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### Chapter 3

Time Value of Money

The Time Value of Money

- Interest Rate
- Simple Interest
- Compound Interest
- Future Value (Compounding)
- Present Value (Discounting)
- Annuities
- Loan Amortization
- Bond Valuation

Obviously, $10,000 today.

You already recognize that there is TIME VALUE TO MONEY!!

Which would you prefer -- $10,000 today or $10,000 in 5 years?

The Interest RateTIME allows you the opportunity to postpone consumption and earn INTEREST.

Why is TIME such an important element in your decision?

Why Time?TIME VALUE OF MONEY

- THE UNIVERSAL PREFERENCE FOR A DOLLAR TODAY OVER A DOLLAR AT SOME FUTURE TIME
- UNCERTAINTY (RISK)
- ALTERNATIVE USES
- INFLATION

INTEREST RATES

- THE PRICING MECHANISM FOR THE TIME VALUE OF MONEY
- REFLECT INVESTORS’ TIME PREFERENCES FOR MONEY
- MAY ALSO ACCOUNT FOR RISK AND INFLATION

INTEREST RATES REFLECT THE OPPORTUNITY COST OF NOT PUTTING MONEY TO ITS BEST USE.

SIMPLE INTEREST

- MEANS THAT ONLY THE ORIGINAL PRINCIPAL EARNS INTEREST OVER THE LIFE OF THE TRANSACTION.
- THE PRODUCT OF THE PRINCIPAL, THE TIME IN YEARS, AND THE ANNUAL INTEREST RATE.

Simple Interest Formula

FormulaSI = P0(i)(n)

SI: Simple Interest

P0: Principal Deposited today (t=0)

i: Interest Rate per Period

n: Number of Time Periods

SI = P0(i)(n)= $5,000(.08)(3) = $1,200

Assume that you deposit $5,000 in an account earning 8% simple interest for 3 years. What is the accumulated interest at the end of the 3rd year?

Simple Interest ExampleFV = P0 + SI = $5,000+ $1,200 =$6,200

Future Valueis the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

What is the Future Value (FV) of the deposit?

Simple Interest (FV)The Present Value is simply the $5,000 you originally deposited. That is the value today!

Present Valueis the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

What is the Present Value (PV) of the previous problem?

Simple Interest (PV)COMPOUND INTEREST

- WHEN INTEREST IS EARNED AND CONVERTED TO PRINCIPAL MORE THAN ONCE DURING THE TIME OF THE INVESTMENT.
- THE INTERVAL BETWEEN SUCCESSIVE CONVERSIONS IS CALLED THE CONVERSION PERIOD.
- MONTHLY
- QUARTERLY
- DAILY
- SEMI-ANNUALLY
- ANNUALLY

COMPOUND AMOUNT - THE TOTAL AMOUNT AT THE END OF THE CONVERSION PERIOD.

- COMPOUND INTEREST - IS THE DIFFERENCE BETWEEN THE COMPOUND AMOUNT AND THE BEGINNING PRINCIPAL.

COMPOUND INTEREST RATE (PERIODIC RATE) IS THE RATE PER CONVERSION PERIOD THAT IS CHARGED ON THE OUTSTANDING BALANCE AT THE BEGINNING OF THAT PERIOD.

- Example: an annual rate of 12% is converted to a quarterly rate by dividing the annual rate by the number of conversions periods in a year; therefore, 12% ÷ 4 = 3% quarterly rate.

NOMINAL ANNUAL RATE - THE PERIODIC RATE CONVERTED TO AN ANNUAL BASIS.

- Example: a monthly rate of 1.5%/month is converted to an annual rate by multiplying by the number of conversions periods in a year; therefore, the annual rate is 12 x 1.5 = 18%
- EFFECTIVE ANNUAL RATE - THE RATE OF INTEREST ACTUALLY EARNED IN A YEAR.
- With compounding the effective annual rate will be greater than the nominal annual rate.

Why Compound Interest?

Future Value (U.S. Dollars)

COMPOUNDING

- FUTURE VALUE OF A PRESENT SUM
- FUTURE VALUE OF A SERIES OF PAYMENTS

Future Value of a Present Sum (graphic)

Assume that you deposit $5,000 at a compound interest rate of 8% for 2 years.

0 12

8%

$5,000

$5,832

$5,400

FV2

COMPOUNDING

FUTURE VALUE OF A PRESENT SUM

FV n = PVO (1+i)n

OR

FUTURE VALUE = PRESENT VALUE *

(1 + COMPOUND RATE) CONVERSION PERIODS

Future Value of a Present Sum (formula)

FV1 = P0 (1+i)1 = $5,000(1.08) = $5,400

Compound Interest

You earned $400 interest on your $5,000 deposit over the first year.

This is the same interest you would earn under simple interest.

Future Value of a Present Sum (formula)

FV1 = P0(1+i)1 = $5,000 (1.08) = $5,400

FV2 = FV1 (1+i)1 = {P0 (1+i)}(1+i) = P0(1+i)2

=$5,000(1.08)(1.08)

= $5,000(1.08)2 = $5,832.00

You earned an EXTRA$32.00 in Year 2 with compound over simple interest.

General Formula for Future Value

FV1 = P0(1+i)1

FV2 = P0(1+i)2

General Future Value Formula:

FVn = P0 (1+i)n

or FVn = P0 (FVD in) -- See Table A1

etc.

Valuation Using Table A1

FVD I,nis found in Table A1

Using Future Value Tables

FV2 = $5,000 (FVD 8%,2) = $5,000 (1.166) = $5,830 [ due to rounding]

PROBLEM:

$5000 @ 8% COMPOUNDED ANNUALLY FOR 3 YEARS

FV n = 5000*(1.08)3

FV n =5000(1.259712) = 6,298.56

PROBLEM:

$5000 @ 8% COMPOUNDED QUARTERLY FOR 3 YEARS

FV n = 5000*(1.02)12

FV n =5000(1.2682418) = 6,341.21

Example Problem

Julie Miller wants to know how large her $10,000 deposit will become at a compound interest rate of 10% for 5 years.

0 1 2 3 4 5

10%

$10,000

FV5

Problem Solution

- Calculation based on general formula:FVn = P0 (1+i)nFV5= $10,000 (1+ 0.10)5 = $16,105.10

- Calculation based on Table A1: FV5= $10,000(FVD 10%, 5)= $10,000(1.6105) = $16,105

DISCOUNTING

- PROCEDURE WHEREBY THE PRESENT VALUE OF FUTURE INCOME IS DETERMINED.
- PRESENT VALUE OF A FUTURE PAYMENT
- PRESENT VALUE OF A SERIES OF PAYMENTS

PRESENT VALUE OF A FUTURE PAYMENT

PVO = FVN /(1+i)n

OR

PRESENT VALUE = FUTURE VALUE /

(1 + COMPOUND RATE) CONVERSION PERIODS

Present Value of a Single Deposit (graphic)

Assume that you need $5,000in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7%.

0 12

7%

$5,000

PV0

PV1

Present Value of a Single Deposit (formula)

PV0 = FV2 / (1+i)2 = $5,000/ (1.07)2 = FV2 / (1+i)2 = $4367.19

0 12

7%

$5,000

PV0

General Formula for Present Value

PV0= FV1 / (1+i)1

PV0 = FV2 / (1+i)2

General Present Value Formula:

PV0 = FVn / (1+i)n

or PV0 = FVn (PVD i,n) -- See Table A2

etc.

Valuation Using Table A2

PVD i,nis found on Table A2

Using Present Value Tables

PV2 = $5,000 (PVD 7%,2) = $5,000 (.873) = $4365.00

Example Problem

Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000in 5 years at a discount rate of 10%.

0 1 2 3 4 5

10%

$10,000

PV0

Problem Solution

- Calculation based on general formula: PV0 = FVn / (1+i)nPV0= $10,000/ (1+ 0.10)5 = $6,209.21
- Calculation based on Table A2: PV0= $10,000(PVD 10%, 5)= $10,000(.6209) = $6,209.00

CALCULATOR

- PV = PRESENT VALUE
- FV = FUTURE VALUE
- I/YR (I/Y) = INTEREST RATE OR DISCOUNT RATE
- N = PERIODS
- PMT = PAYMENTS
- P/YR = PAYMENTS PER YEAR

Types of Annuities

- An Annuity represents a series of equal payments (or receipts) occurring over a finite period of time.

- Ordinary Annuity: Payments or receipts occur at the end of each period.
- Annuity Due: Payments or receipts occur at the beginning of each period.

FORMULA TO CALCULATE THE FUTURE VALUE OF AN ORDINARY ANNUITY

- FV = Pmt * [{(1+i)n – 1}/i]
- EXAMPLE:
- Pmt = $1000/year
- 40 years
- 8% annual compounding

FV = 1000 * [{(1.08)40 – 1}/0.08]

- FV = 1000 * [{(21.72452) – 1}/0.08]
- FV = 1000 * [20.72452/0.08]
- FV = 1000 * 259.05652 = 259,056.52

WITH THE CALCULATOR

- PV = 0
- PMT = -1000
- I/Y = 8
- N = 40
- P/Y = 1
- FV= ? 259,056.52

ANNUITY VS. A PERPETUITY

- AN ANNUITY IS A CONSTANT INCOME STREAM THAT CONTINUES FOR A FINITE PERIOD.
- A PERPETUITY IS A CONSTANT INCOME STREAM THAT CONTINUES FOR A INFINITE PERIOD.

Examples of Annuities

- Student Loan Payments
- Car Loan Payments
- Insurance Premiums
- Mortgage Payments
- Retirement Savings

AMORTIZED LOAN

- A LOAN THAT IS REPAID IN A SERIES OF PAYMENTS THAT COVER INTEREST AND PRINCIPAL.
- IN OTHER WORDS, EACH PAYMENT INCLUDES BOTH PRINCIPAL AND INTEREST.
- MAY BE LEVEL PAYMENTS OR DECREASING PAYMENTS

FULLY AMORTIZED LOAN

- ONE WHERE THE PERIODIC LOAN PAYMENTS ARE SUFFICIENT TO PAY OFF THE ENTIRE PRINCIPAL AMOUNT OF THE LOAN OVER THE TERM OF THE LOAN

PARTIALLY AMORTIZED LOAN

- THE PERIODIC LOAN PAYMENTS MAKE SOME REDUCTION IN THE PRINCIPAL BALANCE BUT DO NOT FULLY PAY OFF THE ENTIRE PRINCIPAL OVER THE TERM OF THE LOAN

BALLOON PAYMENT

- A LUMP SUM PAYMENT OF PRINCIPAL DUE AT THE END OF THE TERM OF THE LOAN.
- REPRESENTS THE REMAINING UNPAID PRINCIPAL BALANCE.

AMORTIZATION SCHEDULE

- A TABLE THAT DETAILS THE PAYMENTS, PRINCIPAL PAID, INTEREST PAID, AND REMAINING PRINCIPAL BALANCE FOR A PROMISSORY NOTE.

AMORTIZATION PROBLEM

- YOU HAVE PURCHASED A FARM AND FINANCED $50,000 FOR 5 YEARS AT 8%.
- YOU HAVE TWO OPTIONS:
- LEVEL PAYMENTS
- DECREASING PAYMENTS

BOND VALUATION

- BONDS ARE DEBT OBLIGATIONS OF CORPORATIONS, AND FEDERAL, STATE, AND LOCAL GOVERNMENTS.
- FACE VALUE = AMOUNT THAT WILL BE PAID AT MATURITY. MOST BONDS HAVE A FACE VALUE OF $1,000
- COUPON RATE = THE RATE AT WHICH INTEREST IS PAID

MATURITY DATE - THE DATE WHEN THE BOND WILL PAY THE FACE VALUE

- YIELD TO MATURITY - THE ANNUAL PERCENT RETURN THE BOND WILL GIVE THE INVESTOR WHEN HELD TO MATURITY. TAKES INTO ACCOUNT THE INTEREST PAID AND ANY CAPITAL GAIN OR LOSS.

PROBLEM:

A 5% BOND THAT MATURES IN 6 YEARS AND PAYS SEMI-ANNUAL INTEREST.

WHAT IS THE VALUE OF THE BOND IF IT IS PRICED TO YIELD 6%?

PMT =(1000*.05)/2 = $25

P/Y = 2

N = 6 YEARS * 2 = 12

I/Y = 6.0 %

FV = $1,000

PV = ?

ZERO COUPON BONDS

- BONDS WHERE THE COUPON RATE IS ZERO.
- THE SAME BOND IN THE PREVIOUS PROBLEM WITH A ZERO COUPON.

PMT = 0

P/Y = 2

N = 6 YEARS * 2 = 12

I/Y = 6.0 % FV = $1,000

PV = ?

APR Annual Percentage Rate

- APR is the true or effective interest rate for a loan. It is the actual yield to the lender.
- The APR is calculated using the stated interest rate, any prepaid interest (points) or other lender fees.

Points

- Points are loan fees that are viewed as prepaid interest and raise the APR of the loan. One point is 1% of the loan amount.

Calculation of APR with Points

- Your are purchasing a residence that has a purchase price of $250,000. You plan on making a down payment of 20%. Your mortgage lender has agreed to finance the loan at 6% for 30 years, monthly payments, and wants 2 points.

Calculate the monthly payment on the loan amount after making the down payment of $50,000.

- PV = 200,000
- FV = 0
- PMT = ? -1,199.10
- I/Y = 6.0
- N = 30x12 = 360
- P/Y = 12

The amount of the points that is being required is $200,000 x 0.02 = $4,000.

Therefore the amount of the funded loan is $200,000 less the $4,000 = $196,000.

Calculate the APR (I/Y) based on the calculated payment and a funded loan amount of $196,000.

- PV = 196,000
- FV = 0
- PMT = -1,609.25
- I/Y = ? 6.18948% APR
- N = 30x12 = 360
- P/Y = 12

REFINANCE ANALYSIS

- THE PROPER PERSPECTIVE FOR REFINANCING IS TO WEIGH THE DISCOUNTED CASH FLOW SAVINGS OF THE NEW, LOWER PAYMENT AGAINST THE COST OF THE TRANSACTION

EXAMPLE PROBLEM FROM TEXTBOOK

- ORIGINAL LOAN$200,000 AT 9% FOR 30 YEARS WITH MONTHLY PAYMENTS
- CALCULATE MONTHLY PAYMENTS PV=200,000 FV=0 I/Y=9.0 N=360 P/Y=12
- PMT= -$1,609.25

REFINANCE THE BALANCE AFTER 5 YEARS AT 8% WITH 2 POINTS AND $1,000 IN OTHER LOAN FEES FOR 25 YEARS WITH MONTHLY PAYMENTS. THE LENDER WILL FINANCE THE COST OF THE POINTS AND FEES.

- WHAT IS THE PAYOFF AMOUNT FOR THE ORIGINAL LOAN?USING THE AMORTIZATION FUNCTION THE PRINCIPAL BALANCE FOLLOWING THE 60TH PAYMENT IS $191,760.27 WHISH IS ≈$191,760

AMOUNT OF THE POINTS:191,760*0.02=$3,835

- LOAN FEES =$1,000
- TOTAL =$4,835
- AMOUNT OF NEW LOAN = $191,760 = 4,835TOTAL OF NEW LOAN = $196,595

CALCULATE THE MONTHLY PAYMENT FOR THE NEW LOAN

PV=196,595 FV=0 I/Y=8.0 N=300

P/Y=12

- PMT = -$1,517.35
- SINCE THE NEW LOAN IS PAID OFF AT THE SAME TIME AS THE ORIGINAL LOAN, THE FACT THAT THE NEW MONTHLY PAYMENT IS LESS MEANS THE REFINANCE WOULD BE PROFITABLE.

CALCULATE THE PRESENT VALUE OF THE SAVINGS FROM REFINANCING

- ORIGINAL PAYMENT = $1,609.25
- NEW PAYMENT = $1,517.35 91.90
- FV=0 PMT=91.90 I/Y=8.0 N=300 P/Y=12
- PV= -$11,906.98

BUT, WHAT IF THE NEW LOAN IS FOR A TERM THAT EXTENDS THE ORIGINAL TERM OF THE LOAN?

- IF THE NEW LOAN IS FOR 30 YEARS AT 8.0% WITH 2 POINTS THE NEW LOAN WOULD EXTEND THE PAYOFF DATE BY 5 YEARS.
- THE MONTHLY PAYMENT WOULD BE

PV=196,595 FV=0 I/Y=8.0 N=360

P/Y=12

- PMT = -$1,442.54

THE NEW LOAN WOULD REDUCED THE PAYMENT BY $166.71 PER MONTH FROM THE ORIGINAL LOAN OVER 25 YEARS OR 300 PAYMENTS.

- HOWEVER, THERE WOULD BE AN ADDITIONAL 5 YEARS OR 60 PAYMENTS IN THE AMOUNT OF $1,442.54.

TO EVALUATE THE REFINANCE IN THIS SITUATION, WE NEED TO USE DISCOUNTING.

- FOR PAYMENTS 1 – 300 (25 YEARS)
- FV=0 PMT=166.71 I/Y=8.0 N=300 P/Y=12
- PV= -$21,599.70
- THIS REPRESENTS THE PRESENT VALUE OF THE SAVINGS OVER THE 25 YEARS

NEXT WE NEED TO CALCULATE THE PRESENT VALUE OF THE ADDITIONAL PAYMENTS.

- FOR PAYMENTS 301 – 360 (5 YEARS)
- FV=0 PMT= -1,442.54 I/Y=8.0 N=60 P/Y=12
- PV= $71,143.81
- THIS REPRESENTS THE PRESENT VALUE OF THE ADDITIONAL PAYMENTS BACK TO YEAR 25.

NEXT WE NEED TO DISCOUNT THIS AMOUNT ($71,143.81) TO THE PRESENT.

- FV= -71,143.81 PMT=0 I/Y=8.0 N=300 P/Y=12
- PV= $9,692.38
- THE PRESENT VALUE (BACK TO YEAR 0) OF THE ADDITIONAL PAYMENTS IS $9,692.38.

SO, WHAT IS THE NET RESULT?

- LETS EXPRESS THE PV IN TERMS WHERE A SAVINGS IS POSITIVE AND AN ADDITIONAL COST IS NEGATIVE.
- PV OF SAVINGS FOR 25 YEARS =$21,599.70
- PV OF ADDITIONAL PAYMENTS FOR 5 YEARS = -$9,692.38

THEREFORE, THE NET RESULT IS A BENEFIT FROM REFINANCING OF $11,907.32

- WHICH MEANS THE REFINANCING SHOULD BE DONE.

10. Techsans think “The Corps” refers to the gift shop at the Lubbock Apple Orchard.

9. Most Aggies think that “Raider Red” is that Viking guy that discovered Greenland.

8. Most Techsans would say that a “cadet” is a small riding lawnmower.

7. Aggies think that “the Strip” is a dance they’ve heard about, (but certainly never seen) done at the Men’s Club in Houston.

6. 70% of Techsans, when asked, said “TAMU” was the whale that does the main show at SeaWorld.

5. Aggies think that “Bobby Night” is a Tech fraternity event where all the Brothers dress up like English policemen.

4. Most Techsans would say that “Reveile” is a French wine that they aren’t supposed to buy anymore.

3. Aggies actually like East Texas.

2. At Texas Tech they really understand how to name athletic complexes to personify the highest in values: A phone company and a grocery store.

1. Techsans don’t kiss their dates after every touchdown because, after so many, it gets boring.

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