Chapter Two. Determinants of Interest Rates. McGraw-Hill /Irwin. Copyright 2001 by The McGraw-Hill Companies, Inc. All rights reserved.

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(1)

Chapter Two

Determinants of

Interest Rates

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Interest Rate Fundamentals

• Nominal interest rates - the interest

rate actually observed in financial

markets

– directly affect the value (price) of most

securities traded in the market

– affect the relationship between spot and

forward FX rates

• Nominal interest rates - the interest

rate actually observed in financial

markets

– directly affect the value (price) of most

securities traded in the market

– affect the relationship between spot and

forward FX rates

(3)

Time Value of Money and Interest Rates

• Assumes the basic notion that a dollar

received today is worth more than a dollar

received at some future date

• Compound interest

– interest earned on an investment is reinvested

• Simple interest

– interest earned on an investment is not

reinvested

• Assumes the basic notion that a dollar

received today is worth more than a dollar

received at some future date

• Compound interest

– interest earned on an investment is reinvested

• Simple interest

– interest earned on an investment is not

reinvested

(4)

Calculation of Simple Interest

Value = Principal + Interest

Example:

$1,000 to invest for a period of two years at 12 percent

Value = $1,000 + $1,000(.12)(2)

= $1,240

Value = Principal + Interest

Example:

$1,000 to invest for a period of two years at 12 percent

Value = $1,000 + $1,000(.12)(2)

(5)

Value of Compound Interest

Value = Principal + Interest + Compounded interest

Value = $1,000 + $1,000(12)(2) + $1,000(12)(2)

= $1,000[1 + 2(12) + (12)

2

]

= $1,000(1.12)

2

= $1,254.40

Value = Principal + Interest + Compounded interest

Value = $1,000 + $1,000(12)(2) + $1,000(12)(2)

= $1,000[1 + 2(12) + (12)

2

]

= $1,000(1.12)

2

(6)

Present Values

• PV function converts cash flows received over a future

investment horizon into an equivalent (present) value

by discounting future cash flows back to present using

current market interest rate

– lump sum payment

• a single cash payment received at the end of some

investment horizon

– annuity

• a series of equal cash payments received at fixed

intervals over the investment horizon

• PVs decrease as interest rates increase

• PV function converts cash flows received over a future

investment horizon into an equivalent (present) value

by discounting future cash flows back to present using

current market interest rate

– lump sum payment

• a single cash payment received at the end of some

investment horizon

– annuity

• a series of equal cash payments received at fixed

intervals over the investment horizon

(7)

Calculating Present Value (PV) of a Lump

Sum

PV = FV

n

(1/(1 + i/m))

nm

= FV

n

(PVIF

i/m,nm

)

where:

PV = present value

FV = future value (lump sum) received in n years

i = simple annual interest

n = number of years in investment horizon

m = number of compounding periods in a year

PVIF = present value interest factor of a lump sum

PV = FV

n

(1/(1 + i/m))

nm

= FV

n

(PVIF

i/m,nm

)

where:

PV = present value

FV = future value (lump sum) received in n years

i = simple annual interest

n = number of years in investment horizon

m = number of compounding periods in a year

(8)

Calculation of Present Value (PV) of an

Annuity

nm

PV = PMT

(1/(1 + i/m))

t

= PMT(PVIFA

i/m,nm

)

t = 1

where:

PV = present value

PMT = periodic annuity payment received

during investment

i = simple annual interest

n = number of years in investment horizon

m = number of compounding periods in a year

PVIFA = present value interest factor of an annuity

nm

PV = PMT

(1/(1 + i/m))

t

= PMT(PVIFA

i/m,nm

)

t = 1

where:

PV = present value

PMT = periodic annuity payment received

during investment

i = simple annual interest

n = number of years in investment horizon

m = number of compounding periods in a year

(9)

Calculation of Present Value of an Annuity

You are offered a security investment that pays $10,000 on

the last day of every quarter for the next 6 years in

exchange for a fixed payment today.

PV = PMT(PVIFA

i/m,nm

)

at 8% interest - = $10,000(18.913926) = $189,139.26

at 12% interest - = $10,000(16.935542) = $169,355.42

at 16% interest - = $10,000(15.246963) = $152,469.63

You are offered a security investment that pays $10,000 on

the last day of every quarter for the next 6 years in

exchange for a fixed payment today.

PV = PMT(PVIFA

i/m,nm

)

at 8% interest - = $10,000(18.913926) = $189,139.26

at 12% interest - = $10,000(16.935542) = $169,355.42

at 16% interest - = $10,000(15.246963) = $152,469.63

(10)

Future Values

• Translate cash flows received during an

investment period to a terminal (future)

value at the end of an investment horizon

• FV increases with both the time horizon and

the interest rate

• Translate cash flows received during an

investment period to a terminal (future)

value at the end of an investment horizon

• FV increases with both the time horizon and

the interest rate

(11)

Future Values Equations

FV of lump sum equation

FV

n

= PV(1 + i/m)

nm

= PV(FVIF

i/m, nm

)

FV of annuity payment equation

(nm-1)

FV

n

= PMT

(1 + i/m)

t

= PMT(FVIFA

i/m, mn

)

(t = 1)

FV of lump sum equation

FV

n

= PV(1 + i/m)

nm

= PV(FVIF

i/m, nm

)

FV of annuity payment equation

(nm-1)

FV

n

= PMT

(1 + i/m)

t

= PMT(FVIFA

i/m, mn

)

(12)

Relation between Interest Rates and

Present and Future Values

Present

Value

(PV)

Interest Rate

Future

Value

(FV)

Interest Rate

(13)

Equivalent Annual Return (EAR)

Rate returned over a 12-month period

taking the compounding of interest into

account

EAR = (1 + i/m)

m

- 1

At 8% interest - EAR = (1 + .08/4)

4

- 1 = 8.24%

At 12% interest - EAR = (1 + .12/4)

4

- 1 = 12.55%

Rate returned over a 12-month period

taking the compounding of interest into

account

EAR = (1 + i/m)

m

- 1

At 8% interest - EAR = (1 + .08/4)

4

- 1 = 8.24%

(14)

Discount Yields

Money market instruments (e.g., Treasury

bills and commercial paper) that are bought

and sold on a discount basis

i

dy

= [(P

t

- P

o

)/P

f

](360/h)

Where:

P

f

= Face value

P

o

= Discount price of security

Money market instruments (e.g., Treasury

bills and commercial paper) that are bought

and sold on a discount basis

i

dy

= [(P

t

- P

o

)/P

f

](360/h)

Where:

P

f

= Face value

(15)

Single Payment Yields

Money market securities (e.g., jumbo CDs,

fed funds) that pay interest only once during

their lives: at maturity

i

bey

= i

spy

(365/360)

Money market securities (e.g., jumbo CDs,

fed funds) that pay interest only once during

their lives: at maturity

(16)

Loanable Funds Theory

• A theory of interest rate determination that

views equilibrium interest rates in financial

markets as a result of the supply and demand

for loanable funds

• A theory of interest rate determination that

views equilibrium interest rates in financial

markets as a result of the supply and demand

for loanable funds

(17)

Supply of Loanable Funds

Interest

Rate

Quantity of Loanable Funds

Supplied and Demanded

(18)

Funds Supplied and Demanded by Various

Groups (in billions of dollars)

Funds Supplied

Funds Demanded

Households $31,866.4

$ 6,624.4

Business -- nonfinancial 7,400.0 30,356.2

Business -- financial 27,701.9 29,431.1

Government units 6,174.8

10,197.9

Foreign participants 6,164.8

2,698.3

Funds Supplied

Funds Demanded

Households $31,866.4

$ 6,624.4

Business -- nonfinancial 7,400.0 30,356.2

Business -- financial 27,701.9 29,431.1

Government units 6,174.8

10,197.9

Foreign participants 6,164.8

2,698.3

(19)

Determination of Equilibrium Interest

Rates

Interest

Rate

Quantity of Loanable Funds

Supplied and Demanded

D

S

I

H

i

I

L

E

Q

(20)

Effect on Interest rates from a Shift in the

Demand Curve for or Supply curve of

Loanable Funds

Increased supply of loanable funds

Quantity of

Funds Supplied

Interest

Rate

DD

SS

SS*

E

E*

Q*

i*

Q**

i**

Increased demand for loanable funds

Quantity of

Funds Demanded

DD

DD*

SS

E

E*

i*

i**

Q* Q**

(21)

Factors Affecting Nominal Interest

Rates

• Inflation

– continual increase in price of goods/services

• Real Interest Rate

– nominal interest rate in the absence of inflation

• Default Risk

– risk that issuer will fail to make promised

payment

• Inflation

– continual increase in price of goods/services

• Real Interest Rate

– nominal interest rate in the absence of inflation

• Default Risk

– risk that issuer will fail to make promised

payment

(22)

• Liquidity Risk

– risk that a security can not be sold at a

predictable price with low transaction cost on

short notice

• Special Provisions

– taxability

– convertibility

– callability

• Time to Maturity

• Liquidity Risk

– risk that a security can not be sold at a

predictable price with low transaction cost on

short notice

• Special Provisions

– taxability

– convertibility

– callability

• Time to Maturity

(23)

Inflation and Interest Rates: The

Fischer Effect

The interest rate should compensate an investor

for both expected inflation and the opportunity

cost of foregone consumption

(the real rate component)

i = Expected (IP) + RIR

Example: 5.08% - 2.70% = 2.38%

The interest rate should compensate an investor

for both expected inflation and the opportunity

cost of foregone consumption

(the real rate component)

i = Expected (IP) + RIR

(24)

Default Risk and Interest Rates

The risk that a security’s issuer will default

on that security by being late on or missing

an interest or principal payment

DRP

j

= i

jt

- i

Tt

Example: DRP

Aaa

= 7.55% - 6.35% = 1.20%

DRP

Bbb

= 8.15% - 6.35% = 1.80%

The risk that a security’s issuer will default

on that security by being late on or missing

an interest or principal payment

DRP

j

= i

jt

- i

Tt

Example: DRP

Aaa

= 7.55% - 6.35% = 1.20%

DRP

Bbb

= 8.15% - 6.35% = 1.80%

(25)

Tax Effects: The Tax Exemption of Interest

on Municipal Bonds

Interest payments on municipal securities are

exempt from federal taxes and possibly state and

local taxes. Therefore, yields on “munis” are

generally lower than on equivalent taxable bonds

such as corporate bonds.

i

m

= i

c

(1 - t

s

- t

F

)

Where:

i

c

= Interest rate on a corporate bond

i

m

= Interest rate on a municipal bond

t

s

= State plus local tax rate

t

= Federal tax rate

Interest payments on municipal securities are

exempt from federal taxes and possibly state and

local taxes. Therefore, yields on “munis” are

generally lower than on equivalent taxable bonds

such as corporate bonds.

i

m

= i

c

(1 - t

s

- t

F

)

Where:

i

c

= Interest rate on a corporate bond

i

m

= Interest rate on a municipal bond

t

s

= State plus local tax rate

t

F

= Federal tax rate

(26)

Term to Maturity and Interest Rates:

Yield Curve

Yield to

Maturity

Time to Maturity

(a)

(b)

(c)

(d)

(a) Upward sloping

(b) Inverted or downward

sloping

(c) Humped

(d) Flat

(27)

Term Structure of Interest Rates

• Unbiased Expectations Theory

– at a given point in time, the yield curve reflects the

market’s current expectations of future short-term

rates

• Liquidity Premium Theory

– investors will only hold long-term maturities if they

are offered a premium to compensate for future

uncertainty in a security’s value

• Market Segmentation Theory

– investors have specific maturity preferences and will

demand a higher maturity premium

• Unbiased Expectations Theory

– at a given point in time, the yield curve reflects the

market’s current expectations of future short-term

rates

• Liquidity Premium Theory

– investors will only hold long-term maturities if they

are offered a premium to compensate for future

uncertainty in a security’s value

• Market Segmentation Theory

– investors have specific maturity preferences and will

demand a higher maturity premium

(28)

Forecasting Interest Rates

Forward rate is an expected or “implied” rate

on a security that is to be originated at some

point in the future using the unbiased

expectations theory

_

_

R

2

= [(1 + R

1

)(1 + (f

2

))]

1/2

- 1

where

f

2

= expected one-year rate for year 2, or the implied

forward one-year rate for next year

Forward rate is an expected or “implied” rate

on a security that is to be originated at some

point in the future using the unbiased

expectations theory

_

_

R

2

= [(1 + R

1

)(1 + (f

2

))]

1/2

- 1

where

f

2

= expected one-year rate for year 2, or the implied

forward one-year rate for next year

Figure

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