Chapter 2. Time Value of Money

Chapter 2 Time Value of Money Interest: The Cost of Money Economic Equivalence Interest Formulas – Single Cash Flows Equal-Payment Series Dealing with Gradient Series Composite Cash Flows.

Decision Dilemma—Take a Lump Sum or Annual Installments A suburban Chicago couple won the Power-ball. They had to choose between a single lump sum $116.5 million, or $198 million paid out over 25 years (or $7.92 million per year after taxes). The winning couple opted for the lump sum. Did they make the right choice? What basis do we make such an economic comparison?

Option A (Lump Sum) Option B (Installment Plan) 1 2 3 25 $116.5 M $7.92 M Total $198M

What Do We Need to Know? To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different points in time. To do this, we need to develop a method for reducing a sequence of benefits and costs to a single point in time. Then, we will make our comparisons on that basis.

Time Value of Money Money has a time value because it can earn more money over time (earning power). Money has a time value because its purchasing power changes over time (inflation). Time value of money is measured in terms of interest rate. Interest is the cost of having money available for use — a cost to the borrower and an earning to the lender.

Delaying Consumption

Elements of Transactions Involving Interest P (principal): initial amount of money invested or borrowed i (interest rate): cost or price of money; expressed as a percentage per period of time n (interest period): determines how frequently interest is calculated N (number of interest periods): duration of transaction An (a plan for receipts or disbursements): a particular cash flow pattern F (future amount): results from the cumulative effects of the interest

Which Repayment Plan? End of Year Receipts ($) Payments ($) Plan 1 20,000.00 200.00 Year 1 5,141.85 Year 2 Year 3 Year 4 Year 5 30,772.48 The amount of loan = $20,000, origination fee = $200, interest rate = 9% APR (annual percentage rate)

Cash Flow Diagram

End-of-Period Convention Interest Period 1 End of interest period Beginning of Interest period 1

Methods of Calculating Interest Simple interest: the practice of charging an interest rate only to an initial sum (principal amount). Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.

Simple Interest P = Principal amount i = Interest rate N = Number of interest periods Example: P = $1,000 i = 8% N = 3 years End of Year Beginning Balance Interest earned Ending Balance $1,000 1 $80 $1,080 2 $1,160 3 $1,240

Simple Interest Formula

Compound Interest Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.

Compound Interest P = Principal amount i = Interest rate N = Number of interest periods Example: P = $1,000 i = 8% N = 3 years End of Year Beginning Balance Interest earned Ending Balance $1,000 1 $80 $1,080 2 $86.40 $1,166.40 3 $93.31 $1,259.71

Compounding Process $1,080 $1,166.40 $1,259.71 1 $1,000 2 3 $1,080 $1,259.71 1 $1,000 2 3 $1,080 $1,166.40

$1,259.71 1 2 3 $1,000

Compound Interest Formula

“The greatest mathematical discovery of all time” Albert Einstein Compound Interest The Fundamental Law of Engineering Economy “The greatest mathematical discovery of all time” Albert Einstein

Practice Problem: Warren Buffett’s Berkshire Hathaway Went public in 1965: $18 per share Worth today (August 22, 2003): $76,200 Annual compound growth: 24.58% Current market value: $100.36 Billion If he lives till 100 (current age: 73 years as of 2003), his company’s total market value will be ?

Solution Assume that the company’s stock will continue to appreciate at an annual rate of 24.58% for the next 27 years.

Example In 1626 the Indians sold Manhattan Island to Peter Minuit of the Dutch West Company for $24. If they saved just $1 from the proceeds in a bank account that paid 8% interest, how much would their descendents have now (year 2003)? As of Year 2003, the total US population would be close to 275 millions. If the total sum would be distributed equally among the population, how much would each person receive?

Solution F=1(1+0.08)377= $3,988,006,142,690

Excel Worksheet A B C 1 P 2 i 8% 3 N 377 4 FV 5 FV(8%,377,0,1) = $3,988,006,142,690

Practice Problem If you deposit $100 now (n = 0) and $200 two years from now (n = 2) in a savings account that pays 10% interest, how much would you have at the end of year 10?

Solution F 0 1 2 3 4 5 6 7 8 9 10 $100 $200

Practice problem Consider the following sequence of deposits and withdrawals over a period of 4 years. If you earn 10% interest, what would be the balance at the end of 4 years? ? $1,210 3 1 2 4 $1,500 $1,000 $1,000

? $1,210 1 3 2 4 $1,000 $1,000 $1,500 $1,100 $1,000 $1,210 $2,981 $2,100 $2,310 + $1,500 -$1,210 $1,100 $2,710

Solution n = 0 n = 1 n = 2 n = 3 n = 4 End of Period Beginning balance Deposit made Withdraw Ending n = 0 $1,000 n = 1 $1,000(1 + 0.10) =$1,100 $2,100 n = 2 $2,100(1 + 0.10) =$2,310 $1,210 $1,100 n = 3 $1,100(1 + 0.10) =$1,210 $1,500 $2,710 n = 4 $2,710(1 + 0.10) =$2,981 $2,981

Economic Equivalence

Economic Equivalence What do we mean by “economic equivalence?” Why do we need to establish an economic equivalence? How do we establish an economic equivalence?

Economic Equivalence Economic equivalence exists between cash flows that have the same economic effect and could therefore be traded for one another. Even though the amounts and timing of the cash flows may differ, the appropriate interest rate makes them equal. Any cash flow – whether a single payment or a series of payments – can be converted to an equivalent cash flow at any point in time.

Economic Equivalence

Equivalence from Personal Financing Point of View If you deposit P dollars today for N periods at i, you will have F dollars at the end of period N.

Alternate Way of Defining Equivalence P F dollars at the end of period N is equal to a single sum P dollars now, if your earning power is measured in terms of interest rate i. N = F N

= Practice Problem F At 8% interest, what is the equivalent worth of $2,042 now 5 years from now? $2,042 If you deposit $2,042 today in a savings account that pays 8% interest annually. how much would you have at the end of 5 years? 1 2 3 4 5 3 F = 5

Solution

would these two amounts be equivalent? Example At what interest rate would these two amounts be equivalent? i = ? $3,000 $2,042 5

Solution F=P(1+i)N 3000=2042(1+i)5 1.469=(1+i)5 1.4691/5=1.07995=1+i