Chapter 4 The Time Value of Money

Essentials of Chapter 4 Why is it important to understand and apply time value to money concepts? What is the difference between a present value amount and a future value amount? What is an annuity? What is the difference between the Annual Percentage Rate and the Effective Annual Rate? What is an amortized loan? How is the return on an investment determined?

Time Value of Money The most important concept in finance Used in nearly every financial decision Business decisions Personal finance decisions

Time 0 is today Time 1 is the end of Period 1 or the beginning of Period 2. Graphical representations used to show timing of cash flows Cash Flow Time Lines Time: FV = ? % PV=100 Cash Flows:

Future Value The amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate.

PV = Present value, or the beginning amount that is invested r = Interest rate the bank pays on the account each year INT = Dollars of interest you earn during the year FV n = Future value of the account at the end of n periods n = number of period interest is earned Terms Used in Calculating Future Value

Future Value In general, FV n = PV (1 + r) n

Four Ways to Solve Time Value of Money Problems Use Cash Flow Time Line Use Equations Use Financial Calculator Use Electronic Spreadsheet

Time(t): % Account balance: x (1.10) The Future Value of $100 invested at 10% per year for 3 years Time Line Solution

Numerical (Equation) Solution FV 3 =$100(1.10) 3 =$100(1.331) =133.10

Financial Calculator Solution INPUTS OUTPUT ? NI/YR PV PMTFV

Spreadsheet Solution Click on Function Wizard and choose Financial/FV Set up Problem

Spreadsheet Solution Reference cells: Rate = interest rate, r Nper = number of periods interest is earned Pmt = periodic payment PV = present value of the amount

Present Value Present value is the value today of a future cash flow or series of cash flows. Discounting is the process of finding the present value of a future cash flow or series of future cash flows; it is the reverse of compounding.

What is the PV of $100 due in 3 years if r = 10%? Time(t): % 75.13Account balance: x (1.10)

What is the PV of $100 due in 3 years if r = 10%?  $75.13 =0.7513$100 =

Financial Calculator Solution INPUTS OUTPUT 3 10 ? 0100 NI/YR PV PMTFV

Spreadsheet Solution

What interest rate would cause $100 to grow to $ in 3 years? 100 (1 + r ) 3 = = FV 0123 r = ? PV = = / (1 + r ) 3 ][

$100 (1 + r ) 3 = $ INPUTS OUTPUT 3 ? % NI/YRPV PMT FV What interest rate would cause $100 to grow to $ in 3 years?

Spreadsheet Solution

How many years will it take for $68.30 to grow to $100 at an interest rate of 10%? (1.10 ) n = = FV 012n-1 10% PV = = $ (1.10 ) n ][... n = ?

How many years will it take for $68.30 to grow to $100 if interest of 10% is paid each year? FV n = PV(1+r) n $ = $68.30 (1.10) n FV 4 = 68.30(1.10) 4 = $100.00

INPUTS OUTPUT ? NI/YRPV PMT FV How many years will it take for $68.30 to grow to $100 if interest of 10% is paid each year?

Spreadsheet Solution

Future Value of an Annuity Annuity: A series of payments of equal amounts at fixed intervals for a specified number of periods. Ordinary (deferred) Annuity: An annuity whose payments occur at the end of each period. Annuity Due: An annuity whose payments occur at the beginning of each period.

PMT 0123 r% PMT 0123 r% PMT Ordinary Annuity Annuity Due Ordinary Annuity Versus Annuity Due

% FV= What’s the FV of a 3-year Ordinary Annuity of $100 at 5%?

Numerical Solution:

Financial Calculator Solution INPUTS OUTPUT ? NI/YRPV PMT FV

Spreadsheet Solution

Find the FV of an Annuity Due % x[(1.05) 0 ](1.05) x[(1.05) 1 ](1.05) x[(1.05) 2 ](1.05) FVA(DUE) 3 =

Numerical Solution = $100[ x1.05] = $100[ ] =

Financial Calculator Solution Switch from “End” to “Begin”. INPUTS OUTPUT ? NI/YRPV PMT FV

Spreadsheet Solution

Present Value of an Annuity PVA n = the present value of an annuity with n payments. Each payment is discounted, and the sum of the discounted payments is the present value of the annuity.

= PV % What is the PV of this Ordinary Annuity?

Numerical Solution

Financial Calculator Solution INPUTS OUTPUT 310 ? NI/YRPV PMT FV

Spreadsheet Solution

% (1.05)x[1/(1.05) 1 ]x PVA(DUE) 3 = (1.05)x[1/(1.05) 2 ]x (1.05)x[1/(1.05) 3 ]x Find the PV of an Annuity Due

Financial Calculator Solution INPUTS OUTPUT 35 ? NI/YRPV PMT FV

r = ? You pay $ for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year. What interest rate will you earn on this investment? Solving for Interest Rates with Annuities

Use trial-and-error by substituting different values of r into the following equation until the right side equals $ Numerical Solution $  $ (1  r) 4 r        

Financial Calculator Solution INPUTS OUTPUT 4 ? N I/YR PV PMT FV 7.0

Spreadsheet Solution

Annuities that go on indefinitely Perpetuities

Uneven Cash Flow Streams A series of cash flows in which the amount varies from one period to the next: Payment (PMT) designates constant cash flows— that is, an annuity stream. Cash flow (CF) designates cash flows in general, both constant cash flows and uneven cash flows.

% = PV What is the PV of this Uneven Cash Flow Stream?

Numerical Solution

Financial Calculator Solution Input in “CF” register: CF0 =0 CF1 =100 CF2 =300 CF3 =300 CF4 =-50 Enter I = 10%, then press NPV button to get NPV = (Here NPV = PV)

Spreadsheet Solution

Semiannual and Other Compounding Periods Annual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added once a year. Semiannual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added twice a year.

If compounding is more frequent than once a year—for example, semi-annually, quarterly, or daily—interest is earned on interest—that is, compounded—more often. LARGER! Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated r constant? Why?

% Annually: FV 3 = 100(1.10) 3 = % Semi-annually: FV 6/2 = 100(1.05) 6 = Compounding Annually vs. Semi-Annually

Simple (Quoted) Rate r SIMPLE = Simple (Quoted) Rate used to compute the interest paid per period Effective Annual Rate r EAR = Effective Annual Rate the annual rate of interest actually being earned Annual Percentage Rate APR = Annual Percentage Rate = r SIMPLE periodic rate X the number of periods per year Distinguishing Between Different Interest Rates

Comparison of Different Types of Interest Rates r SIMPLE : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. r PER : Used in calculations, shown on time lines. r EAR : Used to compare returns on investments with different payments per year.

Simple (Quoted) Rate r SIMPLE is stated in contracts Periods per year (m) must also be given Examples: 8%, compounded quarterly 8%, compounded daily (365 days)

Periodic Rate Periodic rate = r PER = r SIMPLE /m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Examples: 8% quarterly: r PER = 8/4 = 2% 8% daily (365): r PER = 8/365 = %

The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: 10%, compounded semiannually: r EAR = (1 + r SIMPLE /m) m = (1.05) = = 10.25% Effective Annual Rate

How do we find r EAR for a simple rate of 10%, compounded semi-annually?

FV of $100 after 3 years if interest is 10% compounded semi-annual? Quarterly?

% FV = ? Example: $100 deposited in a bank at EAR = 10% for 0.75 of the year INPUTS OUTPUT ? N I/YR PV PMT FV Fractional Time Periods

Spreadsheet Solution

Amortized Loans Amortized Loan: A loan that is repaid in equal payments over its life Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, and so forth to determine how much of each payment represents principal repayment and how much represents interest

PMT % -1,000 Construct an amortization schedule for a $1,000, 10 percent loan that requires three equal annual payments.

PMT % Step 1: Determine the required payments INPUTS OUTPUT ? 0 NI/YRPV PMT FV

INT t = Beginning balance t (r) INT 1 = 1,000(0.10) = $ Step 2: Find interest charge for Year 1

Repayment= PMT - INT = $ $ = $ Step 3: Find repayment of principal in Year 1

Ending bal.=Beginning bal. - Repayment =$1,000 - $ = $ Repeat these steps for the remainder of the payments (Years 2 and 3 in this case) to complete the amortization table. Step 4: Find ending balance after Year 1

Spreadsheet Solution

Loan Amortization Table 10 Percent Interest Rate * Rounding difference

Chapter 4 Essentials Why is it important to understand and apply time value to money concepts? To be able to compare various investments What is the difference between a present value amount and a future value amount? Future value adds interest - present value subtracts interest What is an annuity? A series of equal payments that occur at equal time intervals

Chapter 4 Essentials What is the difference between the Annual Percentage Rate and the Effective Annual Rate? APR is a simple interest rate quoted on loans. EAR is the actual interest rate or rate of return. What is an amortized loan? A loan paid off in equal payments over a specified period How is the return on an investment determined? The amount to which the investment will grow in the future minus the cost of the investment