MATHEMATICS OF INVESTMENT -...

MATHEMATICS OF INVESTMENT AN INTRODUCTION

Prepared by: Francis Joseph H. Campeña 1

CHAPTER 1 Simple and Discount Interest

In financial transactions, interest is the amount paid by a borrower to a lender for the

use of money over a period. Interest that is paid as a percent of amount borrowed or invested is called simple interest. The formula for simple interest is given by the following:

Example 1. Suppose and amount Php 500.00 is invested for 2 years at 6% per year. How much is the earning of the investment after two years? Solution:

The following can be obtained from the problem: 500P , 060.r , 2t .

602060500 .tPrI . From this we conclude that, the investment earned Php60.00.

2. If Php4, 000.00 is borrowed at an annual interest rate of 16% for 0.75 years. How much is the interest due in borrowing the amount of money? Solution:

The following can be obtained from the problem: 0004,P , 160.r , 750.t .

4807501600004 ..,tPrI . From this we conclude that, the interest due is

Php480.00.

Simple Interest

Where,

( )

( )



Sometimes if we are the investor, we consider the value of our investment after a given period. In this case we introduce the concept of future values or accumulated values or maturity value. Example 1. If Php4, 000.00 is borrowed at an annual interest rate of 16% for 0.75 years, what is the value of the investment after 0.75 years? Solution: Since the interest earned by the amount invested for 0.75 years is Php480.00, the value of Php4,000.00 after 0.75 years is Php4,480.00. 2. What is the simple interest rate applied if an investment of Php37,500 accumulates to Php45,937.00 in the period of 1.5 years? Solution:

We note that the interest earned by the investment is Php8, 437, that is, 8437I . From the

formula tPrI , we have

%..,

,

Pt

Ir 15150

5150037

4378

3. The repayment on a loan was Php12,100. If the loan was for 15 months or 1.25 years at 16.8% interest a year, how much was the principal?

( )

Future Value

Where,

( )

( )



The Bankers Rule or Ordinary Simple Interest is applied whenever a given

problems does not specify the time factor to be used.

Solution:

Based from the given we have the following: 10012,F , 1680.r , and 251.t

Since rtPF 1 , we have

0001025116801

10012

1,

..

,

rt

FP

.

Different ways of expressing time/term of a loan or investment. Sometimes the term of investment is not given in years. The term or time frame given in certain problems maybe stated in days or months. In cases where the time is expressed in months it is easy to express it in years. But when the term/time is given in days we use a time factor such as the following:

Sometimes the term or time frame may be drawn from the specified origin and repayment dates. The following indicates how to compute for the actual time and approximate time. Actual time – Number of days until the repayment date except the origin date. Approximate time – Assumes that every month contains 30 days.

Ordinary Simple Interest or Bankers Rule

Exact Simple Interest

Remark



Example 1. Find the actual and approximate time from May 1, 1983 to September 15, 1983.

2. Find the actual and approximate time from April 15, 2008 to December 21, 2008.

3. Find the actual and approximate time from June 25, 2008 to Nov 18, 2008.

Approximate Time June 5 July 30 Aug 30 Sept 30 Oct 30 Nov 18 143

Approximate Time April 15 May 30 June 30 July 30 Aug 30 Sept 30 Oct 30 Nov 30 Dec 21 246

Actual Time May 30 June 30 July 31 Aug 31 Sept 15 137

Approximate Time May 29 June 30 July 30 Aug 30 Sept 15 134

31-1=30 30-1=29

Actual Time April 15 May 30 June 30 July 31 Aug 31 Sept 30 Oct 31 Nov 30 Dec 21 250

30-15=15

Actual Time June 5 July 31 Aug 31 Sept 30 Oct 31 Nov 18 146

30-25=5 30-25=5

30-15=15



Discount Interest.

Similar to simple interest, discount interest is an amount paid for borrowing money. Unlike simple interest, however, discount interest is charged at the time the loan has been negotiated and executed. Whereas, simple interest is paid on the maturity date when it is added to the amount of the loan applied for on the origin date, discount interest is charged in advance and is taken from the amount of the loan applied for on the origin date. Example Francis borrows Php 10,000 from SSS for a one year term. He was charged a 5% discount interest rate. Determine his proceeds. Solution: ( )( )( ) ( )

( )

Discount Interest

Where,



EXERCISES 1. Determine the Actual and Approximate number of days in the given origin and repayment

dates.

Origin Date Repayment Date Actual Time Approximate Time

1 May 22, 1995 July 09, 1995

2 January 06, 1997 November 06, 1997

3 March 03, 2007 October 11, 2007

4 February 04, 1990 November 05, 1992

5 March 02, 2005 November 05, 2006

2. Justin borrowed Php 8,600 on November 2, 1992, which is to be repaid on May 21, 1993 at

16.2% interest per year. Find the amount to be repaid. How much will the interest be at the repayment date using the following time factors? a. Bankers Rule b. Exact Simple Interest

c.

d.

3. How much should Mark pay to Michele if he borrowed Php 10,000 on June 25, 2008 and if

the principal and interest are to be paid on November 18, 2008 at 15% simple interest per year? Use the following time factors. a. Bankers Rule b. Exact Simple Interest

c.

d.

4. At what simple interest rate will a sum of money double itself in 15 years? 5. How long will it take for Php 4,000 to grow Php 14,000 if the simple interest rae is 12.5%? 6. How much will Php 22,500 become if invested at 9% simple interest rate for 45 days? 7. April issues a check for Php 4,950 to settle a 4 month loan of Php 4,500. How much simple

interest rate was she charged?



Promissory Notes These are written commitments by a person (also called the drawer) or business to pay a certain sum to another person (also called the drawee) or business within a specified time. A type of promissory note is the simple interest note which has the following characteristics:

The note is drawn on the origin date

The note is redeemed on the maturity date

The stated value of the note (or the face value ) corresponds to the principal

The face value plus the interest is called the maturity value. Example From the note we have the following:

Drawer Mark Tan

Drawee Justin Cruz

Face Value Php 5,000

Interest Rate 16.2%

Term 75 days

Maturity Date March 27, 2008 (75 days after January 11, 2008)

Interest ( )( ) (

)

Maturity Value

Php 5, 000.00 Quezon City, Philippines January 11, 2008

Seventy five days after the above date, the undersigned promises to pay to the order of Justin Cruz Five thousand pesos with simple interest at 16% per annum payable at BCC Bank of QC, Philippines.

Mark O. Tan



Another type of promissory note is the discount note. In this case, the amount stated in the note is the maturity value on which the interest is computed and charged in advance. The money received by the drawer on the origin date is the proceeds. In this case, the interest charged in advance is first computed. Then it is subtracted from the maturity value. The difference is the proceeds. Example From the note we have the following:

Drawer John O. Javier

Drawee BBC Bank

Maturity Value Php 7,000

Discount Interest Rate

14.4%

Term 100 days

Maturity Date January 3, 2009 (100 days after September 25, 2008)

Interest in advance ( )( ) (

)

Proceeds

Remark:

Php 7, 000.00 Manila City, Philippines September 25, 2008

One hundred days after the above date, for value received with interest at 14.4% per annum discounted to maturity, the undersigned promises to pay to the order of Justin Cruz Seven thousand pesos payable at BCC Bank of QC, Philippines.

John O. Javier



If the term in the note is given in months, the day of the origin date and the maturity date coincide. Thus, a 5 month note with an origin date of May 12, 2008 will mature on October 12, 2008.

Exercises

1. A simple interest note for 120 days at 13.8% per annum has a maturity value of Php 5,753. What is the face value of this note?

2. Chard signed a Php 2,800 bank discount note on April 16, 2008.If the note was for 10 months a 18% per annum, find the following:

a. Maturity date b. Interest deducted in advance c. proceeds

3. Determine the interest and maturity value of a simple interest note signed on October 29, 2008 and due February 11, 2009. The note has a face value of Php 10,000 and an interest rate of 16.5% per annum.

Discounting Notes Example 1 Mike bought office furniture from Jay and gave him a Php 50,000 bank discount note due on August 15, 1993, for the balance. Jay needs money on June 16, 1993 and sells his note to a bank, which discounts it a 13.8%. Find the proceeds of Jay. Solution: The number of days from June 16, 1993 – August 15, 1993 is 60. Finding the discount interest we have:

( )( ) (

)

Thus the proceeds of Jay is . Illustration

June 16 Aug 15

60 days

50,000



Example 2 Bert holds a Php 7,500, 120-day simple interest note from Nestor Flores. The interest rate is 15% and the note was made on August 18, 1993. On September 17, 1993, Bert wanted to encash the note. He found a bank which would buy the note that day at 16% interest, collectible in advance. How much will Bert receive from the bank on September 17? Solution

The value of Php 7,500 on the maturity date is ( ) (

)

The interest in advance that the bank will take is ( ) (

)

Therefore the proceeds that Bert will receive is just Exercises Find the proceeds of the following ban discount notes:

Maturity Value Maturity Date Date of Discount Interest Rate

1. 30,000 Nov. 25, 1993 July 3, 1993 10.2%

2. 23,200 May 16, 1994 Dec. 15, 1993 12%

3 4,000 Sept. 17, 1993 Feb. 19, 1993 9.75%

4. 6,600 June 5, 1994 Jan. 11, 1994 8%

5. 12,000 April 16, 1993 Oct. 25, 1992 12.6%

1. Mia holds a Php 4, 000 note at 12.75%, simple interest, payable in a year, from Banco de

Oro. In need of cash, Mia sells it back to the bank after 4 months. How much will he receive if the bank charges 15% interest to be drawn in advance?

2. Caloy receives a 60-day Php 6,200 bank discount note on September 4, 2008. How much will he received if it is discounted at 9% 15 days later?

Aug 18 Dec 16

15% in 120 days

Aug 18

16% in 90 days

7,500



CHAPTER 2 Compound Interest Exercises Compound Interest

1. Find the compound amount and interest: a. If Php 2,500 is invested at 13% compounded quarterly for 12 years

Solution Given:

( ) ( )

b. If Php 3,700 is invested at 12% compounded semi-annually for 5 years Solution Given:

( ) ( )

2. Find the present value of Php 2,850 due in 5 years if money is worth 10% compounded quarterly.

Solution Given:

( ) ( )

3. How much must be invested today in a savings account to realize Php 9,000 in 4 years, if money earns at the rate of 4% compounded quarterly?

Solution Given:



( ) ( )

4. What rate compounded annually will double any sum in 6 years?

Solution Let be the amount to be invested. . We want to find j.

( ) (

)

( ) √

5. What nominal rate converted semi-annually will make Php 20,000 amount to Php

50,000 in 8.5 years? Solution We want to find j. Given: .

( ) (

)

(

)

( √

)

6. When will Php 30,000 earn interest of Php 15,000 if it is invested at the rate of 7.5% converted annually? Solution We want to find t. Given:

( ) (

)

( )

( )

( )

7. When will a principal double itself if the interest rate is 14% compounded quarterly?

Solution We want to find j. Given:

( ) (

)

( ) ( )

( )

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