Excel Companion Chapter 3
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Chapter 3 Index
EC_3   TIME VALUE of MONEY:
    Terminolgy and Notation
  TVM1.xls Simple versus Compound Interest Future Value, Doubling Time, Present Value
  TVM1.xls Financial Calculator 1
  TVM1.xls Continuous Compounding
  TVM2.xls FV of Ordinary Annuity
  TVM2.xls PV of Ordinary Annuity
  TVM2.xls Financial Calculator 2
  TVM2.xls Car Loan Amortization
  TVM2.xls Home Loan Amortization
     

Introduction to the Time Value of Money:Terminology and Notation

When dealing with cash flows and other financial applications we must keep in mind the fundamental principle that money has time value. One dollar several weeks, months, or years from now does not have the same value as one dollar today. A certain number of dollars in the present can earn interest in the future, usually compound interest. In practice, interest rates are generally quoted on an annual basis (often called the nominal rate). Suppose the nominal rate is 9%. If the periods are semi-annual (six months), then the interest rate must be a semi-annual rate. Since there are two six-month periods in a year, we divide the annual rate by two to obtain the semi-annual rate. If the periods are months, then the interest rate must be a monthly rate. Since there are twelve months in a year, we must divide the annual rate by twelve to obtain the monthly rate. Thus, when the interest periods are given in units other than years, we must convert the annual interest rate into an interest rate per period by dividing the annual interest rate by the number of periods in a year.

nominal rate 9 %

2 periods per year

 semi-annual rate = = 4.5 %

nominal rate 9 %

12 periods per year

 monthly rate = = 0.75 %

The correspondence between the interest rate (% i) and the number of periods is particularly important when the interest rate is compounded at a rate other than annual. Many situations deal with daily compounding using a 365 day year. In these cases, we divide the annual rate by 365 to obtain the daily rate and convert time units to days. For example, if we have a 10 % annual rate compounded daily for two years, then we multiply 2 times 365 to obtain the number of periods and divide the 10% annual rate by 365 to obtain the daily rate. Leap years have 366 days. [In certain situations, such as overnight loans between banks, a 360 day year is often used along with simple interest formulas.] You also need to pay attention to the form that your particular calculator or spreadsheet requires for the entry of an interest rate. For example, if the interest rate for the period is 4% (four percent) then some devices require you to enter the decimal form, .04. Others allow you to enter 4 followed by the % key, and certain financial calculators allow you to enter 4 followed by the % i key.

Patterns that lead to formulas for simple interest and compound interest

P = Principal or Present Value   i = interest rate for one conversion period   n = number of conversion period

SIMPLE INTEREST

Initial amount Year 1 Year 2 Year 3 · · · Year n
P P + Pi P(1+i)+Pi P(1+2i)+Pi    
    =P(1+i+i) =P(1+2i+i)    
  =P(1+i) =P(1+2i) =P(1+3i) · · · =P(1+ni)

COMPOUND INTEREST
Initial amount Year 1 Year 2 Year 3 · · · Year n
P P + Pi P(1+i)+P(1+i)i P(1+i)2+P(1+i)2 i    
    =P(1+i)(1+i) =P(1+i)2(1+i)    
  =P(1+i) =P(1+i)2 =P(1+i)3 · · · =P(1+i)n

Commonly Used Symbolism, Notation, and Key Formulas

Symbols Definitions
j nominal rate ( annual rate, quoted rate, APR) as a decimal
%j nominal rate ( annual rate, quoted rate, APR) as a percent
m Frequency of compounding (number of interest periods/ year)
i interest rate for one interest period as a decimal
%i interest rate for one interest period as a percent
EFF effective rate ( interest on $1 in one year as a percent ). Banks often call this the " effective yield" ( usually to nearest hundredth of a percentage)
t time period, usually in years
n or N total number of periods ( days, quarters, months, years)
PV present value of a lump sum ( usually in dollars)
PVA present value of an annuity ( usually in dollars)
FV future value of a lump sum ( usually in dollars)
FVA future value of an annuity ( usually in dollars)
PMT size of the periodic payment ( usually in dollars)

Formula (1) is the key formula that underlies most compound interest calculations. The formulas for both types of annuities, FVA and PVA, can be derived using this key formula and the mathematical pattern for summing a series of numbers in which any two consecutive terms have a common ratio (geometric series).

1) Compound Interest -- Lump Sum
  where and n = m*t
2) Future Value of an Annuity
equivalent to
3) Present Value of an Annuity
equivalent to

 

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Copyright © Joseph F. Aieta, Babson College 1997