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Chapter 4 The Time Value of Money

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**Chapter Outline 4.1 The Timeline 4.2 The Three Rules of Time Travel**

4.3 Valuing a Stream of Cash Flows 4.4 Calculating the Net Present Value 4.5 Perpetuities, Annuities, and Other Special Cases

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Example 4.1 You have just taken out a five-year loan from a bank to buy an engagement ring. The ring costs $5000. You plan to put down $1000 and borrow $4000. You will need to make annual payments of $1000 at the end of each year. Show the timeline of the loan from your perspective. How would the timeline differ if you created it from the bank’s perspective.

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**4.2 The Three Rules of Time Travel**

Comparing and Combining Values Moving Cash Flows Forward in Time Moving Cash Flows Back in Time Applying the Rules of Time Travel

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**Table 4.1 The Three Rules of Time Travel**

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Chapter 4, problem 3a, 4a 3a. Calculate the future value of $200 in 5 years at an interest rate of 5% per year. 4a. What is the present value of $10,000 received 12 years from today when the interest rate is 4% per year?

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**The Power of Compounding**

This graph illustrates the future value of $1000 invested at a 10% interest rate. Because interest is paid on past interest, the future value grows exponentially—after 50 years, the money grows 117-fold and in 75 years (only 25 years later), it is 1272 times larger than the value today.

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**Present Value of a Stream of Cash Flows Chapter 4, problem 13**

You have a loan outstanding. It requires making three annual payments at the end of the next three years of $1000 each. Your bank has offered to allow you to skip making the next two payments in lieu of making one large payment at the end of the loan’s term in three years. If the interest rate on the loan is 5%, what final payment will the bank require you to make so that it is indifferent between the two forms of payment?

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**Example 4.5 Net Present Value of an Investment Opportunity Chapter 4, problem 15**

Marian Plunket owns her own business and is considering an investment. If she undertakes the investment, it will pay $4000 at the end of each of the next three years. The opportunity requires an initial investment of $1000 plus an additional investment at the end of the second year of $5000. What is the NPV of this opportunity if the interest rate is 2% per year? Should Marian take it?

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**4.5 Perpetuities, Annuities, and Other Special Cases**

Historical Examples of Perpetuities Common Mistake: Discounting One Too Many Times Annuities Growing Cash Flows Growing Perpetuity Growing Annuity

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Chapter 4, problem 18 The British government has a consol bond outstanding paying £100 per year forever. Assume the current interest rate is 4% per year. What is the value of the bond immediately after a payment is made? What is the value of the bond immediately before a payment is made?

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**Present Value of an Annuity**

Suppose that we now have a stream of equal cash flows of $100 that last for 10 years (and not forever). What is the PV of such a stream if interest rates are 4%?

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Chapter 4, problem 20 You are head of the Schwartz Family Endowment for the Arts. You have decided to fund an arts school in the San Francisco Bay area in perpetuity. Every five years, you will give the school $1 million. The first payment will occur fiver years from today. If the interest rate is 8% per year, what is the preset value of your gift?

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**Growing Perpetuity and Annuity**

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Chapter 4, problem 26 You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last 17 years, You expect that the drug’s profits will be $2 million in its first year and that this amount will grow at a rate of 5% per year for the next 17 years. Once the patent expires, other pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the present value of the new drug if the interest rate is 10% per year?

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**4.8 Solving for Variables Other Than Present Value or Future Value Chapter 4, problem 48**

You are shopping for a car and read the following advertisement in the newspaper: “Own a new Spitfire! No money down. Four annual payment of just $10,000.” You have shopped around and know that you can buy a Spitfire for cash for $32,500. What is the interest rate the dealer is advertising (what is the IRR of the loan in the advertisement?). Assume that you must make the annual payments at the end of each year.

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**4.8 Solving for Variables Other Than Present Value or Future Value**

Your grandmother bought an annuity from Rock Solid Life Insurance Company for $200,000 when she retired. In exchange for the $200,000, Rock Solid will pay her $25,000 per year until she dies. The interest rate is 5%. How long must she live after the day she retired to come out ahead (that is, to get more in value than what she paid in)?

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Chapter 5 Interest Rates

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**5.1 Interest Rate Quotes and Adjustments**

The Effective Annual Rate Adjusting the Discount Rate to Different Time Periods Annual Percentage Rates Application: Discount Rates and Loans Computing Loan Payments Computing the Outstanding Loan Balance

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**Table 5.1 Effective Annual Rates for a 6% APR with Different Compounding Periods**

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**Interest Rate Quotes Chapter 5, problem 1**

Your bank is offering you an account that will pay 20% interest in total for a two-year deposit. Determine the equivalent discount rate for a period length of Six months One year. One month.

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Chapter 5, problem 14 You have decided to refinance your mortgage. You plan to borrow whatever is outstanding on your current mortgage. The current monthly payment is $2356 and you have made every payment on time. The original term of the mortgage was 30 years, and the mortgage is exactly four years and eight months old. You have just made your monthly payment. The mortgage interest rate is 6 3/8% (APR). How much do you owe on the mortgage today?

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Chapter 5, problem 22 You need a new car and the dealer has offered you a price of $20,000, with the following payment options: (a) pay cash and receive a $2000 rebate, or (b) pay a $5000 down payment and finance the rest with a 0% APR loan over 30 months. But having just quit your job and started and MBA program, you are in debt and you expect to be in debt for at least the next 2.5 years. You plan to use credit cards to pay your expenses; luckily you have one with a low (fixed) rate of 15% APR (monthly). Which payment option is best for you?

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**5.3 The Determinants of Interest Rates**

Inflation and Real Versus Nominal Rates Investment and Interest Rate Policy The Yield Curve and Discount Rates The Yield Curve and the Economy

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**Figure 5.1 U.S. Interest Rates and Inflation Rates,1960–2009**

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**Figure 5. 2 Term Structure of Risk-Free U. S**

Figure 5.2 Term Structure of Risk-Free U.S. Interest Rates, November 2006, 2007, and 2008

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**Figure 5. 3 Short-Term Versus Long-Term U. S**

Figure 5.3 Short-Term Versus Long-Term U.S. Interest Rates and Recessions

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Chapter 5, problem 32 Suppose the current one-year interest rate is 6%. One year from now, you believe the economy will start to slow and the one-year interest rate will fall to 5%. In two years, you expect the economy to be in the midst of a recession, causing the Federal Reserve to cut interest rates drastically and the one-year interest rate to fall to 2%. The one year interest rate will then risk to 3% the following year, and continue to rise by 1% per year until it return to 6%, where it will remain from then on. If you were certain regarding these future interest rate changes, what two-year interest rate would be consistent with these expectations. What current term structure of interest rates, for terms of 1 to 10 years, would be consistent with these expectations? Plot the yield curve in this case. Howe does the one-year interest rate compare to the ten year interest rate.

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5.3 Risk and Taxes

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**Equation 5.8 After-Tax Interest Rate**

5.5 The Opportunity Cost of Capital The best available expected return offered in the market on an investment of comparable risk and term to the cash flow being discounted.

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More on interest rates Interest rates are typically quoted as APR “r%” per year, compounded n times per year. That has an interpretation that you will earn an interest rate of r/n per period, for n periods. In introductory finance we learn that if you invest $1, in 1 year you will have.

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Interest rates We also say, that the EAR (Effective interest rate) is The bottom line is that, the more compounding periods in a year, the larger the EAR.

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**Compounding Periods for 10% APR**

Compounding Number of times Effective period compounded annual rate Year % Quarter Month Week Day Hour 8, Minute 525,

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**Continuous compounding**

If compounding is continuous – that is , if interest accrues every instant, then we can use the exponential function to compute future values. For example with a 10% continuous compound rate, after 1 year we will have a future value of Thus, our EAR= %

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**Continuous compounding**

By definition so that is why we can use exp() to calculate future value or EAR. Note also that if you know how much you earned from a $1 investment (i.e., the future value) – you can easily figure out the corresponding continuously compounded rate of return of your investment by taking the natural log.

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**Example: continuous rate**

Suppose you have a zero-coupon bond that matures in 5 years. The price today is $ for a bond that pays $100. What is the annually compounded rate of return (EAR)? What is the continuously compounded rate of return?

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**Two nice properties of continuously compounded returns.**

Additive property of rate being in the power Symmetry for increases and decreases Result: When trying to explain dynamics of asset prices (especially of derivatives), it is much simpler and more intuitive to use the quoted continuous rate.

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Examples If the continuous rate is 10%, what will be the future value of $100 in 3 years and 6 months. What is the present value of a $500 payment that will be paid 2 years from now, if the continuous rate is 15%.

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