Chapter 4

The Time Value of Money

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Learning OutcomesChapter 4

Identify various types of cash flow patternsCompute the future value and the present value of different cash flow streamsCompute the return on an investment and how long it takes to reach a financial goalExplain the difference between the Annual Percentage Rate and the Effective Annual RateDescribe an amortized loan. Compute amortized loan payments and the amount that must be paid at a specific point during the life of the loan

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Time Value of Money

The principles and computations used to revalue cash payoffs from different times so they are stated in dollars of the same time period.

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Graphical representations used to show timing of cash flows

Cash Flow Time Lines

PV = Present Value – the beginning amount that can be invested. PV also represents the current value of some future amount.FV = Future Value – the value to which an amount invested today will grow at the end of n periods.

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Types of Cash Flow Patterns

Lump Sum Amount – a single payment (received or made) that occurs either today or at some date in the future.Annuity – Multiple payments of the same amount over equal time periods.Uneven Cash Flows – Multiple payments of different amounts over a period of time.

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Future Value

Compounding – To compute the future value of an amount we push forward the current amount by adding interest for each period in which the money can earn interest in the future.

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Future Value of a Lump-Sum AmountFVn

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Four Ways to Solve Time Value of Money Problems

Use Cash Flow Time LineUse EquationsUse Financial CalculatorUse Electronic Spreadsheet

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The Future Value of $700 invested at 10% per year for 3 years

Time Line Solution

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Equation Solution

FV3 = $700(1.10)3

= $700(1.33100)= $931.70

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Financial Calculator Solution

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Spreadsheet Solution - MS Excel1. Set up a table that

containsthe data used to solve the problem

2. Click fx and choose function

3. Click the cells containing the appropriate data to calculate the answer

Rate (B2)Number of periods

(B1)Present value (B3)Payment (B4) (not used)Type (B5) (not used)

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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.

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What’s the FV of a 3-year Ordinary Annuity of $400 at 5%?

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Equation Solution:

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Financial Calculator Solution

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Find the FV of an Annuity Due

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Equation Solution

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Financial Calculator Solution

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Find the FV of an Uneven Cash Flow Stream – FVCFn

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Equation Solution

Problem 1:

If you invest $500 today in an account that pays 6% interest compounded annually, how much will be in your account after two years?

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Problem 2:What is the PV of an investment that promises to pay you $1,000 in five years if you can earn 6% interest compounded annualy?

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Problem 7:If you invest $600 per year for the next 10 years, how much will your investment be worth at the end of 10 years if your opportunity cost is 10%? The first $600 investment will be at the end of this year.

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Problem 8If you invest $600 per year for the next 10 years, how much will your investment be worth at the end of 10 years if your opportunity cost is 10%? The first $600 investment will be made today.

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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.

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Present Value of a Lump-Sum AmountPV

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PV of a Lump-Sum AmountFinancial Calculator

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Present Value of an Annuity (Ordinary)

PVAn = 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.

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What is the PV of $400 due in 3 years if r = 5%?

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Equation Solution

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Financial Calculator Solution

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Present Value of an Annuity Due

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Equation Solution

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Financial Calculator Solution

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Streams of equal payments that are expected to go on forever

Perpetuities

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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.

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PV of an Uneven Cash Flow Stream – PVCFn

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Equation Solution

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Financial Calculator Solution

Input in “CF” register:CF0 = 0CF1 = 400CF2 = 300CF3 = 250

Enter I = 5, then press NPV button to get NPV = -869.02.

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You pay $78.35 for an investment that promises to pay you $100 per year for the next five years. What annual rate of return will you earn on this investment?

Solving for Interest Rates (r)

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Financial Calculator Solution

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A security costing $68.30 will provide a return of 10% per year and you want to keep the investment until it grows to a value of $100. How long will it take the investment to grow to $100?

Solving for Time (n)

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Financial Calculator Solution

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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.

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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.

The FV of a lump sum be larger if we compound more often, holding the stated r constant? Why?

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rSIMPLE = Simple (Quoted) Rate used to compute the interest paid per period rEAR = Effective Annual Ratethe annual rate of interest actually being earned

APR = Annual Percentage Rate = rSIMPLE periodic rate X the number of periods per year

Distinguishing Between Different Interest Rates

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Comparison of Different Types of Interest Rates

rSIMPLE: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.rPER: Used in calculations, shown on time lines.rEAR: Used to compare returns on investments with different payments per year.

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Simple (Quoted) Rate

rSIMPLE is stated in contracts Periods per year (m) must also be given

Examples:8%, compounded quarterly8%, compounded daily (365 days)

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Periodic Rate

Periodic rate = rPER = rSIMPLE/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: rPER = 8/4 = 2%8% daily (365): rPER = 8/365 = 0.021918%

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The annual rate that causes PV to grow to the same FV as under multi-period compounding.

Effective Annual Rate

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rEAR = 1 + rSIMPLEm

m - 1

10.25% = 0.1025 =1.0 - 1.05 =

1.0 - 20.10 +1 =

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

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Amortized Loans

Amortized Loan - A loan that is repaid in equal payments over its lifeAmortization 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

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An amortization schedule for a $15,000, 8 percent loan that requires three equal annual payments.

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Financial Calculator Solution

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Amortization Schedule