Pricing of BondsPricing of Bonds

Chapter 2Chapter 2

Time of ValueTime of Value

Future ValueFuture Value

where:where:

n = number of periodsn = number of periods

PPnn = future value = future value nn periods from now (in dollars) periods from now (in dollars)

PPo o = original principal (in dollars)= original principal (in dollars)

r = interest rate per period (in decimal form)r = interest rate per period (in decimal form)

Future Value of on Ordinary AnnuityFuture Value of on Ordinary Annuity

non rPP )1(

r

rAP

n

n

1)1(

Time Value of MoneyTime Value of Money

Present ValuePresent Value

Present Value of a Present Value of a Series of Future Series of Future ValuesValues

n

no

rPP

)1(

1

n

tt

t

r

PPV

1 )1(

Time Value of MoneyTime Value of Money

Present Value of an Ordinary AnnuityPresent Value of an Ordinary Annuity

r

rAPV

n)1(

11

Bond PricingBond Pricing

price = PV of all future cash flowsprice = PV of all future cash flows to find price, you needto find price, you need

expected CFsexpected CFs coupon paymentscoupon payments par valuepar value

yieldyield

n

n

t t r

M

r

CP

)1()1(1

n

n

t t r

M

r

CP

)1()1(1

n

n

t t r

M

r

CP

)1()1(1

Bond PricingBond Pricing price – yield relationshipprice – yield relationship

inverse because as yield falls, PV of CFs increases inverse because as yield falls, PV of CFs increases and price of bond increasesand price of bond increases

price-yield curve is convex for noncallable bondsprice-yield curve is convex for noncallable bonds discount / premium to par when coupon rate discount / premium to par when coupon rate

< / > required yield< / > required yield market price changes whenmarket price changes when

yield changes due to credit quality of issueryield changes due to credit quality of issuer change in price of bond selling at discount/premium change in price of bond selling at discount/premium

because getting closer to maturitybecause getting closer to maturity change in yield required by market for bonds of change in yield required by market for bonds of

similar risksimilar risk

Bond PricingBond Pricing floater – security that has coupon rate that floater – security that has coupon rate that

changes when market rates changechanges when market rates change coupon = reference rate plus spreadcoupon = reference rate plus spread reset periodicallyreset periodically cap / floor cap / floor

inverse floaterinverse floater created using a fixed rate bond (collateral)created using a fixed rate bond (collateral) create floater and inverse floater so thatcreate floater and inverse floater so that

total coupon interest paid to two bonds in each period is total coupon interest paid to two bonds in each period is less than or equal to collateral’s coupon interest in each less than or equal to collateral’s coupon interest in each periodperiod

total par value of two bonds is less than or equal to total par value of two bonds is less than or equal to collateral’s total par valuecollateral’s total par value

ie., two securities created so that CFs from collateral can ie., two securities created so that CFs from collateral can satisfy obligation of floater and inverse floatersatisfy obligation of floater and inverse floater

(1)Price Quote

(2)Converted to a Decimal [ =

(1)/100]

(3)Par Value

(4)Dollar Price[ = (2) x (3)]

97 0.9700000 $ 10,000 $ 9,700.00

85 1/2 0.8550000 100,000 85,500.00

90 1/4 0.9025000 5,000 4,512.50

80 1/8 0.8012500 10,000 8,012.50

76 5/32 0.7615625 1,000,000 761,562.50

86 11/64 0.8657588 100,000 86,172.88

100 1.0000000 50,000 50,000.00

109 1.0900000 1,000 1,090.00

103 3/4 1.0375000 100,000 103,750.00

105 3/8 1.0537500 25,000 26,343.75

103 19/32 1.0358375 1,000,000 1,035,937.50

Prices Converted into a Prices Converted into a Dollar PriceDollar Price