SIMPLE and COMPOUND

INTEREST

Module 3

INTEREST

the money paid for the use of borrowed capital

for the lender viewpoint, represents the income produced by the money he lent.

the income produced by the money which has been loaned

CASH FLOW DIAGRAM

the graphical representation of each flow in the scale

Cash-in flow (+) Cash-out flow (-)

VIEWPOINT OF THE VIEWPOINT OF THE

BORROWER LENDER

0 1 2 3 . . . n 0 1 2 3 … n

F P

P F

A. SIMPLE INTEREST

the interest is based on the original amount of the loan or principal

I = Pin

F = P + I or F = P (1 + in)

Where: F = Future worth or accumulated amountP = Present worth or original valueI = Total interesti = Interest rate per periodn = no. of periods

Sample Problems

1. Find the interest and final amount due on P5,500 at a simple interest rate of 15½% for 9 months.

2. The interest on a loan of P15,000 is P4,500. If the rate is 15%, when is the loan due?

3. Mr. Garcia paid P1,222.50 on a loan made 3 months before at 12 ½% simple interest. Find the original amount of the loan and interest generated.

4. Determine the simple interest rate if an investment of P20,000 accumulates to P23,300 in 18 months.

TYPES OF SIMPLE INTEREST

ORDINARY SIMPLE INTEREST - computed using banker’s calendar (1 year = 360 days)

EXACT SIMPLE INTEREST - based on the exact number of days of the given year

Leap year = 366 days; Non leap year = 365 days

Divide the last 2 digits of the year by 4, if the result is a whole number, then it is a leap year.

All century years are leap years.

Sample Problems

1. Find the exact interest and the final amount due on P28,000 at 12% for 120 days.

2. Using ordinary interest, determine the final amount due on P10,800 at 15 ½% for 100 days.

3. Find the approximate and the actual number of days from March 15,1993 to Dec. 20 of the same year.

BANK DISCOUNT

Used if the charge for business loans is based on the final amount rather than on the principal or present value.

like simple interest, it is also an amount paid for borrowing money. However, unlike simple interest, the discount is charged in advanced.

Proceeds - the money which the borrower receives.

Discount rate - percentage used in computing the discount

BANK DISCOUNT

Id = Fdt

P = F – Id or P = F (1 – dt)

Where: Id = DiscountF = Final amountP = Discounted valued = Discount rate t = Time or term of discount

Sample Problems

1. A man borrows P800 due in 6 months from a lender who charged a discount rate of 8%. How much was the discount and the money the borrow gets?

2. Find the amount due at the end of 18 months whose present value is P2,000 at 14% simple discount.

3. Discount P10,900 for 9 months and find the discount at:

a. 12% simple interest

b. 12% simple discount

B. COMPOUND INTEREST

the interest for an interest period is calculated on the principal plus the total amount of interest accumulated in previous periods

the interest on top of interest

in most business transactions, the practice is to state or quote an annual interest rate and this is called the nominal rate of interest

B. COMPOUND INTEREST

F = P ( 1 + i )n

P = F ( 1 + i )-n

i = r

mWhere: (1+i)n = Single payment compound amount factor

(1+i)-n = Single payment present worth factori = Rate of interest per interest period r = Nominal rate of interestm = No. of compounding periods per year

Sample Problems

1. Find the compound interest earned and the compound amount due at the end of 5 years if P10,000 is invested at 12% compounded quarterly.

2. Accumulate P12,000 for 10 years at 7% compounded (a) annually (b) semi-annually (c) quarterly (d) monthly.

EFFECTIVE RATE OF INTEREST (ERI)

for two or more nominal rates to be equal, their effective rates must be equal

ieff = ( 1 + i )n - 1

Sample Problems

1. Find the effective rate equivalent to 8% compounded semi-annually.

2. What rate converted quarterly yields the effective rate 8%.

3. Which is better to invest at 7% compounded monthly or at 71/2% compounded semi-annually?

AMOUNT OF CHANGING RATES

the amount at the end of an interest rate which becomes the principal for new rate

AMOUNT OF CHANGING RATES

P = F ( 1 + i )-n

P = F (1 + ieff)-n

P = F [ 1 + (r/m)]-nm

For Present Worth:

F = P ( 1 + i )n

F = P (1 + ieff)n ; m=12

F = P [ 1 + (r/m)]nm

For Future Worth:

Sample Problems

1. Find the amount in 6 years if P2,500 is invested at 16% compounded semi-annually in the first 4 years and at 14% compound quarterly in the last 2 years.

2. A man invested P15,000 for 10 years. How much will be the final amount at the end of 10 years if the interest rate is 14% compounded quarterly for the first 3 years and 12% compounded semi-annually for the next 4 years and 11% compounded quarterly for the remaining 3 years.

EQUATION OF VALUE

obtained by setting the sum of the values on a certain comparison of focal date of one set of obligations equal to the sum of the values on the same date of another set of obligations

DISCRETE AND CONTINUOUS COMPOUNDING

Discrete Compounding

Present Worth

Future Worth

Continuous Compounding - interest is continuous throughout the year

DISCRETE AND CONTINUOUS COMPOUNDING

F = Pern

P = Fe-rn

Where: F = Future worth P = Present worthr = Nominal rate of interestn = no. interest periode = 2.71828

Sample Problems

1. Find the accumulated value of P5,000 at the end of 5 years if it is invested at 12% converted continuously.

2. Find the present value of P9,000 due in 4 years at 10% converted continuously.

3. Mr. Tan owes Mr. Sy P5,000 due at the end of 3 years and P8,000 due at the end of 7 years. Mr. Tan is allowed to replace these obligations by a single payment at the end of 5 years. How much should he pay on the 5th year if money is worth 14%, m = 2.

4. A man owes P5,000 due in 3 years with interest at 10 % compounded quarterly and P10,000 due in 5 years with interest at 12% compounded semi-annually. If money is worth 8% effective, what single payment at the end of 6 years will equitably replace these 2 debts?