MONEY & BOND MARKETSMONEY & BOND MARKETS

AN INTRODUCTION TO MONETARY ECONOMICSAN INTRODUCTION TO MONETARY ECONOMICS

Interest Rate consists of 3 components: Interest Rate consists of 3 components:

1) inflation 1) inflation

2) reward for postponing consumption 2) reward for postponing consumption

3) risk3) risk

R ≈ E(e) + time preference + risk premiumR ≈ E(e) + time preference + risk premium

Real = Inflation-adjusted RReal = Inflation-adjusted RR R ≈≈ R − e R − e

RRRR = [(1+R)/(1+e)] − 1 = [(1+R)/(1+e)] − 1

So, real returns reflect (time preference + risk).So, real returns reflect (time preference + risk).

DEFINITIONS:DEFINITIONS:

In the money market, we use the concept of interest rate; in risky In the money market, we use the concept of interest rate; in risky assets, we use the concept of return.assets, we use the concept of return.

ReturnReturn: R: Rtt = ( V = ( Vtt / V / Vt-1t-1 ) – 1 ) – 1 (% change in value(% change in value))

Log Return = ln (VLog Return = ln (Vtt / V / Vt-1t-1))

Interest RateInterest Rate: : i i oror r r

rrss : annualized simple interest rate : annualized simple interest rate

rrcc : annualized compound interest rate : annualized compound interest raterrpp : periodic interest rate ( : periodic interest rate (the interest to be accrued the interest to be accrued

between tbetween t00 and T, the holding period and T, the holding period))

In money market transactions, bankers always quote rIn money market transactions, bankers always quote rs s , however , however the effective annual rate of return is rthe effective annual rate of return is rcc. .

MONEY MARKETS – INTEREST CALCULATIONSMONEY MARKETS – INTEREST CALCULATIONS

Calculating rCalculating rhphp : : rrhphp = (1+r = (1+rss/n/n))kk – 1 – 1

k is the number of compoundingk is the number of compoundingss over the holding period over the holding period (how many times the initial money is reinvested), r(how many times the initial money is reinvested), rss is is the interest rate per unit period of time (typically 1 year).the interest rate per unit period of time (typically 1 year).

If the interest is not reinvestedIf the interest is not reinvested, you do not need to , you do not need to compound (to take power), rather you should simply compound (to take power), rather you should simply multiply: multiply: rrpp = r = rss * k * k

Converting rConverting rss into r into rcc: : rrcc = (1+r = (1+rss/n)/n)n n –1–1

where n is the number of compounding (reinvestment) per where n is the number of compounding (reinvestment) per year.year.

As the compounding frequency (n) increases, the effective As the compounding frequency (n) increases, the effective annual rate grows.annual rate grows.

The limit is continuous compounding:The limit is continuous compounding: rrcc= e= ersrs rrpp= = eerstrst

With compounding (reinvesting at the same rate), With compounding (reinvesting at the same rate), money grows exponentially (geometrically).money grows exponentially (geometrically).

Example:Example: (1+0.07) (1+0.07)5050 = 29.46 (that is, if you invest $1000 = 29.46 (that is, if you invest $1000 at 7% for 50 years reinvesting the interest, you end up at 7% for 50 years reinvesting the interest, you end up with $29,460 at the end of 50th year) (1+0.09)with $29,460 at the end of 50th year) (1+0.09)5050 = = 74.36 (at 9% you end up with $74,360; so a small 74.36 (at 9% you end up with $74,360; so a small difference in rate makes a huge differencedifference in rate makes a huge difference))

Money Markets:Money Markets:

Fed Funds Market, LIBOR, Repo, Money Fed Funds Market, LIBOR, Repo, Money Market Mutual FundsMarket Mutual Funds

Bank discount: P = F(1-rBank discount: P = F(1-rpp))

Present Value of a Future Monetary Value:Present Value of a Future Monetary Value:PVPV00(C(Ctt) = C) = Ctt / (1+r / (1+rhphp)) (1+r) is called gross return (1+r) is called gross return

General Formula for Valuation of All Financial Assets:General Formula for Valuation of All Financial Assets:ΣΣnn

t=1t=1 PV PV00(CF(CFtt) = PV) = PV00(C(C11) + PV) + PV00(C(C22) +….+ PV) +….+ PV00(C(Cnn) ) = Σ= Σnn

t=1t=1[C[Ctt / (1+r) / (1+r)tt]]

General Rule: The value of any asset should equal to the General Rule: The value of any asset should equal to the sum of the PV’s of its all cash flows.sum of the PV’s of its all cash flows.

Under specific assumptions about the cash flow pattern, this Under specific assumptions about the cash flow pattern, this calculation is tractablecalculation is tractable::

Perpetuity: A constant periodic cash flow C forever.Perpetuity: A constant periodic cash flow C forever.PV = C / rPV = C / r

Growing Perpetuity: A periodic cash flow growing at a Growing Perpetuity: A periodic cash flow growing at a constant rate g, forever. constant rate g, forever.

PV = C / (r-g) (r>g)PV = C / (r-g) (r>g)

Annuity: A constant periodic cash flow C until a specified Annuity: A constant periodic cash flow C until a specified date T in the future.date T in the future.

Perpetuity (Consol)Perpetuity (Consol)

A constant stream of cash flows that lasts foreverA constant stream of cash flows that lasts forever

0

…1

C

2

C

3

C

32 )1()1()1( r

C

r

C

r

CPV

r

CPV

Growing PerpetuityGrowing Perpetuity

A growing stream of cash flows that lasts foreverA growing stream of cash flows that lasts forever

0

…1

C

2

C×(1+g)

3

C ×(1+g)2

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

CPV

gr

CPV

BONDSBONDS

• Two main types of Bonds:Two main types of Bonds:

A) Pure Discount BA) Pure Discount Billills: s: no interim payments, one single repayment at the no interim payments, one single repayment at the

maturitymaturity

B) Coupon Bonds:B) Coupon Bonds:Pays coupons C at regular intervals + principal at Pays coupons C at regular intervals + principal at

the maturitythe maturity T. T.

Pure Discount BillsPure Discount Bills Make no periodic interest payments (coupon Make no periodic interest payments (coupon

rate = 0%)rate = 0%)

The entire yield to maturity comes from the The entire yield to maturity comes from the difference between the purchase price and difference between the purchase price and the par value.the par value.

Cannot sell for more than par valueCannot sell for more than par value

Treasury Bills and principal-only Treasury Treasury Bills and principal-only Treasury strips are good examples.strips are good examples.

generally generally usedused for short term borrowing for short term borrowing

Pure Pure DDiscount iscount BillBillss::

N: par value days to maturity= T-tN: par value days to maturity= T-t

rrss: simple annualized rate: simple annualized rate

rrcc: compound annualized rate : compound annualized rate

n = 365/(T-t) n = 365/(T-t) i.e.: number of compounding per yeari.e.: number of compounding per year

hpt r

NB

1n

cst r

N

nr

NB

/1)1()1(

Coupon BondsCoupon Bonds: :

BBtt= =

coupon ratecoupon rate (C%) (C%): indicates the amount of annual : indicates the amount of annual coupon payments.coupon payments.

Current yieldCurrent yield= C / B= C / Btt

ytmytm: indicates the annual market interest rate that : indicates the annual market interest rate that applies to the bondapplies to the bond

nytm

FC

ytm

C

ytm

C

ytm

C

)1(........

)1()1()1( 32

Yield to Maturity: BondsYield to Maturity: Bonds

Coupon Bond (Coupon rate = 10% = Coupon Bond (Coupon rate = 10% = CC//FF))

P $100

1 i

$100

1 i 2 $100

1 i 3 ... $100

1 i 10 $1000

1 i 10

P C

1 i

C

1 i 2 C

1 i 3 ... C

1 i n F

1 i n

Consol: Fixed coupon payments of $C forever

P C

ii

C

P

Relationship Between Price and Relationship Between Price and Yield to MaturityYield to Maturity

Three interesting facts in Table 3-1Three interesting facts in Table 3-11.1. When bond is at par, yield equals coupon rateWhen bond is at par, yield equals coupon rate2.2. Price and yield are negatively relatedPrice and yield are negatively related3.3. Yield greater than coupon rate when bond price Yield greater than coupon rate when bond price

is below par valueis below par value

YTM and Bond ValueYTM and Bond Value

800

1000

1100

1200

1300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Discount Rate

Bon

d V

alu

e

6 3/8

When the YTM < coupon, the bond trades at a premium.

When the YTM = coupon, the bond trades at par.

When the YTM > coupon, the bond trades at a discount.

The returns from holding a bond for less than The returns from holding a bond for less than until its maturity consists of two until its maturity consists of two components:components:

1.1. Interest gainsInterest gains2.2. Capital gains or losses (the price of a bond Capital gains or losses (the price of a bond

is inversely related to ytm, i.e. market is inversely related to ytm, i.e. market interest rates).interest rates).

If you hold a bond until its maturity, you If you hold a bond until its maturity, you receive a certain return (which equals to receive a certain return (which equals to its interest rate, ytm), hence fixed income.its interest rate, ytm), hence fixed income.

Yield Curve = Term Structure of Interest RatesYield Curve = Term Structure of Interest Rates

A Graph showing the effective annualized interest A Graph showing the effective annualized interest rates of different terms.rates of different terms.

Duration: Duration:

The weighted average maturity terms of cash flows of The weighted average maturity terms of cash flows of a bond. It indicates the responsiveness of the bond a bond. It indicates the responsiveness of the bond price to changes in ytm. D = (∂B/B) / ∂r . As such, it price to changes in ytm. D = (∂B/B) / ∂r . As such, it is a measure of the riskiness of a bond. is a measure of the riskiness of a bond.

EurobondsEurobonds

Callable Bond:Callable Bond:

Convertible Bond:Convertible Bond:

Inflation-linked bonds: e.g. TIPSInflation-linked bonds: e.g. TIPS

DUR tCPt

1 i tt 1

n

CPt

1 i tt 1

n

DurationDuration

Key facts about durationKey facts about duration1.1. All else equal, when the maturity of a All else equal, when the maturity of a

bond lengthens, the duration rises as bond lengthens, the duration rises as wellwell

2.2. All else equal, when interest rates rise, All else equal, when interest rates rise, the duration of a coupon bond fallthe duration of a coupon bond fall

Risk Structure of Long Bonds in the U.S.Risk Structure of Long Bonds in the U.S.

Bond RatingsBond Ratings

Liquidity Premium TheoryLiquidity Premium Theory

2222

Market Market Predictions Predictions

of Future of Future Short Short RatesRates

Case: Interpreting Yield Curves, 1980–2008Case: Interpreting Yield Curves, 1980–2008