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Pre University-Mathematics for Business

Session 1

Statistical Thinking and Methods for Describing

Sets of Data

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Learning Objectives

• solve problems involving the time value of money.

• solve problems with interest is compounded continuously.

• introduce the notions of ordinary annuities and annuities due.

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It is really simple

–When people are feeling good about their circumstances, they spend more. When people are worried about their futures, they save more.

Why don’t

we save,

and how

much

should we

worry???

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Why is Interest Interesting?

• Simple Interest – Interest calculated only on the money you’ve

deposited.

• Earned Interest– Payment you receive

(because you are a LENDER)

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Why is Interest Interesting?

• Compound interest– It is interest calculated on both your deposits

made and prior interest earned. – “Interest on Interest”

One of the most powerful

principles in personal finance

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The Power of Compounding

8%

4% $10.40 $10.82

Interest Rate 1 Year 2 Years 4 Years 6 Years

$10.80 $15.87$13.60$11.66

$12.65$11.70

$ 10

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Compound Interest

• Compound amount S at the end of n interest periods at the periodic rate of r is as

nrPS 1

Suppose that $500 amounted to $588.38 in a savings account after three years. If interest was compounded semiannually, find the nominal rate of interest, compounded semiannually, that was earned by the money.

Example 1 – Compound Interest

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Solution:

There are 2 × 3 = 6 interest periods.

The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.

0275.01500

38.588

500

38.5881

500

38.5881

38.5881500

6

6

6

6

r

r

r

r

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Effective Rate

• The effective rate of interest is the amount of money that one unit (one dollar) invested at the beginning of a (the first) period will earn during the period, with interest being paid at the end of the (first) period.

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Example 5 – Effective Rate

Effective Rate• The effective rate re for a year is given by

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n

e n

rr

To what amount will $12,000 accumulate in 15 years if it is invested at an effective rate of 5%?

Solution:

14.947,24$05.1000,12 15 S

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Comparing Interest Rates

If an investor has a choice of investing money at 6% compounded daily or % compounded quarterly, which is the better choice?

Solution:

Respective effective rates of interest are

The 2nd choice gives a higher effective rate.

%27.614

06125.01

and %18.61365

06.01

4

365

e

e

r

r

8

16

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Present Value

• P that must be invested at r for n interest periods so that the present value, S is given by

Find the present value of $1000 due after three years if the interest rate is 9% compounded monthly.

Solution:

For interest rate, .

Principle value is .

nrSP 1

15.764$0075.11000 36 P

0075.012/09.0 r

Example 1 – Present Value

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Example 5 – Net Present Value

You can invest $20,000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table.

Assume an interest rate of 7% compounded annually and find the net present value of the cash flows.

Net Present Value investment Initial - luespresent va of Sum NPV ValuePresent Net

Year Cash Flow

2 $10,000

3 8000

5 6000

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Solution:

31.457$

000,2007.1600007.1800007.1000,10NPV 532

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Example

Year Project A ($) Project B ($)

1 140000 20000

2 80000 40000

3 60000 60000

4 20000 100000

5 20000 180000

Each of following mutually exclusive projects involve an initial cash outlay of $240,000. The estimated net cash flows for the projects are:

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• The company’s required rate of return is 11 percent. Calculate the NPV for both projects. Which project should be chosen? Why?

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Solution

NPVA =

5432 1.11

000 20

1.11

000 20

1.11

000 60 +

1.11

000 80 +

1.11

000 140 + 000 240$

= $20 000 (to nearest thousand)

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Solution

NPVB =

543

2

1.11

000 180 +

1.11

000 100 +

1.11

000 60 +

1.11

000 40 +

1.11

000 20 + 000 $240

= $27 000 (to nearest thousand)Using the NPV method, project B should be selected.

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ExercisesABC firm is considering to invest in one of three mutually exclusive projects X, Y and Z. The initial investment and annual net cash inflows over the life of each project are shown in the following table:

  Project X Project Y Project ZInitial investment

$78,000 $52,000 $66,000

Year Net cash flows1 $17,000 $28,000 $15,0002 $25,000 $38,000 $15,0003 $33,000 ------ $15,0004 $41,000 ------ $15,0005 ------ ------ $15,0006 ------ ------ $15,0007 ------ ------ $15,0008 ------ ------ $15,000

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All the projects have equal risk and firm’s required rate of return for these projects is 14%. Calculate the NPV for each project over its life. Rank the projects in descending order based on NPV.