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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

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**Chapter 4 Contents Compounding Frequency of Compounding**

Present Value and Discounting Alternative Discounted Cash Flow Decision Rules Multiple Cash Flows Annuities Perpetual Annuities Loan Amortization Exchange Rates and Time Value of Money Inflation and Discounted Cash Flow Analysis Taxes and Investment Decisions

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**Financial Decisions Costs and benefits being spread out over time**

The values of sums of money at different dates The same amounts of money at different dates have different values.

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Time Value of Money Interest Purchasing power Uncertainty

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**Compounding Present value (PV) Future value (FV)**

Simple interest: the interest on the original principal Compound interest: the interest on the interest Future value factor

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**Value of Investing $1 at an Interest Rate of 10%**

Continuing in this manner you will find that the following amounts will be earned:

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

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Value of $5 Invested If we can earn 10% interest on the principal $5, then after 4 years

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**Future Value of a Lump Sum**

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**Example: Future Value of a Lump Sum**

Your bank offers a CD (Certificate of Deposit) 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?

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**Example: Reinvesting at a Different Rate**

You have $10,000 to invest for two years. Two years CDs and one year CDs are paying 7% and 6% per year respectively. What should you do? Reinvestment rate? You are sure the interest rate on one-year CDs will be 8% next year. With the two-year CD With the sequence of two one-year CDs

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**Frequency of Compounding**

Annual percentage rate (APR) Effective annual rate (EFF) Suppose you invest $1 in a CD, earning interest at a stated APR of 6% per year compounded monthly. General formula

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**Effective Annual Rates of an APR of 18%**

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**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?

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Present Value In order to reach a target amount of money at a future date, how much should we invest today? Discounting Discounted-cash-flow (DCF)

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**Present Value of a Lump Sum**

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**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?

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**Solving Lump Sum Cash Flow for Interest Rate**

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**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?

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**Solving Lump Sum Cash Flow for Number of Periods**

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**NPV (Net Present Value) Rule**

NPV rule: Accept a project if its NPV is positive. Opportunity cost of capital: The rate (of return) we could earn somewhere else if we did not invest in the project under evaluation. Yield to maturity or Internal Rate of Return (IRR)

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**Example: Evaluate a Project**

A five-year savings bond with face value $100 is selling for a price of $75. Your next-best alternative for investing is an 8% bank account. Is the savings bond a good project?

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**Example: Evaluate a Project**

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**Example: Borrowing You need to borrow $5,000 to buy a car.**

A bank can offer you a loan at an interest rate of 12%. A friend says he will lend the $5,000 if you pay him $9,000 in four years. Should you borrow from the bank or the friend?

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PV of Annuity Formula

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**Example: Buying an Annuity**

You are 65 years old and expect to live until age 80. For a cost of $10,000, an insurance company will pay you $1,000 per year for the rest of your life. You can earn 8% per year on your money in a bank account. Does it pay to buy the insurance policy?

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**Perpetual Annuities / Perpetuities**

Recall the annuity formula: Let n -> ∞ with i > 0:

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Loan Amortization Home mortgage loans or car loans are repaid in equal periodic installments. Part of each payment is interest on the outstanding balance of the loan. Part is repayment of principal. The portion of the payment that goes toward the payment of interest is lower than the previous period’s interest payment. The portion that goes toward repayment of principal is greater than the previous period’s.

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Calculator Solution This is the yearly repayment

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**Amortization Schedule for 3-Year Loan at 9%**

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**Example: Exchange Rates**

Investing $10,000 in dollar-denominated bonds offering an interest rate of 10% per year Investing in yen-denominated bonds offering an interest rate of 3% per year The exchange rate for the yen is now $0.01 per yen. Which is the better investment for the next year?

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**Time U.S.A. Japan 0.01 $/¥ 3% ¥ / ¥ ? $/¥ $10,000 $11,000 ¥ 1,000,000¥**

1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥ / ¥ ? $/¥ U.S.A. Japan

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**Time U.S.A. Japan 0.01 $/¥ 3% ¥/¥ 0.0108 $/¥ $10,000 $11,124 $11,000 ¥**

1,000,000¥ 1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥/¥ $/¥ U.S.A. Japan

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**Time U.S.A. Japan 0.01 $/¥ 3% ¥ / ¥ 0.0106 $/¥ $10,000 $10,918 ¥**

$11,000 ¥ 1,000,000¥ 1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥ / ¥ $/¥ U.S.A. Japan

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**Time U.S.A. Japan 0.01 $/¥ 3% ¥ / ¥ 0.01068 $/¥ $10,000 $11,000 ¥**

1,000,000¥ 1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥ / ¥ $/¥ U.S.A. Japan

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**The Real Rate of Interest**

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Switch to a Gas Heat ? You currently heat your house with oil and your annual heating bill is $2,000. By converting to gas heat, you estimate that this year you could cut your heating bill by $500. You think the cost differential between gas and oil is likely to remain the same for many years. The cost of installing a gas heating system is $10,000. Your alternative use of the money is to leave it in a bank account earning an interest rate of 8% per year. Is the conversion worthwhile?

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Switch to a Gas Heat ? Assume that the $500 cost differential will remain forever. The investment in switch of heating is a perpetuity, i.e. paying $10,000 now for getting $500 per year forever. If the $500 cost differential will increase over time with the general rate of inflation, then the 5% rate of return is a real rate of return. The conversion is not worthwhile unless the rate of inflation is greater than 2.875% per year.

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Taxes

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**Taking Advantage of a Tax Loophole**

You are in a 40% tax bracket and currently have $100,000 invested in municipal bonds earning a tax-exempt rate of interest of 6% per year. Now you buy a house at a cost of $100,000. A bank offers a loan for you at an interest rate of 8% per year. Does it pays for you to borrow?

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