WFU Law School
Law & Valuation
Chapter 1 - Time Value of Money

Chapter 1 - Problems

Future Value
Time value of money (delayed judgment)
Power of compounding
Different compounding methods
Time to double/tenfold
Annuities
Present Value
Annuities discount rate
Annuity valuation
Mixed returns (bond valuation)
Investment choices
Internal rates of return
Loan amortization
Solutions
  Attached spreadsheet

Tables for your use:


Your client is interested in buying a business. You help structure the transaction and draw up the appropriate dcouments. The client has been offered to ways to pay the $400,000 purchase price. One is for cash, which your client could finance by borrowing from the bank with a 9% loan payable over 10 years -- annuial payments due on December 31 of each year. THe other is a strcutrued purchase as follows:  $40,000.00/yr. for 10 years, payable by December 31 of each year, beginning next year plus each year accrued interest on the outstanding balance at an interest rate of 6%/year for the first five years, then 10% for years 6 through 10.

Which is the better deal for your client?

1 - Time value of money (delayed judgment)

Your client ran over Missy's poodle. You believe that your client will be held liable in the amount of $20,000, but you guess that it will take three years before your client will have to pay. How much should your client set aside right now and invest at 6.2% annual interest to cover this likely judgment?

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2 - Power of compounding

You are 25 years old and start investing $2,000 every year (on January 1) in a tax-deferred IRA. The IRA has annual returns of 12%. After ten years you stop investing and let the IRA continue to grow at 12%. Your twin sister waits for ten years and then starts investing $2,000 every year in an IRA that also has 12% annual returns. She continues until the end of her 65th year. At this point, whose IRA is larger?

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3 - Different compounding methods

You are in a car accident on April 15. Three years later you win a $100,000 judgment that calls for "interest as provided by statute." The statute allows successful plaintiffs to collect interest on any judgment at a "rate of 12% per annum" from the time they were injured. The statute does not mention whether and how interest is compounded. How much interest are you entitled to, if:

a. interest is not compounded?

b. interest is compounded annually?

c. interest is compounded continuously?

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4 - Time to double/tenfold

At a growth rate of 8.4 percent, how long does it take a sum to double? for it to increase ten times?

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5 - Annuities

Which 10-year annuity will be worth more --

a. Payments of $2,500 per year, earning 8% per year.

b. Payments of $2,200 per year, earning 11% per year.

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6 - Annuities discount rate

Your brother-in-law asks for advice. He has won the $1,000,000 West Virginia lottery. He will receive $100,000 at the end of each year for 10 years. He has gotten offers to sell his winning ticket and has been surprised that they are all for less than $1 million.

a. He asks you how much he should sell it for. Without considering the tax implications, what is the ticket worth assuming 6.7% percent return on current 10-year annuities sold by insurance companies?

b. Your brother-in-law gets a serious offer of $500,000. What rate of return, or yield, does this offer entail?

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7 - Annuity valuation

Your client wants to make sure she will have a comfortable retirement. She has been offered an annuity that will pay $24,000 per year for the next 25 years. (If your client dies before the term ends, her estate will receive the annuity's value at death.) The annuity company would have her purchase the annuity for a lump-sum amount. Ignoring taxes, what is the most she should pay if she could invest her money in investments with a similar risk as the annuity company that pay 7.5%?

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8 - Mixed returns (bond valuation)

RealmBank is considering purchasing debt. Three companies, with identical risk profiles, offer three series of newly issued bonds, each with the same level of priority and a par (face amount) of $1,000. One, with a five-year maturity that pays $120 interest annually, sells in the market at par (face value). RealmBank consider the other two bonds:

a. A six-year bond that pays only $60 interest annually. What should RealmBank pay for this bond?

b. A zero-coupon bond (that pays no interest) with a maturity of three years. What should RealmBank pay for this bond?

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9 - Investment choices

You are considering two investment opportunities A and B. A is expected to pay $400 a year for the first 10 years, $600 a year for the 15 years thereafter, and then nothing. B is expected to pay $1,000 a year for 10 years and nothing thereafter. You find that other investments of similar risk to A yield 8% and to B yield 14%.

a. Find the present value of each investment.

b. Which is the more risky investment? Why?

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10 - Internal rates of return

A bakery is considering buying a dough-making machine. There are two options, each promising a five-year useful life. Machine A will produce steady cash flow; Machine B will be expensive to install and learn, but will eventually produce greater cash flows. Each machine's purchase price is $25,000.

Year
Machine A
Machine B
1
$10,000
$3,000
2
10,000
5,000
3
10,000
16,000
4
10,000
16,000
5
10,000
16,000

a. Why is the payback method (which measures how long its takes for the original cost to be recouped) a poor measure of the machine's value?

b. What is the present value of each machine, assuming the bakery's cost of capital (discount rate) is 12%? Which machine seems the better buy?

c. What is the internal rate of return (the discount rate) that would produce the anticipated cash flows? Now which machine seems the better buy?

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11 - Loan amortization

You are thinking of refinancing your house. You will take out a loan for $120,000.

a. What would be your monthly payments with a 30-year loan that carries an annual interest rate of 6.5%?

b. How much interest would you pay, and thus be able to deduct, during the first year of the loan?

c. Assuming you are in a 28% tax bracket, what is the after-tax rate you are paying in the first year for this 6.5% loan?

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Chapter 1 - Time Value of Money

©2003 Professor Alan R. Palmiter

This page was last updated on: November 7, 2007