CHAPTER THREECHAPTER THREE

THE INTEREST RATE FACTOR

IN FINANCING

Chapter ObjectivesChapter Objectives

• Present value of a single sum• Future value of a single sum• Present value of an annuity• Future value of an annuity• Calculate the effective annual yield for

a series of cash flows• Define what is meant by the internal

rate of return

Compound InterestCompound Interest

• PV= present value• i=interest rate, discount rate, rate of

return• I=dollar amount of interest earned• FV= future values• Other terms:

– Compounding– Discounting

Compound InterestCompound Interest

• FV=PV (1 + i)n

• When using a financial calculator:– n= number of periods– i= interest rate– PV= present value or deposit– PMT= payment– FV= future value– n, i, and PMT must correspond to the same

period:– Monthly, quarterly, semi annual or yearly.

The Financial CalculatorThe Financial Calculator

• n= number of periods• i=interest rate• PV= present value, deposit, or mortgage

amount• PMT= payment• FV= future value• When using the financial calculator three

variables must be present in order to compute the fourth unknown.– PV or PMT must be entered as a negative

Future Value of a Lump SumFuture Value of a Lump Sum

• FV=PV(1+i)n

• This formula demonstrates the principle of compounding, or interest on interest if we know:– 1. An initial deposit

– 2. An interest rate

– 3. Time period

– We can compute the values at some specified time period.

Present Value of a Future SumPresent Value of a Future Sum

• PV=FV 1/(1+i)n

• The discounting process is the opposite of compounding

• The same rules must be applied when discounting– n, i and PMT must correspond to the

same period• Monthly, quarterly, semi-annually, and

annually

Future Value of an AnnuityFuture Value of an Annuity

• FVA=P(1+i)n-1 +P(1+i)n-2 ….. + P

• Ordinary annuity (end of period)

• Annuity due (begin of period)

Present Value of an AnnuityPresent Value of an Annuity

• PVA= R 1/(1+i)1 + R 1/(1+i)2…..

R 1/(1+i)n

Future Value of aFuture Value of a Single Lump Sum Single Lump Sum

• Example: assume Astute investor invests $1,000 today which pays 10 percent, compounded annually. What is the expected future value of that deposit in five years?

• Solution= $1,610.51

Future Value of an AnnuityFuture Value of an Annuity

• Example: assume Astute investor invests $1,000 at the end of each year in an investment which pays 10 percent, compounded annually. What is the expected future value of that investment in five years?

• Solution= $6,105.10

AnnuitiesAnnuities

• Ordinary Annuity– (e.g., mortgage payment)

• Annuity Due– (e.g., a monthly rental payment)

Sinking Fund PaymentSinking Fund Payment

• Example: assume Astute investor wants to accumulate $6,105.10 in five years. Assume Ms. Investor can earn 10 percent, compounded annually. How much must be invested each year to obtain the goal?

• Solution= $1,000.00

Present Value of aPresent Value of a Single Lump Sum Single Lump Sum

• Example: assume Astute investor has an opportunity that provides $1,610.51 at the end of five years. If Ms. Investor requires a 10 percent annual return, how much can astute pay today for this future sum?

• Solution= $1,000

Payment to Amortize Payment to Amortize Mortgage LoanMortgage Loan

• Example: assume Astute investor would like a mortgage loan of $100,000 at 10 percent annual interest, paid monthly, amortized over 30 years. What is the required monthly payment of principal and interest?

• Solution= $877.57

Remaining Loan Remaining Loan Balance CalculationBalance Calculation

• Example: determine the remaining balance of a mortgage loan of $100,000 at 10 percent annual interest, paid monthly, amortized over 30 years at the end of year four.– The balance is the PV of the remaining

payments discounted at the contract interest rate.

• Solution= $97,402.22