Bonds part 1

Bond Prices and YieldsBond Prices and Yields

Part 1 Part 1

Learning Objectives Learning Objectives

• Types of bonds• Bond parameters including yield • Bond price as discounted expected cash flow

o Applications • Compute bond yield from a known price • Compute bond price from a known yield• Plot bond price v. yield • Compute bond price when yield is not known• Plot bond price v. time to maturity • Compute mortgage payments• Bond formulas

• Bond price quotes2

Bond OverviewBond Overview• Corporations and governments can raise capital by selling bonds

o Long term liability (accounting)o Debt capital (finance)

• The bond has o Principal, par, or face value: Fo Price: Po Yield: y (also the discount rate ) o Maturity date, time to maturity, term, or tenor: T

• Date at which the bond principal, F, is returned to investors• This is different than the duration

o In the case of a coupon bond (not a zero coupon bond)• Coupon rate: c (annual, simple, nominal rate) • Annual payment frequency: m; or period t

o In the U.S. semiannual coupons is typical: m = 2 or t = .5

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Zero Coupon BondsZero Coupon Bonds• ZCBs do not pay a coupon

o No intermediate cash flow • The rate of return or ‘yield’ is due to the bond being bought,

at P, a discount to face value, F • U.S. Treasury bills (T – bills) are zero coupon bonds

o Time-to-maturity at issue is 4, 13, 26, 52 weeks o Face value $100 to $5,000,000

• A ZCB yield is the interest rate, (and the discount rate) denoted z

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F

P

t=0

t=T

Zero Coupon BondZero Coupon Bond• For T ≤ 1 year:

where z is the annual yield – a simple rate

• For T > 1 year

where z is the annualized yield – an effective rate

If a bond has a term of a year or less, simple interest is used, otherwise compound annual interest is used

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F

P

t=0

t=T

Zero Coupon Bond ExampleZero Coupon Bond Example• The face value is $1000, the market price is $850, and the

time to maturity is 3.5 years. What is the annualized yield ?

• The face value is $1000, the market price is $975, and the time-to-maturity is 0.5 years. What is the annualized yield?

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Zero Coupon Bond Zero Coupon Bond • A bond dealer can split a coupon bond into ZCBs

o one for the principal and o one for each coupono This is called ‘stripping’ the bond

• The advantage of a ZCB is that there is no reinvestment risk

• For a ZCB, the yield, y, is the zero coupon rate denoted as z

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Coupon BondCoupon Bond

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Coupon Bond: MSFTCoupon Bond: MSFT

9June 30


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June 30


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Coupon BondCoupon Bond

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P = current price

C = coupon paymentc = coupon rate (simple interest)

F = face or par value

t=0.0 t=t t=2∙t t=m∙N∙t=T

i=0 i=1 i=2 i=m∙N

t0=0.0 t1=t t2=2t tm∙N= m∙N∙t =T

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Coupon PaymentCoupon Payment• Bond coupon cash flows, C, are defined by a nominal,

simple coupon rate, c, and a compounding frequency per year, m, or coupon period measured in years, to m is always 1 or 2 for a bondo t = .5 or 1 year

• The total cash flow at time ti, CFi, is defined as:

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Current interest and yield summaryCurrent interest and yield summary• Bloomberg• Yahoo• Bankrate

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Coupon Bond YieldCoupon Bond Yield• Yield to maturity is the actual yield achieved for a coupon bond if

o The bond is held to maturity, and o Each coupon payment is reinvested at the rate of return y through time

T• The risk that coupons cannot be reinvested at a rate greater than or equal

to y due to market conditions is called “reinvestment risk”

• The yield to maturity, y, is the investor’s expected return on the investment of P and is thus the issuer’s rate costo It’s the issuer’s ‘cost of debt’, kD, for the bond

• The yield reflects both the time value of money and the credit risk of the borrowero The expected variance of the cash flows is reflected in the yield, y

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Bond PriceBond Price• The discount rate y is the yield to maturity or simply the

yield on a coupon bond• It’s the internal rate of return that sets the discounted cash

flow on the right hand side to the market price of the bond, P, on the left hand sidex4

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y is the nominal annual yield to maturity y is effective annual

yield to maturity

M is the number of periods or “cash flows”M = N * m if N is an integer number of yearsM = T * m if T is floating point years

Homework 10Homework 10

• Compute the price, P, of the following bond and the nominal yield to maturity, yo Time to maturity 4.5 yearso Annual, nominal coupon rate is 7% o Semi annual coupons and compounding o Par value is $1000o Yield to maturity 8% (effective annual rate)

• Submit a knitr pdf with explanation and echoed codeo Print the data frame or table for clarity

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Homework 11Homework 11

• Compute the nominal and effective yields to maturity for this bond o Time to maturity 4.5 yearso Annual, nominal coupon rate is 7% o Semi annual coupons and compounding o Par value is $1000o Price is $900.00


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• For a fractional initial coupon period: t1 < ∆t

Fractional Initial Time PeriodFractional Initial Time Period

For a bond with semi-annual coupons, assume that the next coupon payment is in 3 months. The coupon payments occur at t0=0.0, t1=0.25, t2=0.75, t3=1.25, t4 = 1.75, …

i=0 i=1 i=2 i=M

t0=0.0 t1 t2=t1+t tM= T

C = coupon payment F = face or par value

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Homework 12 Homework 12

• Compute the price of this bondo Time to maturity 4.25 yearso The next coupon is paid 3 months from present o Annual, nominal coupon rate is 7% o Semi annual coupons and compounding o Par value is $1000o Yield to maturity 8% (effective annual rate)


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Bond Price to MaturityBond Price to Maturity

$825

$850

$875

$900

$925

$950

$975

$1,000

$1,025

$1,050

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Time

Pric

e

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• For a bond with price $840.34 at time 0, here’s a plot of price as time progresses from 0 to 4.5 years assuming a constant yield to maturity of 12%

• The plot is computed using the DCF formula. These prices are referred to as ‘dirty’ prices. There are also ‘clean’ prices.

Homework 13Homework 13• Plot price v. time for the following bond

o Time to maturity 10 yearso Annual, nominal, simple coupon rate is 5% o Par value is $1000o Effective annual yield to maturity is a constant 6%o (It’s a saw tooth curve)


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Bonds part 1

Economy & Finance

Transcript of Bonds part 1