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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

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

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 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

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

- lump sum payment
- PVs decrease as interest rates increase

Calculating Present Value (PV) of a Lump Sum

PV = FVn(1/(1 + i/m))nm = FVn(PVIFi/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

Calculation of Present Value (PV) of an Annuity

nm

PV = PMT (1/(1 + i/m))t = PMT(PVIFAi/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

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(PVIFAi/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

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

Future Values Equations

- FV of lump sum equation
- FVn = PV(1 + i/m)nm = PV(FVIF i/m, nm)

- FV of annuity payment equation
- (nm-1)
- FVn = PMT (1 + i/m)t = PMT(FVIFAi/m, mn)
- (t = 1)

Relation between Interest Rates and Present and Future Values

Present

Value

(PV)

Future

Value

(FV)

Interest Rate

Interest Rate

Equivalent Annual Return (EAR) Values

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%

Discount Yields Values

Money market instruments (e.g., Treasury

bills and commercial paper) that are bought

and sold on a discount basis

idy = [(Pt - Po)/Pf](360/h)

Where:

Pf = Face value

Po = Discount price of security

Single Payment Yields Values

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

fed funds) that pay interest only once during

their lives: at maturity

ibey = ispy(365/360)

Loanable Funds Theory Values

- 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

Supply of Loanable Funds Values

Demand

Supply

Interest

Rate

Quantity of Loanable Funds

Supplied and Demanded

Funds Supplied and Demanded by Various Groups (in billions of dollars)

Funds SuppliedFunds 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

Determination of Equilibrium Interest Rates of dollars)

D

S

Interest

Rate

I H

i

E

I L

Q

Quantity of Loanable Funds

Supplied and Demanded

Effect on Interest rates from a Shift in the Demand Curve for or Supply curve of Loanable Funds

Increased supply of loanable funds

Increased demand for loanable funds

DD*

Interest

Rate

SS

SS

DD

DD

SS*

i**

E*

E

i*

E

i*

E*

i**

Q*

Q**

Q*

Q**

Quantity of

Funds Supplied

Quantity of

Funds Demanded

Factors Affecting Nominal Interest Rates for or Supply curve of Loanable Funds

- 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

(continued)

- Liquidity Risk for or Supply curve of Loanable Funds
- 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

Inflation and Interest Rates: The Fischer Effect for or Supply curve of Loanable Funds

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%

Default Risk and Interest Rates for or Supply curve of Loanable Funds

The risk that a security’s issuer will default

on that security by being late on or missing

an interest or principal payment

DRPj = ijt - iTt

Example: DRPAaa = 7.55% - 6.35% = 1.20%

DRPBbb = 8.15% - 6.35% = 1.80%

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.

im = ic(1 - ts - tF)

Where: ic = Interest rate on a corporate bond

im = Interest rate on a municipal bond

ts = State plus local tax rate

tF = Federal tax rate

Term to Maturity and Interest Rates: Yield Curve Bonds

(a) Upward sloping

(b) Inverted or downward

sloping

(c) Humped

(d) Flat

Yield to

Maturity

(a)

(d)

(c)

(b)

Time to Maturity

Term Structure of Interest Rates Bonds

- 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

Forecasting Interest Rates Bonds

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

__

R2 = [(1 + R1)(1 + (f2))]1/2 - 1

where

f2 = expected one-year rate for year 2, or the implied

forward one-year rate for next year

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