Time Value of Money 1: Analyzing Single Cash Flows Chapter 04 Time Value of Money 1: Analyzing Single Cash Flows McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Introduction Time Value of Money (TVM) The values of money change depend on time. Powerful financial decision-making tool Used by financial and nonfinancial business managers Key to making sound personal financial decisions 4-2

> $Today Introduction (cont.) TVM Basic Concept: $1 today is worth more than $1 next year TVM Decision Based on: Size of cash flows Time between cash flows Rate of return $Today $ Next Year > 4-3

Organizing Cash Flows Cash flows (cash inflow & outflow) timing key to successful business operations Cash flow analysis Time line shows magnitude of cash flows at different points in time Monthly Quarterly Semi-annually Annually 4-4

Organizing Cash Flows Cash flow analysis *Inflow = Cash received a positive number *Outflow = Cash going out a negative number Inflow Positive # Outflow Negative # Organization 4-5

Time Line Example Outflow Inflow A helpful tool for analysis of cash flows is the time line, which shows the magnitude of cash flows at different points in time Cash we receive is called an inflow and is represented by a positive number Cash that leaves us is called an outflow and is represented by a negative number Outflow Inflow 4-6

Future Value Value of an investment after one or more periods For example: the $105 payment your bank credits to your account one year from the original $100 investment at 5% annual interest 4-7

Single-period Future Value Concept: Interest is earned on principal Today’s cash flow + Interest = Value in 1 year Formula: 4-8

Single-period Future Value Example Assumptions: Invest $100 today Earn 5% interest annually (one period) 4-9

Compounding & Future Value Concept: Compounding Interest is earned on both principal and interest Today’s cash flow + Interest on Principal and Interest on Interest = Value in 2 years Formula: To compute the two-year compounded future value, simply use the one-year equation (4-1) twice. Suppose you leave the money invested for two years. What is the Future Value? The total interest of $10.25 represents $10 earned on the original $100 investment plus $0.25 earned on the $5 first year’s interest This represents compounding, i.e. earning interest on interest 4-10

Compounding & Future Value Example Assumptions: Invest $100 today Earn 5% interest for more than one period To compute the two-year compounded future value, simply use the one-year equation (4-1) twice. Suppose you leave the money invested for two years. What is the Future Value? The total interest of $10.25 represents $10 earned on the original $100 investment plus $0.25 earned on the $5 first year’s interest This represents compounding, i.e. earning interest on interest 4-11

The Power of Compounding Compound interest is powerful wealth-building tool  exponential growth Let’s illustrate the power of compounding. Figure 4.2 shows the original $100 deposited, the cumulative interest earned on that deposit, the cumulative interest-on-interest earned. By the 27th year, the money from the interest-on-interest exceeds the interest earned on the original deposit. By the 40th year, interest-on-interest contributes more than double the interest on the deposit. 4-12

Present Value Opposite of Future Value Future Value = Compounding Present Value = Discounting 4-13

Present Value Concept: Discounting Value today of sum expected to be received in future Next period’s valuation ÷ One period of discounting Formula: Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much? This process is called discounting. Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV) 4-14

Present Value Example Assumptions: Banks pays $105 in 1 year Interest rate = 5% interest Example: If I want to end up with $100 in an account at the end of one year at 5% per year, how much would I need to deposit? This process is called discounting. Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV). 4-15

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Present Value Over Multiple Periods Concept: Discounting Reverse of compounding over multiple periods Formula: Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much? This process is called discounting. Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV) 4-17

Present Value Over Multiple Periods Example Assumptions: $100 payment five years in the future Interest rate = 5% interest Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much? This process is called discounting. Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV) 4-18

Present Value with Multiple Rates Concept: Discounting Value today of sum expected to be received in future -- variable rates of interest over time Formula: Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much? This process is called discounting. Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV) 4-19

Present Value with Multiple Rates Example Assumptions: Banks pays $2,500 at end of 3rd year Interest rate year 1 = 7% Interest rate year 2 = 8% Interest rate year 3 = 8.5% Example: If I want to end up with $100 in an account at the end of one year at 5% per year, I would need to deposit how much? This process is called discounting. Notice that this isn’t a new equation. It is really the same as the future value equation we saw before (just rearranged to solve for PV instead of FV) 4-20

Present Value & Future Value Concepts: Discounting & Compounding Move cash flows around in time Use PV Calculation to discount the Cash Flow Use FV Calculation to compound the Cash Flow Businesses often need to move cash flows around in time. That is no problem – we can use FV and PV to do that. Example: We expect to receive a cash flow of $200 at the end of 3 years. What is the value of the cash flow if we move it to year 2 using a discount rate of 6%? Since we are moving the cash flow one year earlier, we use the present value calculation. 4-21

PV & FV Example Assumptions PV: Expected cash flow of $200 in 3 years Decision: change receipt of CF to 2 years (one year earlier) Discount rate = 6% PV Calculation to Discount the Cash Flow for 1 year: Businesses often need to move cash flows around in time. That is no problem – we can use FV and PV to do that. Example: We expect to receive a cash flow of $200 at the end of 3 years. What is the value of the cash flow if we move it to year 2 using a discount rate of 6%? Since we are moving the cash flow one year earlier, we use the present value calculation. 4-22

PV & FV Example Assumptions FV: Expected cash flow of $200 in 3 years Decision: change receipt of CF to 5 years later Compound rate = 6% FV Calculation to Compound the Cash Flow for 5 years: What about moving the $200 cash flow to year 5? Since this requires moving the cash flow later in time by two years, we use the future value equation. 4-23

Rule of 72 Concept: Compound Interest How much time for an amount to double? Formula: 72 / i = Time for amount to double The Rule of 72 is based on compounding and shows that the amount of time needed for an amount to double can be approximated by dividing 72 by the interest rate. For example, at 6% it should take 12 years for any amount to double. 12 years = 72/6 4-24

Rule of 72 Example Assumptions: Rule of 72 calculation: Interest rate = 6% interest Rule of 72 calculation: 72 = Amount of time for amount to double 6 72 / 6 = 12 years The Rule of 72 is based on compounding and shows that the amount of time needed for an amount to double can be approximated by dividing 72 by the interest rate. For example, at 6% it should take 12 years for any amount to double. 12 years = 72/6 4-25

Interest Rate to Double an Investment Remember that this rule provides only a mathematical approximation. It’s more accurate with lower interest rates. After all, with a 72 percent interest rate, the rule predicts that it will take one year to double the money. However, we know that it actually takes a 100 percent rate to double money in one year. 4-26

Computing Interest Rates Concept: Solving for Interest Rate Complex Calculation – Use financial calculator Formula: 4-27

Computing Interest Rates Example Assumptions: Bought asset for $350 Sold asset for $475 Timeframe: 3 years Interest Rate Computation – Use financial calculator 4-28

Solving for Time Concept: Solving for Time Assumptions/Known Data: Starting Cash Flow Interest Rate Future Cash Flow Complex calculation – use financial calculator 4-29

Solving for Time Example Question: When interest rates are 9%, how long will it take $5,000 to double? Assumptions: Interest = 9% PV = -5,000 PMT = 0 FV =10,000 Solution: 8.04 years 4-30