Chapter 10 Section 1 Interest

Terms Interest : Fee that is paid for the use of money Principal : Amount of initial deposit or initial/current balance Compound Amount : Amount to which the principal grows (after the addition on interest). Alternate term: Balance Compounded : Computed

Compound Periods The number of times interest is compounded in a single year Denoted by : m Table 2 (page 470)

Compound Periods Table Interest Compounded … Number of Interest Periods Per Year Annuallym = 1 Semiannuallym = 2 Quarterlym = 4 Monthlym = 12 Weeklym = 52 Dailym = 365

Annual Interest Rate Denoted by : r Also known as Nominal Rate or Stated Rate. Number which is stated / advertised and used to calculate the interest rate per period. Use decimal form when calculating by hand.

Interest Rate Per Compound Period Denoted by : i Number which is used to calculate interest for each compounding period. Use decimal form when calculating. Formula on next slide (page 470 – Blue- gray box).

Interest Rate Per Period Formula Formula: i = r / m where r = annual interest rate ( in decimal form) m = number of compound periods in a year

Example of Interest Rate Per Period Find the interest rate per period of an account that earns 6.25% interest compounded weekly. Solution: Given: r = and m = 52 i = r / m = / 52 ~ Interest rate per period is approximately 0.12 %

Compound Interest Problems Basic idea for compound interest accounts 1.Deposit an initial amount of money into an account. 2.Step back and watch it grow. 3.You do not deposit or withdraw any additional money while interest is accumulating.

Diagram for Compound Interest … P B1B1 B2B2 B3B3 B4B4 B0B0 B = Balance P = Principal = Initial Deposit Each tick mark represents a compound period Balances: Interest: Deposits or Withdraws i ·B 0 i ·B 1 i ·B 2 i ·B 3

Balance for Compound Interest New balance based on the old balance B new = B previous + i·B previous which simplifies to B new = (1+ i)B previous (Note that this is in the form of a difference equation) Note that i·B previous represents the amount of interest that one earns for the compound period

Balance after n interest/compounding periods F = ( 1 + i ) n ·P Where: F = compounded amount after n compounding periods. P = Principal (in the form of an initial deposit or current balance).

Notational Differences BookCalculatorTerm nNNumber of compounding periods FFVFuture Value PPVPrincipal Value mP/Y & C/Y Number of compounding periods in a year RPMTRent / Payment per period

Accessing the TVM Solver 1.Hit APPS key 2.Select 1:Finance function (Hit ENTER key) 3.Select 1:TVM Solver …function (Hit ENTER key)

TVM Solver Variables N = Number of compound periods I% = Annual Interest Rate (in percent form ( r% )) PV = Principal Value (or) “Previous/Current” Balance PMT = Rent / Payment Per Compound Period FV = Future Value P/Y = Payments Per Year = m C/Y = Compounding Periods Per Year = m PMT:END = Payments(/Interest) made(/calculated) at the end of the compounding period

Using the TVM Solver Enter the numbers for each variable of interest. Move the cursor to the variable that you want to solve for. Hit the ALPHA (green) key and then the ENTER (/solve) key. The answer will appear next to the variable that you are solving for.

When using the TVM Solver on the calculator Think: 1.Outflow = NEGATIVE cash flow (i.e. You DO NOT have the instantaneous use of your money ) 2.Inflow = POSITIVE cash flow (i.e. You do have the instantaneous use of your money )

Exercise 5 (page 477) Formula Solution Calculate the compound amount of $1,000 after 2 years if deposited at 6% interest compounded monthly. Solution: n = 2 ·12 = 24 i = r/m = 0.06/12 = F = ( 1 + i ) n ·P F = ( ) 24 ·1000 F = ( …) ·1000 F = Answer :$1,127.16

Exercise 5 (page 477) TVM Solver Solution Calculate the compound amount of $1,000 after 2 years if deposited at 6% interest compounded monthly. Solution: N = 2 ·12 = 24 I% = 6 PV = – 1000 PMT = 0 FV = P/Y = C/Y = 12 Set cursor on FV and Solve ( ALPHA key and then ENTER key) Answer :$1, Note the negative sign!!!

Effective Rate of Interest Page 474 Used to 1.Compares two annual interest rates that have two different yearly compounding periods. 2.When money from the interest is reinvested in the account, will tell you the ‘true’ interest rate that you are earning.

Effective Rate of Interest Formula Formula: r eff = ( 1 + i ) m – 1 where: r eff = Effective Rate of Interest i = Interest Rate Per Period = r / m m = Number of compounding periods in a single year

Effective Rate of Interest on the Calculator Access –Hit APPS key –Select 1:Finance function –Use down (or up) arrow key to select C: Eff( function Syntax Eff( r%, m ) ( r% = Annual Interest Rate in % form)