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Compound Interest
Interest on interest

Future Value (FV)
Projecting cash flows forward on the basis of an appropriate compound interest rate to the end of the investment's life.

Present Value (PV)
Brings cash flows from investment back to beginning of the investment's life based on appropriate compound rate of return.

PV & FV
Allow for comparing investment alternatives based on investment's cash flows at a common point in time usually at the end of the investment's life (FV) or the beginning of the investment horizon (PV)

Time Line Key Points
Point 0
Cash Outflows
Cash Inflows
 Point 0 = Cash flow that occurs in the present (Today)
 Outflows are given a negative sign
 Inflows are given a positive sign

Discounting
Process to move cash flows to beginning of time line (computation of Present Value)

Compounding
Process to move cash flows to end of time line (computation of Future Value)

Time Line Example:
INSERT IMAGE HERE (PAGE 104)
Cash flows occur at the end of the period depicted on the time line. End of one period = beginning of next period.
 I.e. t=2 = end of second year AND beginning of 3rd year.
 Cash flow at beginning of Year 3 same as t=2

Interest Rates
Measure of time value of money. Risk differences in financial securities lead to differences in equilibrium interest rate.
ALSO KNOWN AS DISCOUNT RATE
ALSO KNOWN AS OPPORTUNITY COST OF CURRENT CONSUMPTION

Required Rate of Return
Equilibrium interest rate. Market rate of return is return that investors and savers require to get them to willingly lend their funds.

Interest Rate Examples
Discounting:
vs.
Opportunity Cost
Discounting  If you can burrow funds at 10% interest, you can discount those payments to be made in the future at that rate in order to get their equivalent value in today's dollars.
Opportunity Cost  If market interest rate on 1 yr securities is 5%, earning an additional 5% is the opportunity forgone by spending the money instead of saving it over that year.

Interest Rate Components (4)
Nominal Risk Free Rate + Default Risk Premium + liquidity risk Premium + Maturity Risk Premium

Real RiskFree Rate
Theoretical risk free rate w/ no inflation expectation.

Nominal Risk Free Rate
Risk Free rate with inflation built in. Contain inflation premium.
Nominal Risk Free Rate = Rate on short term tbills.

Default Risk
Risk that borrower will not make the promised payments in a timely manner

Liquidity Risk
Risk of receiving less than fair value for an investment if it must be sold for cash quickly.

Maturity Risk
Prices for long term more volatile than shorter term. This premium covers the risk of that volatility. (short term bonds vs. long term bonds)

Effective Annual Rate (EAR)
Rate of interest that investors actually realize as a result of the compounding (i.e. 8% stated annual rate compounded quarterly is actually 2% per quarter which is not exactly equal to 8% annually)
Represents the annual rate of return actually being earned after adjustments have been made for different compounding periods.
Greater the compounding periods (i.e. semiannual, quarterly, monthly), the greater the EAR > stated annual rate.

Effective Annual Rate (EAR)
FORMULA
 Rate = periodic rate = stated annual return/m
 m = number of compounding periods in the year.
Formula helpful when comparing investments that have different compounding periods to allow for applestoapples comparison

Future Value of Single Sum
FORMULA
Amount which investment will grow if placed in an account paying compound interest.
 PV = today's amount
 I/Y = rate of return per compounding period
 N = total number of compounding periods

Future value factor / Future value interest factor
Factor
Represents the compounding rate of an investment. This is the value that appears in interest factor tables.

Present Value of a Single Sum
FORMULA
Today's value of a cash flow that is to be received at some point in the future.
 I/Y = rate of return per compounding period
 N = total number of compounding periods

Present Value Factor / Present Value interest factor / discount factor
In the PV equation

Annuity
Stream of equal cash flows that occurs at equal intervals over a given period.

Ordinary Annuity
Most common type of annuity. Cash flows that occur at the end of each compounding period.

Annuity Due
Payments or receipts occur at the beginning of each period (i.e. first payment is today at t=0)

Future Value of Ordinary Annuity
CALCULATOR FORMULA
What is the future value of an ordinary annuity that pays 150 per year at the end of each of the next 15 years given the investment to earn a 7% rate of return
 N = 15
 I/Y = 7
 PMT = 150
 CPT > FV
= $3,769.35

Present Value of Ordinary Annuity
CALCULATOR FORMULA
What is the PV of an annuity that pays $200 per year at the end of each of the next 13 years given a 6% discount rate?
 N = 13
 I/Y = 6
 PMT = 200
 CPT > PV = $1,770.54

Present Value of Ordinary Annuity w/ beginning later than t=1
CALCULATOR FORMULA
What is the PV of four $100 endofyear payments if the first payment is to be received three years from today and the appropriate rate of return is 9%
1. Find present value of annuity as of the end of year 2 (PV2) ; N=4 , I/Y = 9 ; PMT = 100 ; FV = 0 ; CPT > PV = $323.97
 2. Find Present value (t=0) of PV2 from above
 N = 2 ; I/Y = 9; PMT = 0; FV = 323.97 ; CPT > PV = PV0 = $272.68

PV Annuity Function on Calculator in END mode
Gives you the value ONE PERIOD BEFORE THE ANNUITY BEGINS. For example, if the annuity begins at t = 3 (3rd year start), we discount the result for only two years to get present (t=0) value.

PV of Bond's Cash Flows
CALCULATOR FORMULA
A bond will make coupon interest payments of 70 euros at the end of each year and will also pay its face value of 1,000 euros at maturity in five years. Discount rate = 8%. What is the PV of the bond's promised cash flows?
PV = PV of 5 70 euro payments (ordinary annuity) + PV of 1,000 lump sum five years from now.
 N = 5
 PMT = 70
 I/Y = 8
 FV = 1000
 CPT PV = 960

Future Value of Annuity Due
CALCULATOR Change
Have to set calculator to BGN mode (beginning) vs. END mode
Payments are made at beginning of period, but FV of annuity due is calculated as of the end of the last period.

FV of Annuity Due
FORMULA
 FVA_D = FV of annuity due
 FVA_O = FV of ordinary annuity

FV of Annuity Due
CALCULATOR FORMULA
What is the FV of an annuity that pays 100 per year at the beginning of each of next 3 years commencing today. Cash flows invested at annual rate of 10%.
 MODE = BGN
 N = 3
 I/Y = 10
 PMT = 100
 CPT > FV = 364.10

PV of Annuity Due
With an annuity due, there is one less discounting period since the first cash flow starts at t=0 vs. t=1
PV of annuity due > PV of ordinary annuity

PV of Annuity Due
FORMULA
 PVA_D = PV Annuity Due
 PVA_O = PV ordinary Annuity

PV of Annuity Due
CALCULATOR FORMULA
Given a discount rate of 10%, what is the PV of a 3 year annuity that makes a series of 100 payments at the beginning of the next three years, starting today?
 Mode = BGN
 N = 3
 I/Y = 10
 PMT = 100
 CPT > PVA_D = $273.55

Perpetuity
Financial instrument that pays fixed amount of money at set intervals forever. (British Consol Bonds and Preferred Stocks)

PV of Perpetuity
FORMULA
Fixed periodic cash flow / periodic rate of return

PV of deferred perpetuity
FORMULA
Preferred stock will be paid 4.50 dividend in four years and does not pay any dividends in years 13. 8% required rate of return. What is PV of deferred perpetuity?
Perpetuity = 4.5/.08 = 56.25
56.25 @ t=4
Discount for remaining 3 periods = 56.25/(1.08)^3 = 44.65

PV of series of unequal cash flows
CALCULATOR FORMULA
Insert page 120 from notes

PV and FV with compounding periods <> annual
Example = quarterly
Divide the annual stated rate by # periods (i.e. quarterly would divide rate by 4).
 Calculator:
 I/Y = interest rate / 4
 N = # of years * 4

Loan amortization
Process of paying off a loan with a series of periodic loan payments. Portion of outstanding loan is paid off (amortized) with each payment.

Loan Payments  Annual
CALCULATOR CALCULATION
Company borrows 50k for five years. Bank lends money @ 9% rate and requires that loan be paid off in five equal end of year payments. Calculate payment amount.
 N = 5
 I/Y = 9
 PV = 50000
 CPT > PMT = 12,854.62
(FV = 0 as the loan will be completely paid off in five years)

Loan Payments  Quarterly
CALCULATOR CALCULATION
Company borrows 50k for five years. Bank lends money @ 9% rate and requires that loan be paid off in quarterly payments over five year period. Calculate payment amount.
 N = 5 * 4 = 20
 I/Y = 9/4 = 2.25
 PV = 50000
 CPT > PMT = 3,132.10

Amortization Schedule
IMAGE AND INFO FROM PAGE 125

Principal @ interest component of a specific loan payment
CALCULATION
Suppose you borrowed 10k @ 10% interest to be paid semiannually over ten years. Calculate the amount of the outstanding balance after the second payment is made.
 Part 1  calculate payment
 N = 20, I/Y = 5, PV = 10,000 CPT > PMT = 802.43
Part 2  Calculate principal vs. int component:
 Payment 1:
 Interest = (10,000)(.05) = 500
 Principal = 802.43  500 = 302.43
 Payment 2
 Interest = (10,000  302.43)(.05) = 484.88
 Principal = 802.43484.44 = 317.55
Remaining Balance = 10,000  302.43  317.55 = 9,380.02

Compute annuity payment needed to achieve a given FV
At an expected rate of return of 7%, how much must be deposited at end of each year for the next 15 years to accumulate 3,000?
 N = 15
 I/Y = 7
 FV = +3,000
 CPT > PMT = 119.38 (ignore negative)

Compute loan payment
Payment on loan for 2k repaid equal end of year payments with 6% interest paid over 13 years.
 N = 13
 I/Y = 6
 PV = 2000
 CPT > PMT = 225.92

Calculator Sign Convention
PMT & FV must have different signs or your calculator will give error.

Sales at Acme for the first five years have been 4.5m, 5.7m, 5.3m, 6.9m, and 7.1m. Calculate the compound annual growth rate of the sales over the period.
 Calculator:
 FV = 7.1 ; PV = 4.5 ; N = 4 ; CPT > I/Y = 12.08%
Math: (7.1/4.5)^1/4  1 = 12.1%

