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Placing a value on a security
Attempting to determine the worth of a stream of future Cash flows

$ valued higher the earlier it is received
$ now is worth more than $ later

Time Value of Money
Deals with equivalence relationships between cash flows with different dates.

Discount
Reduce value based on how much time passes before money is paid.
I.e. receive 9,500 dollars today in exchange for 10,000 one year from now. You discount the 10,000 based on the one year threshold before the money is repaid.

Interest Rate ( r ) Definition
Rate of return that reflects the relationship between differently dated cash flows.

Interest Rates can be thought of in 3 ways:
 1. Rate of Return  minimum rate of return an investor must receive in order to accept the investment.
 2. Discount rate  rate at which we would discount a future amount to find its value today
 3. Opportunity Cost  If instead of investing the 9,500, the investor decided to spend it, he would loose the interest rate earned over that period.

Opportunity Cost
Value that investors forgo by choosing a particular course of action.

Interest Rate ( r ) Formula
Real Risk Free Interest rate + Inflation Premium + Default Risk Premium + Liquidity Premium + Maturity Premium

Interest Rates (Economics)
Set by the marketplace by the forces of supply (investors) and demand (borrowers) of funds.

Real Riskfree Interest rate
Singleperiod interest rate for a completely riskfree security if no inflation were expected. Reflects the time preferences of individuals for current versus future consumption

Inflation Premium
Compensates investors for expected inflation.Reflects the average inflation rate expected over the maturity of the debt.

Inflation
Reduces purchasing power of a unit of currency  the amount of goods one can buy with a unit of currency.

Nominal riskfree interest rate
 Real risk free interest rate + Inflation premium. Typically the interest rate for short term government debt (US TBILL).
 90 day t bill represents the nominal risk free interest rate over 90 days.

Default risk premium
Compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount.

Liquidity premium
Compensates investors for the risk of loss relative to an investment's fair value if the investment needs to be converted to cash quickly.
US T bills do not have a liquidity premium because they can be bought and sold in large amounts without affecting their market price.

Liquidity premium on financial instruments
Many bonds of small issuers trade infrequently after they are issued. The interest rate on such bonds includes a liquidity premium reflecting the relatively high costs of selling a position (including the impact on price)

Maturity premium
Compensates investors for the increased sensitivity of the MV of debt to a change in market interest rates as maturity is extended.
The difference between the interest rate on longer maturity treasury debt vs. short term treasury debt reflects a positive maturity premium for the longer term debt and possibly a different inflation premium as well.

Future Value of a Single Cash Flow (one period)
FORMULA
 PV = Present value of investment
 FV = future value of investment
 N = # of periods from today
 r = rate of interest per period

Simple Interest
Interest rate * principal (interest earned on principal investment)

Principal
Amount of funds originally invested

Compounding
Interest earned on interest

Future Value of a Like Cash Flow (multiple periods)  Compounding
FORMULA
 PV = present value of investment
 FV = future value of investment
 N = # of compounding periods
 r = rate of interest per period
** Stated interest rate and compounding periods must be compatible (i.e. annual interest rate to annual compounding periods)

Future Value Factor
Links today's present value with tomorrow's future value.
Future value to be received 5 periods from today (N = 5).
PV and FV are separated in time through the factor

PV & FV separated in time has 3 important consequences:
 1. Can add amounts of money only if they are indexed at the same point in time
 2. For a given interest rate, the FV increases with the number of periods
 3. For a given # of periods, the FV increases with the interest rate

Stated Annual Interest Rate ( )
Also known as the quoted annual interest rate. Rate quoted by bank on an annual basis. Normally quoted on annual basis even if instrument compounds more frequently.
Rate is strictly a quoting convention as often times the actual monthly rate x 12 will not be exactly the stated amount.

Future Value w/ more than one compounding period (i.e. semiannual, quarterly):
FORMULA
 = stated annual interest rate
 m = number of compounding periods per year (monthly, daily, etc.)
 N = # of years
*Periodic rate and the # of compounding periods must be compatible (yearly to yearly)

Continuous Compounding Truism
When number of compounding periods is infinite. The more frequent the compounding, the larger ending amount.

Effective Annual Rate (EAR)
The actual rate of interest earned after compounding is taken into account (vs. stated annual rate).
i.e. $1 earning 8.16% interest annual compounding earns the same as $1 earning 8% interest semiannual compounding.

Continuous Compounding
FORMULA
= e (2.7182818) raised to the stated annual interest rate ( ) * the number of years (N)

Effective Annual Rate (EAR)
FORMULA
Periodic interest rate = Stated annual interest rate divided by m
m = # of compounding periods in the year

EAR with continuous compounding
FORMULA

Annuity
finite set of level sequential cash flows

Ordinary Annuity
Annuity that has first cash flow at one period from now (t=1)

Annuity Due
Annuity that has first cash flow that occurs immediately (t=0)

Perpetuity
Perpetual annuity. Set of level neverending sequential cash flows with the first cash flow occurring one period from now.

Future Value of Ordinary Annuity
FORMULA
Term in brackets = future value annuity factor
 A = annuity amount
 r= interest rate per period
 N = number of time periods

Present Value Factor
FORMULA
Discounts future value to present value.

Present Values  Relation to discount rate & # of periods.
1. For a given discount rate, the farther in the future the amount to be received, the smaller that amount's present value.
2. Holding time constant, the larger the discount rate, the smaller the present value of the future amount.

Quoted interest rate.
# of periodic interest rate payments * the number of compounding periods in the year.

Present value w/ compounding
FORMULA
 m = # of compounding periods in the year
 = quoted annual interest rate
 N = number of years

PV of a series of equal cash flows (present value of annuity)
FORMULA
Value in brackets = present value annuity factor
 A = annuity amount
 r = interest rate per period (must correspond to frequency of annuity payments (annual, quarterly)
 N = # of annuity payments

PV of infinite series of equal cash flows (perpetuity)
FORMULA
Only works when interest rates are positive.
Only valid for perpetuity with level payments (some government bonds and preferred stocks can fit perpetuity)

PV of series of unequal cash flows
Must find PV of each cash flow and add the results.

Growth Rate
FORMULA
 1/N = square root
 N = # of years

Cash Flow Additivity Principal
Important for solving for uneven cash flows. Says that dollar amounts indexed at the same point in time are additive.

PV & FV are same measures separated in time.
PV & FV are same measures separated in time.

