

(7)Gain from

Decrease inYield ($)

(8)Price

Volatility(percent)

1 $1,000 $990.57 $ 9.43 -0.94% $1,009.62 $ 9.62 0.96%5 1,000 957.88 42.12 -4.21 1,044.52 44.52 4.45

10 1,000 926.40 73.60 -7.36 1,081.11 81.11 8.1120 1,000 885.30 114.70 -11.47 1,135.90 135.90 13.5940 1,000 849.54 150.46 -15.05 1,197.93 197.93 19.79

100 1,000 833.82 166.18 -16.62 1,245.05 245.05 24.50


Relationship Between Price, Coupon Rate, Yield, and Price Volatility





(1)Coupon

Rate(percent)

(2)Bond Priceat 5 percent

Yield ($)

(3)

BondPrice ($)

(4)Loss fromIncrease in

Yield ($)

(5)Price

Volatility(percent)

(6)

BondPrice ($)

(7)Gain from

Decrease inYield ($)

(8)Price

Volatility(percent)

0% $613.91 $ 558.39 $ 55.52 -9.04% $ 675.56 $ 61.65 10.04%5 1,000.00 926.40 73.60 -7.36 1,081.11 81.11 8.11

10 1,386.09 1,294.40 91.69 -6.62 1,486.65 100.56 7.25


Duration

Duration is a measure of interest rate risk that considers both coupon rate and maturity.Duration is the ratio of the sum of the time-weighted discounted cash flows divided by the price of the bond.Duration equals maturity for zero-coupon bonds.


Duration Calculations

D = duration.

CFt = interest or face value 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).

D

CF ti

CFi

tt

t

n

tt

t

n

( )( )

( )

1

1

1

1


Duration Calculations

Calculate duration of a bond with 3 years to maturity, an 8% coupon rate paid annually, and a yield-to-maturity of 10%.

D years

$80( )

( . )$80

( . )

$80( )

( . )$80

( . )

$1, ( )

( . )$1,

( . )

.

1

110

110

2

110

110

080 3

110080

110

2 781

1

2

2

3

3


Duration for Bonds Yielding 10% (Annual Compounding)

DURATION IN YEARS

MATURITY (YEARS) ZERO COUPON 4 PERCENT COUPON 8 PERCENT COUPON

1 1.00 1.00 1.002 2.00 1.96 1.923 3.00 2.88 2.784 4.00 3.75 3.565 5.00 4.57 4.28


Duration Facts

The longer the duration, the higher is price volatility.

Bonds with higher coupon rates have shorter durations.

Generally, bonds with longer maturities have longer durations.


Duration Facts

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 Bond Price VolatilityThe percentage change in bond price for a given change in the interest rate using duration can be estimated by

1001

% xi

iDPB


Bond Risk ManagementZero-Coupon Approach: Zero-coupon bonds have no reinvestment risk. The duration of a zero equals its maturity. Buy zero-coupon bonds with maturity equal to the desired holding period and lock-in the yield-to-maturity.Duration-Matching Approach: Select a portfolio of bonds with duration matching the desired holding period. Eliminates price risk and reinvestment risk, and lock-in the yield-to-maturity, but may require portfolio rebalancing → can be costly.


Managing Interest Rate Risk with Duration

Maturity-Matching Approach: Buy bonds with maturity equal to the desired holding period - eliminates price risk, but not reinvestment risk.

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CHAPTER 5 BOND PRICES AND RISKS. Copyright© 2003 John Wiley and Sons, Inc. Time Value of Money A...

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