Chapter 5 Introduction

1. Chapter 5Introduction • This chapter introduces the topic of financial mathematics also known as the time value of money. • This is a foundation topic relevant to many finance decisions for a hospitality firm: • Capital budgeting decisions • Cost of capital estimation • Pricing a bond issuance

2. Organization of Chapter • Financial math will be presented in the context of personal finance. Later chapters will apply financial math to various finance applications for a hospitality firm. • Financial math topics covered include: • Computation of future values, present values, annuity payments, interest rates, etc. • Perpetuities and non-constants cash flows • Effective annual rates and compounding periods other than annual

3. Future Value of a Lump Sum • The future value in 2 years of $1,000 earning 5% annually is an example of computing the future value of a lump sum. We can compute this in any one of three ways: • Using a calculator programmed for financial math • Solve the mathematical equation • Using financial math tables (Table 5.1)

4. Solve for the Future Value • The general equation for future value is: • FVn = PV x (1+i)n • Computing the future value in the example: • FV2 = $1,000 x (1+5%)2 = $1,102.50

5. Present Value of a Lump Sum • How much do you need to invest today so you can make a single payment of $30,000 in 18 years if the interest rate is 8%? This is an example of the present value of a lump sum. • Again we can solve it using a programmed calculator, solving the math or using Table 5.2.

6. Solve for the Present Value • The general equation for present value is: • Computing the present value in the example:

7. Annuities • Two or more periodic payments • All payments are equal in size. • Periods between each payment are equal in length.

8. PV Versus FV of an Annuity • The value of an annuity can be expressed as an equivalent lump sum value. • The PV of an annuity is the lump sum value of an annuity at a point in time earlier than the payments. • The FV of an annuity is the lump sum value of an annuity at a point in time later than the payments.

9. Ordinary Annuity Versus Annuity Due • The PV of an ordinaryannuity is located one period before the first annuity payment. • The PV of an annuitydue is located on the same date as the first annuity payment. • The FV of an ordinaryannuity is located on the same date as the last annuity payment. • The FV of an annuitydue is located one period after the last annuity payment.

10. Future Value of an Annuity • Suppose you plan to deposit $1,000 annually into an account at the end of each of the next 5 years. If the account pays 12% annually, what is the value of the account at the end of 5 years? This is a future value of an annuity example. • We can solve this problem using a programmed calculator, solving the math, or using Table 5.3.

11. Solve for the Future Value of an Annuity • The general equation for a FV of an annuity is: • The FV of the annuity in the example is:

12. Present Value of an Annuity • You plan to withdraw $1,000 annually from an account at the end of each of the next 5 years. If the account pays 12% annually, what must you deposit in the account today? This is an example of a present value of an annuity. • We can solve this problem using a programmed calculator, solving the math, or using Table 5.4.

13. Solve for the Present Value of an Annuity • The general equation for PV of an annuity is: • The PV of the annuity in the example is:

14. Perpetuity—An Infinite Annuity • A perpetuity is essentially an infinite annuity. • An example is an investment which costs you $1,000 today and promises to return to you $100 at the end of each forever! • What is your rate of return or the interest rate?

15. The Present Value of a Perpetuity • Another investment pays $90 at the end of each year forever. If 10% is the relevant interest rate, what is the value of this investment to you today? We need to solve for the present value of the perpetuity.

16. Present Value of a Deferred Annuity • There are 3 different PV of annuity computations: • The payments on an ordinary annuity begin one period after the PV. • The payments on an annuity due begin on the same date as the PV. • The payments on a deferred annuity begin 2 or more periods after the PV. Thus it is called a deferredannuity since the payments are deferred more than one period from the present.

17. Computing the PV of a Deferred Annuity • An investment promises to pay $100 annually beginning at the end of 5 years and continuing until the end of 10 years. What is the value of this investment today at a 7% interest rate? Because the payments are deferred 5 years, this is a PV of deferred annuity problem. • 1st step:Compute the PV of an ordinary annuity.

18. Computing the PV of a Deferred Annuity • 2nd step : Discount the PV of the ordinary annuity through deferral period.

19. General Formula for PV of a Deferred Annuity • PMT = $ amount of the perpetuity payment • i = interest rate • n = the number of perpetuity payments • m = the deferral period minus 1

20. PV of a Series of Non-Constant Cash Flows • The PV of a series of non-constant cash flows is just the sum of the individual PV equations for each cash flow. • Where the Cfi’s are a series of non-constant cash flows from year 1 to year n.

21. PV of a Series of Non-Constant Cash Flows • Suppose some new kitchen equipment for your restaurant is expected to save you $1,000 in 1 year, $750 in 2 years, and $500 in 3 years. What is the PV of these cost savings today if 10% is the relevant interest rate?

22. Compounding Periods Other Than Annual • Future value of a lump sum. • inom = nominal annual interest rate • m = number of compounding periods per year • n = number of years

23. Compounding Periods Other Than Annual • A $1,000 investment earns 6% annually compounded monthly for 2 years.

24. Compounding Periods Other Than Annual • PV of a lump sum uses a similar adjustment to the basic equation for non-annual compounding. • inom = nominal annual interest rate • m = number of compounding periods per year • n = number of years

25. Compounding Periods Other Than Annual • Annuity computations require the annuity period and the compounding period to be the same. • For example, suppose a car loan for $12,000 required 48 equal monthly payments and uses a 9% annual rate compounded monthly. • The annuity payments and the compounding periods are both monthly. • The interest rate needs to be expressed as a monthly rate:

26. Compounding Periods Other Than Annual • The car loan payment can be computed with the following equation: • And the car loan payment = $298.62.

27. Effective Annual Rate • An effective annual rate is an annual compounding rate. When compounding periods are not annual, the rate can still be expressed as an effective annual rate using the following: • inom = nominal annual rate • m = number of compounding periods in 1 year

28. Effective Annual Rate • A bank offers a certificate of deposit rate of 6% annually compounded monthly. What is the equivalent effective annual rate?

29. Amortized Loans • Amortized loans are paid off in equal payments over a set period of time and can be viewed as the PV of an ordinary annuity. • An amortization schedule follows for a $120,000 mortgage to be paid of with 360 monthly payments of $965.55 each over 30 years. The interest rate is 9% annually compounded monthly or 0.75% per month.

30. Loan Amortization Schedule

31. Loan Amortization Schedule • A loan amortization schedule shows: • The amount of each payment apportioned to pay interest. The amount paid towards interest declines since the principal balance is declining. • The amount of each payment apportioned to pay principal balance. The amount paid towards principal balance increases as the interest amount declines. • The remaining balance after each payment.

32. Summary • FV & PV of a lump sum • FV & PV of an annuity and PV of a perpetuity • PV of a series of non-constant cash flows • Compounding other than annual and effective annual rates • Loan amortization schedule