**1.a**.Jim makes a deposit of $12,000 in a bank account. The deposit is to earn interest compounded annually at the rate of 6 percent for seven years.

a) How much will Jim have on deposit at the end of seven years? (Hint: What is future value?)

b)Assuming the deposit earned a 9 percent rate of interest compounded quarterly, how much would he have at the end of seven years?

c)In comparing (a) and (b), what are the respective effective annual yields? (Hint: Consider the future value of each deposit after one year only.) Which alternative is better?

**1.b.**What is the essential concept in understanding compound interest?

**2.a.**Would you prefer making a $25,000 investment that will earn interest at the rate of 6 percent compounded monthly or making the same $25,000 investment at 7 percent compounded annually? (Hint: Consider one year only.)

**2.b.** How are the interest factors (IFs) Exhibit 3-3 developed? How may financial calculators be used to calculate IFs in Exhibit 3-3?

**3.a.**Jones can deposit $5,000 at the end of each six-month period for the next 12 years and earninterest at an annual rate of 8 percent, compounded semiannually. What will the value of the investment be after 12 years? If the deposits were made at the beginning of each period, what would the value of the investment be after 12 years?

**3.b.**What general rule can be developed concerning maximum values and compounding intervals within a year? What is an equivalent annual yield?

**4.a.** Suppose you deposit $1,250 at the end of each quarter in an account that will earn interest at an annual rate of 10 percent compounded quarterly. How much will you have at the end of four years?

**4.b.**What does the time value of money (TVM) mean?

**5.a.** Suppose you deposit $2,500 at the end of year 1, nothing at the end of year 2, $750 at the end of year 3, and $1,300 at the end of year 4. Assuming that these amounts will be compounded at an annual rate of 15 percent, how much will you have on deposit at the end of five years?

**5.b. **How does discounting, as used in determining present value, relate to compounding, as used in determining future value? How would present value ever be used?

**6.a.** Suppose you have the opportunity to make an investment in a real estate venture that expects to pay investors $750 at the end of each month for the next eight years. You believe that a reasonable return on your investment should be an annual rate of 15 percent compounded monthly.

**a)**How much should you pay for the investment?

**b)**What will be the total sum of cash you will receive over the next eight years?

**c)**What do we call the difference between (a) and (b)?

**6.b.** What are the interest factors (IFs) in Exhibit 3-9? How are they developed? How may financial calculators be used to calculate IFs in Exhibit 3-9?

**7.a.** An investor is considering an investment that will pay $2,150 at the end of each year for the next 10 years. He expects to earn a return of 12 percent on his investment, compounded annually. How much should he pay today for the investment? How much should he pay if the investment returns are received at the beginning of each year?

**7.b.** What is an annuity? How is it defined? What is the difference between an ordinary annuity and an annuity due?

**8.a.** An investor can make an investment in a real estate development and receive an expected cash return of $45,000 at the end of six years. Based on a careful study of other investment alternatives, she believes that a 9 percent annual return compounded quarterly is a reasonable return to earn on this investment. How much should she pay for it today?

**8.b.** How must one discount a series of uneven receipts to find PV?

**9.a.** Walt is evaluating an investment that will provide the following returns at the end of each of the following years: year l, $12,500; year 2, $10,000; year 3, $7,500; year 4, $5,000; year 5, $2,500; year 6, $0; and year 7, $12,500. Walt believes that he should earn 12 percent compounded annually on this investment. How much should he pay for this investment? What if he expects to earn an annual return of 9 percent compounded monthly? How much should he pay?

**9.b.** What is the sinking-fund factor? How and why is it used?

**10.a.** John is considering the purchase of a lot. He can buy the lot today and expects the price to rise to $15,000 at the end of 10 years. He believes that he should earn an investment yield of 8 percent compounded annually on his investment. The asking price for the lot is $7,000. Should he buy it? What is the internal rate of return compounded annually on the investment if John purchases the property for $7,000 and is able to sell it 10 years later for $15,000?

**10.b.**What is an internal rate of return? How is it used? How does it relate to the concept of compound interest?

**11.**The Dallas Development Corporation is considering the purchase of an apartment project for $100,000. They estimate that they will receive $15,000 at the end of each year for the next 10 years. At the end of the 10th year, the apartment project will be worth nothing. If Dallas purchases the project, what will be its internal rate of return, compounded annually? If the company insists on an 8 percent return compounded annually on its investment, is this a good investment?

**12.**A corporation is considering the purchase of an interest in a real estate syndication at a price of $75,000. In return, the syndication promises to pay $1,000 at the end of each month for the next 25 years (300 months). If purchased, what is the expected internal rate of return, compounded monthly? How much total cash would be received on the investment? How much is profit and how much is return of capital?

**13.** An investment in a real estate venture will provide returns at the end of the next four years as follows: year 1, $5,500; year 2, $7,500; year 3, $9,500; and year 4, $12,500. An investor wants to earn a 12 percent return compounded annually on her investment. How much should she pay for the investment? Assuming that the investor wanted to earn an annual rate of 12 percent compounded monthly, how much would she pay for this investment? Why are these two amounts different?

**14.**A pension fund is making an investment of $100,000 today and expects to receive $1,600 at the end of each month for the next five years. At the end of the fifth year, the capital investment of $100,000 will be returned. What is the internal rate of return compounded annually on this investment?

**15.**A loan of $60,000 is due 10 years from today. The borrower wants to make annual payments at the end of each year into a sinking fund that will earn compound interest at an annual rate of 10 percent. What will the annual payments have to be? Suppose that the monthly payments earn 10 percent interest, compounded monthly. What would the annual payments have to be?

**16.** An investor has the opportunity to make an investment that will provide an effective annual yield of 12 percent. She is considering two other investments of equal risk that will provide compound interest monthly and quarterly, respectively. What must the equivalent nominal annual rate (ENAR) be for each of these two investments to ensure that an equivalent annual yield of 12 percent is earned?

**17.**An investment producing cash flows in the amount of $1,200 per month is undertaken for a period of 28 months. The investor pays $24,000 for the investment and the contract stipulates that investment returns must be reported on a basis equivalent with annual compounding. Given that the investment is sold after 28 months, what would be the equivalent annual compound rate of interest reported to the investor? What would be the annual rate compounded monthly for this investment?

**18.**An investment is expected to produce the following annual year-end cash flows:

year 1: $5,000 year 4: $5,000

year 2: $1,000 year 5: $6,000

year 3: $0 year 6: $863.65

The investment will cost $13,000 today.

a)Will this investment be profitable?

b)What will be the IRR (compounded annually) on this investment?

c)Prove your answer in (b) by showing how much of each year’s cash flow is recovery of the $13,000 investment and how much of the cash flow is return on investment. (Hint: See Concept Box 3.2.)

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