# Chapter 4: Time Value of Money

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Chapter 4: Time Value of Money
Objective Explain the concept of compounding and discounting and to provide examples of real life applications Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc.

Contents Compounding The Frequency of Compounding
Present Value and Discounting Discounted Cash Flow Decision Rules Multiple Cash Flows Annuities, Perpetuities Loan Amortization Exchange Rates and Time Value of Money Inflation and DCF Analysis Taxes and Investment Decisions

Compounding: Value of Investing \$1 at 10%

Compounding: Value of \$5 Invested at 10%
More generally, with an investment of \$5 at 10% we obtain

Compounding: Future Value of a Lump Sum

Example: Future Value of a Lump Sum
Your bank offers a CD with an interest rate of 3% for a 5 year investments. You wish to invest \$1,500 for 5 years, how much will your investment be worth?

Discounting: Present Value of a Lump Sum

Example: Present Value of a Lump Sum
You have been offered \$40,000 for your printing business, payable in 2 years. Given the risk, you require a return of 8%. What is the present value of the offer?

Example: Interest Rate on a Lump Sum Investment
If you invest \$15,000 for ten years, you receive \$30,000. What is your annual return?

‏The Frequency of Compounding
Annual Percentage Rate (APR) Effective Annual Rate (EFF): The equivalent interest rate, if compounding were only once a year.

Effective Annual Rates of an APR of 18%

The Frequency of Compounding
Note that as the frequency of compounding increases, so does the annual effective rate What occurs as the frequency of compounding rises to infinity?

The Frequency of Compounding

The Frequency of Compounding

Annuities A level stream of Cash Flows or Payments
Immediate Annuity: The Cash Flows start immediately. Ordinary Annuity: The Cash Flows start at the end of the current period.

Derivation of PV of Ordinary Annuity Formula

Derivation of PV of Ordinary Annuity Formula

PV of Ordinary Annuity Formula

Annuity Formula: PV of Immediate Annuity

Derivation of FV of Annuity Formula

Perpetual Annuities / Perpetuities
Recall the annuity formula: Let n -> infinity with i > 0:

Alternative Discounted Cash Flow Decision Rules
NPV rule: the NPV is the difference between the present value of all future cash inflows minus the present value of all current and future cash outflows. Accept a project if its NPV is positive.

DCF rules Example: You have the opportunity to buy a piece of land for \$10,000. You are sure that 5 years from now it will be worth \$20,000. If you can earn 8% per year by investing your money in bank, is this investment in the land worthwhile?

NPV rule solution

Alternative Discounted Cash Flow Decision Rules
FV rule: Invest if the future value of the investment is larger than the future value that can be obtained from the next best alternative.

FV rule solution

Alternative Discounted Cash Flow Decision Rules
IRR rule: The IRR is the discount rate at which the NPV is zero. Invest if the IRR is greater than the opportunity cost of capital.

IRR rule solution

Alternative Discounted Cash Flow Decision Rules
Choose the investment alternative with fastest payback.

Payback rule solution

Loan Amortization The process of paying a loan principal gradually over its term Example: \$100,000 mortgage loan, APR: 9%, repaid in 3 annual installments pmt=? pmt=\$

Loan Amortization First Year: Interest: (0.09)(100000)=9000
pmt: principal: Outstanding Balance:

Loan Amortization Second Year: Interest: (0.09)(69494.52)=6254.51
pmt: principal: Outstanding Balance:

Loan Amortization Third Year: Interest:(0.09)(36243.54)=3262
pmt: principal: Outstanding Balance:

Exchange Rate and TVM Time U.S.A. Japan 0.01 \$/¥ 3% ¥ / ¥ ? \$/¥
\$10,000 1,000,000¥ 10% \$/\$ (direct) 3% ¥ / ¥ ? \$/¥ 1,030,000¥ \$11,000 ¥ Time U.S.A. Japan

Exchange Rate and TVM Time U.S.A. Japan 0.01 \$/¥ 3% ¥/¥ 0.0108 \$/¥
\$10,000 1,000,000¥ 10% \$/\$ (direct) 3% ¥/¥ \$/¥ 1,030,000¥ \$11,124 \$11,000 ¥ Time U.S.A. Japan

Exchange Rate and TVM Time U.S.A. Japan 0.01 \$/¥ 3% ¥ / ¥ 0.0106 \$/¥
\$10,000 1,000,000¥ 10% \$/\$ (direct) 3% ¥ / ¥ \$/¥ 1,030,000¥ \$10,918 ¥ \$11,000 ¥ Time U.S.A. Japan

Exchange Rate and TVM Time U.S.A. Japan 0.01 \$/¥ 3% ¥ / ¥ 0.01068 \$/¥
\$10,000 1,000,000¥ 10% \$/\$ (direct) 3% ¥ / ¥ \$/¥ 1,030,000¥ \$11,000 ¥ Time U.S.A. Japan

Computing NPV in Different Currencies
In any TVM calculation, the cash flows and the interest rate must be denominated in the same currency.

Inflation and Future Values
Example: At age 20 you save \$100 and invest it at a dollar interest rate of 8% per year, and you do not take it until age 65. If the inflation is estimated 5% per year, how much will you have accumulated in the account at that time in terms of real purchasing power?

Solution 1:

Solution 2:

Inflation and Present Values
Example: Your daughter is 10 years old, and you are planning to open an account to provide for her college education. Tuition for a year of college is now \$15,000. How much must you invest now in order to have enough to pay for her first year’s tuition 8 years from now, if you think you can earn a rate of interest that is 3% more than the inflation rate of 5%?

Solution: Wrong:

Inflation and Present Values
Never use a nominal interest rate when discounting real cash flows or a real interest rate when discounting nominal cash flows.