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FINANCIAL

MATHEMATICS

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Financial Mathematics

FINM-A Objectives... FINM 1 FINM-B Simple Interest... FINM 2 FINM-C Compound Interest ... FINM 6 FINM-D Effective Interest Rates... FINM 8 FINM-E Equations of Value ... FINM 9 FINM-F Summary... FINM 10 FINM-G Review Exercise... FINM 11 FINM-H Appendix − Transposing Formula... FINM 12 FINM-Y Index... FINM 13

FINM-Z Solutions... FINM 14

FINM-A Objectives

• To perform calculations involving simple interest

• To perform calculations involving compound interest

• To use effective interest rate to compare interest rates

• To determine unknown debts using equations of value.

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FINM-B Simple Interest

When money is borrowed for a loan or invested, interest accumulates. The amount of interest depends on: • the amount of money borrowed or invested, the principal;

• the interest rate; and

• the time.

To calculate the simple interest, I, accumulated on the principal, P, over an interval of t years at an annual interest rate of r, the formula ^I^{= × ×}^{P r t} is used.

The percentage interest rate must be expressed as a decimal for calculations. The symbol p.a. used with interest rates denotes per annum meaning an annual or yearly interest rate. Interest rates stated as flat rates are used in simple interest calculations.

The total amount of money that must be repaid on a loan or the total value of an investment can be called the future value, S. The future value is calculated using ^S ^{= +}^P ^I . The principal is also called the present value.

Examples FINM-B1

1. A student borrows $600 to buy a camera. The loan is over two years, and the simple interest rate is 6% per year. How much interest does the student pay? What is the total amount of money repaid?

Principal = P = $600

Interest rate = 6% per year ∴ r = 0.06 t = 2 years

600 0.06 2 72

I=P r t

= × ×

=

Interest to be paid is $72

600 72 672 S = +P I

= +

=

Total amount ot be paid is $672

• list known information

• write formula

• substitute values

• evaluate

• answer in words

• write formula

• evaluate

• answer in words

2. An investor lent $12000 to a business associate for 6 months at an interest rate of 8% per year.

Calculate the interest the investor earns and how much in total will be repaid.

Principal = P = $12 000

Interest rate = 8% per year ∴ r = 0.08 t = 6 months = ₁₂⁶

= 0 . 5

year

12000 0.08 0.5 480

I=P r t

= × ×

=

Interest to be earned is $480

12000 480 12480 S = +P I

= +

=

Total amount ot be paid is $12 480

• write formula

• evaluate

• answer in words

• write formula

• evaluate

• answer in words

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Examples FINM-B1 continued

3. Calculate the simple interest earned when $500 is invested at 7.5% p.a. for 21 months.

Principal = P = $500

Interest rate = 7.5% per year ∴ r = 0.075 t = 21 months = ₁₂²¹

= 1 . 75

years

500 0.075 1.75 65.625

I =P r t

= × ×

=

Interest to be earned is $65.63

• write formula

• evaluate

• answer in words

Exercise FINM-B1

1. For each of the following situations define principal P, the interest rate r, and the time t.

(a) Barry borrows $15000 in order to buy a second-hand car. The loan is over three years and the interest rate is 12% per year.

(b) Britney wishes to purchase a boat which requires her to take out a $20000 loan at 15% per annum over five years.

2. Find the simple interest due and the total amount owing for each of the following loans.

(i) $2000 at 8% per year for 3 years.

(ii) $18000 at 12.5% per year for 6 months.

(iii) $5000 at 1.5% per month for 3 years.

(iv) $16000 at 8% per year for 36 months.

3. Find the simple interest due, and the total amount to be repaid for each of the following loans.

(i) $3000 at 10% per year for 3 years 6 months.

(ii) $3000 at 8% per year for 18 months.

(iii) $3000 at 12% per year for 2 years 9 months.

(iv) $3000 at 8.5% per year for 36 months.

It is possible to use the simple interest formula, I=P r t to calculate the principal, P, interest rate, r, or the time, t, required to give the specified amount of simple interest, I. To find these values it will be necessary to substitute in given quantities and then rearrange the equation to solve for the unknown.

Sometimes it is necessary to calculate how much money is required to be invested now, (present value) at a given interest rate, to amount to a specified value at some time in the future (future value). To achieve this the formula needs to be manipulated as follows.

(1 ) 1

S P I

S P P r t

S P rt

P S rt

= +

∴ = +

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Examples FINM-B2

1. Calculate the amount of money that would earn $750 simple interest if it was invested at 4.5% p.a. for 5 years.

Principal = P = ? Interest = I = 750

Interest rate = 4.5% per year ∴ r = 0.04.5 t = 5 years

750 0.045 5 750 0.225

750 3333.33 0.225

I P r t P

P P

=

= × ×

=

∴ = =

$3333.33 would need to be invested.

• write formula

• solve for unknown

• answer in words

2. Calculate the time required for $8300 to earn $3087.60 interest at the flat rate of 6.2% p.a.

P = $8300 I = $3087.60 r = 0.062 t = ?

3087.60 8300 0.062 3087.60 514.6

3087.60 514.6 6

I P r t

t t

t

=

= × ×

=

∴ = =

It would take 6 years to earn $3087.60 interest.

• write formula

• answer in words

3. Betty won $245 000 in lotto. She wants to travel with the interest earned by investing her winnings. At what interest rate must betty invest her money so that she will earn $27 500 every 18 months?

P = $245 000 I = $27 500

t = 18 months = ¹⁸₁₂=1.5years r = ?

27500 245000 1.5 27500 367500

27500 367500 0.074829 0.075 7.5%

I P r t

r r r

=

= × ×

=

∴ =

=

≈ =

Interest rate required is 7.5% p.a.

• write formula

• answer in words

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Examples FINM-B2 continued

4. Brendan wants to have $15000 for a new boat in 10 years time. How much should he invest at 5% per year to save this amount (assuming no withdrawals are made and the interest rate does not change)?

S = $15000 t = 10 years r = 0.05 • list known information

(1 ) 1

15000 1 0.05 10 15000

10000 1.5

S P I S P P r t

S P rt

P S rt

= +

∴ = +

= + ×

= =

OR

• write formula

• transpose formula

• evaluate

15000 0.05 10

15000 0.5

15000 1.5 15000

10000 1.5

S P I

S P P r t

P P

= +

= + × ×

= +

=

= =

• write formula

Brendan would need to invest $10 000. • answer in words

When money is borrowed, it is usually paid back on a regular basis over the term of the loan. The total amount of the loan, S, is divided by the required number of payments to give the amount of the regular repayment.

Example FINM-B3

A student borrows $600 to buy a camera. The loan is over two years, and the simple interest rate is 6% per year. How much will her monthly repayments be?

P = $600 Interest rate = 6% per year∴ r = 0.06 t = 2 years = 24 months

600 0.06 2 600 72

72 672

I =P r t S = +P I

= × × = +

= =

Total repaid 672

repayments 28

no. of months 24 S

= =n = =

Her repayments will be $28 per month.

• determine amount to be repaid

• write formula, substitute values and evaluate

• answer in words Exercise FINM-B2

1. Find the principal that was invested to earn $3348 interest in 6 years at 6.2% p.a.

2. How long was $4500 borrowed for at 7.1% p.a. to cost $798.75 interest?

3. At what interest rate was $875 invested for to earn $73.50 interest in 18 months?

4. What amount of money should be invested now at 10% per year to amount to $1800 in 2 years’ time?

5. What amount of money should be invested now at 6% per year to amount to a future value of

$8000 seven months from now?

6. A $120 000 property is purchased with a deposit of $30000. The balance is repayable over 15 years at 7% per year. Find:

(a) the balance of the purchase price (b) the interest due

(c) the total amount repayable (d) the monthly repayments

7. Brian wishes to borrow $15000 for a 6 month holiday in Europe. He has approached two banks and has the following options. The People's Bank quotes an interest rate of 5% per year with the repayments to be made over 3 years, whilst the Friendly Bank interest rate is 4% per year and the payments are to be made over 4 years. Which loan will ultimately be cheaper for Brian?

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FINM-C Compound Interest

In compound interest, the interest earned by an invested amount of money (or principal) is reinvested so that it too earns interest. That is, at the end of each interest period the interest earned for that period is added to the principal so that the interest also earns interest over the next interest period. In the same way, interest due on a debt at the end of a period is subject to interest in the next period.

The accumulated amount or compounded amount, S, has same meaning as future value. The difference between the compounded amount and the original principal is called the compound interest, I= −S P.

For an original principal of P invested at a periodic interest rate of i for n periods, the compounded amount (or future value), S, is given by ^S⁼^P

( )

¹⁺ⁱ ⁿ .

The periodic interest rate can be found by dividing the annual interest rate, r, by the number of periods in a year, N, i.e. r

i= N Example FINM-C1

Clarissa invests $5000 at 6.2% p.a. with interest compounded monthly. What would her investment be worth after five years? What amount of interest has been earned during the five years?

P = $5000

N = 12 (12 months in a year) r = 0.062 p.a.

0.062 12

i r per month

∴ = N =

t = 5 years ∴ = ×n 5 12=60months

( )

60

1

0.062 5000 1

12

$6811.69 S P i n

S

= +

⎛ ⎞

= ⎜ + ⎟

⎝ ⎠

=

The investment will be worth $6811.69.

6811.69 5000 1811.69 S P I

I S P

= +

∴ = −

= − =

The interest earned was $1811.69.

• write formula

• evaluate using calculator

5000 ( 1 + 0.062 ÷ 12 ) x^y 60 =

• answer in words

• write formula

• substitute values and evaluate

• answer in words

It is possible to calculate the principal if the period of investment and compounded amount are known.

By transposing the compound interest formula, ^S⁼^P

(

¹⁺ⁱ

)

ⁿ^, the formula for the principal becomes

(

1

)

(1 )

n n

P S OR P S i

i

= = + −

+

In a similar way the periodic interest rate can be determined by transposing the compound interest formula for i.

1

1 ⁿ 1

n S S

i or

P P

= − ⎛ ⎞⎜ ⎟ −

⎝ ⎠ .

FINM-H Appendix − Transposing Formula, shows the transposition processes.

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Examples FINM-C2

1. What principal should be invested now so that it will compound to

$13 000 in 5 years time at 12% per annum compounded quarterly?

S = $13 000

N = 4 (4 quarters in a year) r = 12 p.a.

0.012 12

i r per quarter

∴ = N =

t = 5 years ∴ = × =n 5 4 20quarters P = ?

• Solution can involve transposing the formula and then substituting values

( )

1

1 (1 )

n

n n

S P i

P S or S i

i

−

= +

+

20 20

13000 0.12

13000 1 0.12 4

1 4

7197.78

P or

P

⎛ ⎞−

= ⎜ + ⎟

⎝ ⎠

⎛ + ⎞

⎜ ⎟

⎝ ⎠

=

• OR substituting values and solving for the unknown.

( )

20

1 13000 1 0.12

4 13000

7197.78 1 0.12

4

S P i n

P P

= +

⎛ ⎞

= ⎜ + ⎟

⎝ ⎠

⎛ + ⎞ = =

⎜ ⎟

⎝ ⎠

The principal required is $7197.78 • write answer in words

2. What annual rate of interest, compounded quarterly, would be required if an initial investment of $3000 is to amount to $3500 after 5 years?

P = $3000 S = $3500 N = 4 (4 quarters in a year) t = 5 years ∴ = × =n 5 4 20quarters r = ? find i first.

( )

1

1 or 1

n

n n

S P i

S S

i P P

= +

= − ⎛ ⎞⎜ ⎟⎝ ⎠ −

1 20

20 3500 3500

1 1

3000 3000

0.007737313

i= ⎛⎜⎝ ⎞⎟⎠− or ⎛⎜⎝ ⎞⎟⎠ −

=

• write formula

• transpose formula

• evaluate using calculator ( 3000 ÷ 3500 ) x^y ( 1 ÷ 20 ) = The periodic rate is 0.774%, so the annual rate would be 4 × 0.774% = 3.096%

Exercise FINM-C1

1. Find the compounded amount and compound interest for the following:

(a) $600 for 8 years at 8% compounded monthly.

(b) $1000 for 5 years at 12% compounded daily.

(c) $750 for 12 months at 6% compounded weekly.

(d) $3000 for 4 years at 9% compounded quarterly.

2. How much should be invested at 8% compounded semi-annually for six years in order to provide a compounded amount of $10000?

3. The day Charlotte was born her father deposited $500 in a bank account. If the account was compounded annually and on Charlotte’s twenty-first birthday there was $1326.65 in the account. What was the annual interest rate?

4. A investor has a choice of investing $4000 at 10% compounded annually or at 9.75%

compounded monthly for one year. Which is the better investment?

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FINM-D Effective Interest Rate

The effective rate of interest is the equivalent rate of simple interest earned over a period of one year for an interest rate which is compounded twice or more over the year. The annual simple interest rate will be greater than the annual compounding interest rate to earn the same amount of interest.

The effective annual interest rate, re, is calculated using:

1 1

N e

r r

N

⎛ ⎞

= +⎜ ⎟ −

⎝ ⎠

where r is the annual interest rate (in decimal form) and N is the number of periods per year.

The annual interest rate, r, for compounding calculations is often called the nominal rate. Effective interest rates can be used to compare investment opportunities.

Examples FINM-D1

1. What effective interest rate is equivalent to a nominal rate of 12% compounded (a) monthly and (b) daily?

(a)

0.12 12

1 1 0.1268

12

12.68%

re = +⎛⎜⎝ ⎞⎟⎠ − ≈

=

(b)

0.12 365

1 1 0.1275

365

12.75%

re = +⎛⎜⎝ ⎞⎟⎠ − ≈

=

• monthly: 12 months in year N=12

• daily: 365 days in year N =365

2. Which is the "best" investment, 6% per annum compounded monthly or 6.20% compounded annually?

0.06 12

1 1 0.0617 6.17%

compunding monthly= +⎛⎜⎝ 12 ⎞⎟⎠ − ≈ =

So, an annual rate of 6.20% is better for investment than 6% per annum compounded monthly.

Exercise FINM-D1

1. Find the compounded amount and the compound interest for the given investment and annual rate.

(i) $6000 for 8 years at an effective rate of 8%.

(ii) $750 for 12 months at an effective rate of 10%.

2. Find the effective rate that corresponds to the given nominal rate.

(i) 10% compounded quarterly.

(ii) 12% compounded monthly.

(iii) 8% compounded daily.

(iv) 12% compounded daily.

3. A $6000 certificate of deposit is purchased for $6000 and is held for 7 years. If the certificate earns an effective rate of 8%, what is it worth at the end of that period?

4. To what sum will $2000 amount in 8 years if invested at a 6% effective rate for the first 4 years and 6% compounded semi-annually thereafter?

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FINM-E Equations of Value

An amount of money can have different values at different times, for a particular interest rate. The value of money is dependant on its ability to earn interest. Equations of value are used to compare the value of money at various times.

The equation ^S⁼^P

(

¹⁺ⁱ

)

ⁿ is used to find the value of money, at various points in the future while

(

1

)

ⁿ

P=S +i ⁻ is used to find the value of money at some point prior to this time. To perform calculations the value of repayments or debts must be determined at only one point in time.

For equations of values the general formula used is repayments = debts (at the same point in time). The value of each repayment or debt should be determined at the point in time when the final repayment is to be paid. To assist with calculations a time line can be drawn showing all repayments and debts.

Examples FINM-E1

1. If Edna owes $200 in 2 years and a further $300 in 5 years, how much does she need to pay at present to account for these debts? (Assume 8% compounding quarterly.)

8% . . 0.08

i= p a = 4 per quarter

1 2

1 1

200 300

2 4 8 5 4 20

S S

n quarters n quarters

= =

= × = = × =

• amounts moved left use ^P⁼^S

(

¹⁺ⁱ

)

⁻ⁿ

1 2

8 20

0.08 0.08

200 1 300 1

4 4

170.69 201.89 372.58

repayments debts debt debt x

− −

=

= +

⎛ ⎞ ⎛ ⎞

= ⎜ + ⎟ + ⎜ + ⎟

⎝ ⎠ ⎝ ⎠

= + =

• list given values

• draw a time line

• write formula

• evaluate Edna needs to pay only $372.58 now to cover her future debts.

2. A debt of $5000, which was due 5 years from now, is instead going to be repaid by three payments: $2000 now, $1000 in 2 years time and the third payment after 4 years. What will be the final payment if the assumed interest rate is 6% p.a. compounding half-yearly?

6% . . 0.06

i= p a = 2 per quarter note: 1 ^0.06 1.03 2

⎛ + ⎞=

⎜ ⎟

⎝ ⎠

( ) ( )

1 2 3

8

4 2

2000 1 0.06 1000 1.03 5000 1.03 2

2533.54 1125.51 4712.98

1053.93

repayments debts

repay repay repay debt x x x

−

=

+ + =

⎛ + ⎞ + + =

⎜ ⎟

⎝ ⎠

+ + =

= The final repayment will be $1053.93

• draw a time line

• move right use ^S⁼^P

(

¹⁺ⁱ

)

ⁿ

• move left use P=S

(

1+i

)

⁻ⁿ

• write formula

• evaluate

• answer in words

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Exercise FINM-E1

1. A debt of $550 due in 4 years and $550 due in 5 years is to be repaid by a single payment now. Find how much the payment is if an interest rate of 10% compounded quarterly is assumed.

2. A debt of $600 due is 3 years and $800 due in 4 years is to be repaid by a single payment 2 years from now. If the interest rate is 8% compounded semiannually, how much is the payment?

3. A debt of $5000 due in 5 years is to be repaid by a payment of $2000 now and a second payment at the end of 6 years. How much should the second payment be if the interest rate is 6% compounded quarterly?

4. A debt of $5000 due 5 years from now and $5000 due 10 years from now is to be repaid by a payment of $2000 in 2 years, a payment of $4000 in 4 years, and a final payment at the end of 6 years. If the interest rate is 7% compounded annually, how much is the final payment?

FINM-F Summary

To calculate the simple interest, I, accumulated on the principal, P, over an interval of t years at an annual interest rate of r, the formula I= × ×P r t.

The total amount of money that must be repaid on a loan or the total value of an investment can be called the future value, S. and is calculated using S= +P I.

It is possible to use the simple interest formula to calculate the principal, P, interest rate, r, or the time, t, required to give the specified amount of simple interest, I. It is also to determine the principal, P, to be invested now to accumulate to some future value, S, using the formula .

1 P S

= rt +

The total amount of the loan, S, is divided by the required number of payments to give the amount of the regular payment.

For an original principal of P invested at a periodic interest rate of i for n periods, the compounded amount (or future value), S, is given by S=P

(

1+i

)

ⁿ . The periodic interest rate can be found by dividing the annual interest rate, r, by the number of periods in a year, N.

It is possible to calculate the principal if the period of investment and compounded amount are known

using:

(

¹

)

(1 )

n n

P S OR P S i

i

= = + −

+ . In a similar way the periodic interest rate can be determined by

1

1 ⁿ 1

n S S

i or

P P

= − ⎛ ⎞⎜ ⎟⎝ ⎠ − .

The effective annual interest rate, re, is calculated using 1 1

N e

r r

N

⎛ ⎞

= +⎜⎝ ⎟⎠ − , where r is the annual interest rate (in decimal form) and N is the number of periods per year.

Equations of value are used to compare the value of money at various times. The equation

(

¹

)

ⁿ

S=P +i is used to find the value of money at various points in the future, while ^P⁼^S

(

¹⁺ⁱ

)

⁻ⁿ^is

used to find the value of money at some point before this time. For equations of value, the general formula used is repayments = debts (at the same point in time).

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FINM-G Review Exercise

1. Calculate the simple interest and accumulated amount for each of the following.

(i) $2000 at 8% p.a. for 5 years (ii) $35 000 at 5.6% p.a. for 30 months.

2. Angela bought a computer for $2100. She paid a deposit of $300 and took out a loan for the remainder at 8 %¹₂ p.a. simple interest over 3 years.

(a) How much interest did she pay?

(b) What was the total cost of buying the computer?

(c) How much did she have to pay per month?

3. How much needs to be invested at 11% p.a. simple interest to have an accumulated amount of $7800 in 5 years from now?

4. Calculate the compounded amount and the compound interest for the following.

(i) $2000 at 8% p.a. for 5 years compounded semi-annually (ii) $35 000 at 5.6% p.a. for 30 months compounded monthly.

5. Find the compound interest on a loan of $600 borrowed at 7.5% compounded quarterly for 3 years.

6. What annual interest rate, compounded monthly, would be required for an investment of

$5000 to amount to $6500 in 4 years?

7. Leading company A offers rates of 8.2% p.a. compounded quarterly, while company B advertises a rate of 8.1% compounded daily. Which company offers the best rate for a loan?

8. Two debts, $2000 due in 3 years and $1500 due in 5 years, are to be repaid by $1000 now and another repayment in 4 years. What is the value of the final repayment if the interest rate is 10% compounded monthly?

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FINM-H Appendix – Transposing Formula

Solving ^S⁼^P

(

¹⁺ⁱ

)

ⁿ for the principal, P.

( )

( ) ( )

1 1

1 1 1

n

n n

S P i

P i

S

i i

P S or P S i x

i x

− −

= +

+ +

⎡ ⎤

∴ = + = + ⎢⎣ = ⎥⎦

Solving ^S⁼^P

(

¹⁺ⁱ

)

ⁿ for the interest rate, i.

( )

( ) ( ( ) )

1 1

1

1 1

1 1 1 1 1 1

1 1

n

n n

S P i

P i

S

P P

S i

P

S S

i OR i

P P

S S

i i

P P

S S

i i

P P

S S

i i

P P

− −

−

= +

= + ⎛ ⎞⎜ ⎟⎝ ⎠ = +

− = + − ⎛ ⎞⎜ ⎟ − = + −

⎝ ⎠

− = ⎛ ⎞⎜ ⎟⎝ ⎠ − =

Solving ^S ⁼^P

(

¹⁺ⁱ

)

ⁿ for the number of investment periods, n.

( )

( ) ( ) ( ) ( )

( ( ) ) ( )

( )

1 1 1

log log 1 ln ln 1

log ln

log 1 ln 1

n

n n

S S

P P

S S

P P

S S

P P

S P i

P i

S

P P

S i

P

i OR i

n i n i

n n

i i

= +

= + = +

= =

+ +

Manipulation of logarithmic equations can be found in LOGS - Logarithms Module.

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FINM-Y Index

Topic Page

Annual Interest Rate ... FINM 2, 6, 8 Compound Amount, S... FINM 6 Compound Interest... FINM 6 Effective Annual Interest Rate... FINM 8 Equations of Value... FINM 9 Future Value, S ... FINM 2 Interest − compound ... FINM 8 Interest − simple... FINM 2, 3 Nominal Rate ... FINM 8 p.a. − per annum ... FINM 2 Periodic Interest Rate... FINM 6 Present Value, P... FINM 4 Principal, P ... FINM 2, 3, 7 Repayments... FINM 5, 9 Simple Interest ... FINM 2, 3 Transposing Formula ... FINM 17

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FINM-Z Solutions

FINM -B Simple Interest

FINM-B1 Simple Interest ... FINM 15 FINM-B1 Simple Interest Manipulation ... FINM 16

FINM -C Compound Interest

FINM-C1 Compound Interest ... FINM 17

FINM -D Effective Interest Rates

FINM-D1 Effective Interest Rates ... FINM 19

FINM -E Equations of Value

FINM-E1 Equations of Value... FINM 20 FINM -G Review Exercise

FINM-G Review Exercise... FINM 21

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FINM-B1 Simple Interest 1

.

(a) $15 000 12% 0.12 3

P r

t years

=

= =

=

(b) $20 000 15% 0.15 5

P r

t years

=

= =

= 2

.

(i) $2000 8% 0.08 3

2000 0.08 3

$480

2000 480

$2480

P r

t years I Prt

Total Amount P I

= = =

=

= = × ×

=

= +

=

(ii) $18 000 12.5% 0.125

6 0.5

18000 0.125 0.5

$1125

18000 1125

$19 125

P r

t months years I Prt

Total Amount P I

= = =

= =

= = × ×

=

= +

=

(iii) $5 000 3

1.5% . . 1.5% 12 . 0.015 12 0.18 5000 0.18 3

$2700

5000 2700

$7700

P t years

r p m p a

I Prt

Total Amount P I

= =

= = ×

= × =

= = × ×

=

= +

=

(iv) $16 000 8% 0.08

36 3

16000 0.08 3

$3840

16000 3840

$19 840

P r

Total Amount P I

= = =

= =

= = × ×

=

= +

=

3 .

(i) $3 000 10% 0.1

3 6 3.5

3000 0.1 3.5

$1050

3000 1050

$4 050

P r

t y months years I Prt

Total Amount P I

= = =

= =

= = × ×

=

= +

=

(ii) $3 000 8% 0.08

18 1.5

3000 0.08 1.5

$360

3000 360

$3 360

P r

Total Amount P I

= = =

= =

= = × ×

=

= +

= (iii)

9 12

$3 000 12% 0.12

2 9 2 2.75

3000 0.12 2.75

$990

3000 990

$3 990

P r

t y months years y

I Prt

Total Amount P I

= = =

= = × ×

=

= +

=

(iv) $3 000 8.5% 0.085

36 3

3000 0.085 3

$765

3000 765

$3 765

P r

t months years

I Prt

Total Amount P I

= = =

= =

= = × ×

=

= +

=

(17)

FINM-B2 Simple Interest Manipulation

1. ? 6.2% 0.062 6 3348

3348 0.062 6

3348 3348 0.372

0.062 6

$9 000 3348 $9 000

0.372

P r t years I

I Prt OR I Prt

I P P

rt

P P

= = = = =

= =

= = × ×

= =

×

∴ = = =

A principal of $9000 would need to be invested.

2. 4500 7.1% 0.071 ? 798.75

798.75 4500 0.071 798.75

798.75 319.5 4500 0.071

798.75

2.5 2.5

319.5

P r t I

I Prt OR I Prt

I t t

Pr

t t

t years t years

= = = = =

= =

= = × ×

= =

×

∴ = = =

The money would be invested for 2.5 years.

3. 875 ? 18 1.5 73.50

73.50 875 1.5 73.50

73.50 1312.5 875 1.5

73.50

0.056 5.6% 0.056 5.6%

1312.5

P r t months years I

I Prt OR I Prt

I r r

Pt

r r

= = = = =

= =

= = × ×

= =

×

∴ = = = = =

The interest rate would be 5.6%.

4.

( )

? 10% 0.1 12 1800

1

1800 1 0.1 2

1 1800 1.2

1800

1 1.2

1800 $1500

1 0.1 2

P r t years Amount

Amount P I OR S P rt

S P Prt P

P rt P

S P P

rt

P P

= = = = =

= + = +

= + = + ×

= + = ×

= =

+

= ∴ =

+ ×

$1500 should be invested.

5.

( )

7 12

? 6% 0.06 7 $8000

1

8000 1 0.06

1 8000 1.035

8000

1 1.035

8000 $7729.47

1 0.06

P r t months years Amount

S P Prt P

P rt P

S P P

rt

P P

= = = = = =

= + = +

= + = + ×

= + = ×

= =

+

= ∴ ≈

+ ×

$7729.47 should be invested.

(18)

FINM-B2 continued

6. (a) $120 000 $30 000

$90 000

Balance= −

=

(b) $90 000 7% 0.07

15 15 12 180

90000 0.07 15

$94 500

I r

t years months

I Prt

= = =

= = × =

=

= × ×

=

(c) 90 000 94 500

$184 500 Total Repayable= +

=

(d) ¹⁸⁴⁵⁰⁰

180

$1025 Monthly Repay=

=

7. $15 000 5% 0.05 3

4% 0.04 4

15000 0.05 3 15000 0.04 4

$2250 $2400

15000 2250 15000 2400

$17250 $17400

people people

friend friend

people friend

P r t years

r t years

I Prt I Prt

Total Total

= = = =

= = =

= =

= × × = × ×

= =

= + = +

= =

The cheaper loan is available from the People's Bank.

FINM-C1 Compound Interest

1. (a)

( )

(

^0.0812

)

⁹⁶

8% 0.08

$600 8 12 96

12 12 1

600 1 $1135.47 1135.47 600 $535.47

n

P n i

S P i

I S P

= = × = = =

= +

= + ≈

= −

= − =

(b)

( )

(

^0.12365

)

¹⁸²⁵

12% 0.12

$1000 5 365 1825

365 365 1

1000 1 $1821.94 1821.94 1000 $821.94

n

P n i

S P i

I S P

= = × = = =

= +

= + ≈

= −

= − =

(c)

( )

(

^0.0652

)

⁵²

12 6% 0.06

$750 52 52

12 52 52

1

750 1 $796.35 796.35 750 $46.35

n

P n i

S P i

I S P

= = × = = =

= +

= + ≈

= −

= − =

(19)

FINM-C1 continued

1. (d)

( )

(

^0.094

)

¹⁶

9% 0.09

$3000 4 4 16

4 4

1

3000 1 4282.86 4282.86 3000 $1282.86

n

P n i

S P i

I S P

= = × = = =

= +

= + ≈

= −

= − =

2.

( )

12

8% 0.08

? $10 000 6 2 12 0.04

2 2

1 1

1 10000 10000 1 0.04

1 0.04

$6245.97

n

P S n i

S P i

P S i OR P S

i

P

−

= = = × = = = =

= +

= + =

+

= + =

+

∴ ≈

3.

( )

1

1 21 21

$500 $1326.65 21 ?

1

1 1

1326.65 1326.65

1 1

600 600

0.0385 3.85% 0.0385 3.85%

n

n n

P S n i

S P i

S S

i OR i

P P

= = = =

= +

= − =⎛ ⎞⎜ ⎟ −

⎝ ⎠

⎛ ⎞

= − =⎜ ⎟ −

⎝ ⎠

≈ = ≈ =

The interest rate was approximately 4.8%.

4.

( ) ( )

10% 9.75%

1 0.0975 12

12

4000

For 10% compounded annually For 9.75% compounded monthly 9.75% 0.0975

10% 0.1 1 12

12 12

1 1

4000 1 0.1 4000 1

$4400 $4407.91

n n

P

i n i n

S P i S P i

=

= = = = = =

= + = +

= =

9.75% compounded monthly is the better investment.

(20)

FINM-D1 Effective Interest Rates 1. (i)

( )

⁸

$6000 8 8% 0.08

1

6000 1 0.08 11105.58 6000

$11105.58 $5105.58

n

P n i

S P i I S P

= = = =

= + = −

≈ =

(ii)

( )

¹

$750 12 1 10% 0.1

12 1

750 1 0.1 825 750

$825 $75

n

P n i

S P i I S P

= = = = =

= + = −

≈ =

2. (i)

4

10% 0.1 4

1 1

1 0.1 1 4

0.1038 10.38%

N e

r N

r r

N

= = =

⎛ ⎞

= +⎜ ⎟ −

⎝ ⎠

⎛ ⎞

= +⎜⎝ ⎟⎠ −

≈ =

(iii)

365

8% 0.08 365

1 1

1 0.08 1 365 0.0833 8.33%

N e

r N

r r

N

= = =

⎛ ⎞

= +⎜ ⎟ −

⎝ ⎠

⎛ ⎞

= +⎜⎝ ⎟⎠ −

≈ =

(ii)

12

12% 0.12 12

1 1

1 0.12 1 12

0.1268 12.68%

N e

r N

r r

N

= = =

⎛ ⎞

= +⎜ ⎟ −

⎝ ⎠

⎛ ⎞

= +⎜⎝ ⎟⎠ −

≈ =

(iv)

365

12% 0.12 365

1 1

1 0.12 1 365 0.1275 12.75%

N e

r N

r r

N

= = =

⎛ ⎞

= +⎜ ⎟ −

⎝ ⎠

⎛ ⎞

= +⎜⎝ ⎟⎠ −

≈ =

3.

( )

⁷

$6000 7 8% 0.08

1

6000 1 0.08 10282.95

n

P n i

S P i

= = = =

= +

≈

∴The certificate is worth $10 282.95 4.

( )

4

0.06 8 2

First 4 years: $2000 6% 0.06 4 1

2000 1 0.06 $2524.95 6% 0.06

Last 4 years: $2524.95 4 2 8

2 2

1

2524.95 1 $3198.53

n

P i n

S P i

P i n

S P i

= = = =

= +

= + ≈

= = = = × =

= +

= + ≈

The $2000 will amount to $3198.53.

(21)

FINM-E1 Equations of Value

1.

The repayment would be $706.14 now.

( ) ( )

1 1 2

16 20

10% 0.1 0.1

1 1.025

4 4 4

repayments debts

550 1.025 550 1.025 370.49 335.65 $706.14 i

repay debt debt

x ⁻ ⁻

⎛ ⎞

= = ⎜⎝ + ⎟⎠=

=

= +

= + =

2.

The repayment would be $1238.57.

( ) ( )

1 1 2

2 4

8% 0.08 0.08

1 1.04

2 2 2

repayments debts

600 1.04 800 1.04 554.73 683.84 $1238.57 i

repay debt debt

x ⁻ ⁻

⎛ ⎞

= = ⎜⎝ + ⎟⎠=

=

= +

= + =

3.

The second repayment should be

$2447.81.

( ) ( )

1 2 1

24 4

6% 0.06 0.06

1 1.015

4 4 4

repayments =debts

2000 1.015 5000 1.015 2859.01 5306.82

$2447.81 i

repay repay debt x x x

⎛ ⎞

= = ⎜⎝ + ⎟⎠=

+ =

+ = + =

= 4.

( )

( ) ( ) ( ) ( )

1 2 3 1 2

4 2 1 4

7% 0.07 1 0.07 1.07 repayments debts

2000 1.07 4000 1.07 5000 1.07 5000 1.07 2621.59 4579.60 5350.00 3814.48

$1963.29 i

repay repay repay debt debt

x x x

−

= = + =

=

+ + = +

= The final repayment will be $1963.29.

(22)

FINM-G Review Exercise

1. (i) $2000 8% 0.08

5

2000 0.08 5 $800 2000 800

$2800

P r

t years I Prt

Total Amount P I

= = =

=

= × × =

= +

=

(ii) $35000 5.6% 0.056

30 2.5

35000 0.056 2.5 $4 900 35000 4900

$39 900

P r

Total Amount P I

= = =

= =

=

= × × =

= +

=

2. (a) 2100 300 $1800 8.5% 0.085 3 36

1800 0.085 3

$459

P r t years months

I Prt

= − = = = = =

=

= × ×

=

(b) 1800 459

$2259

2259 300 $2559 S

Total Cost

= +

=

= + =

(c) 2259

$62.75 Repayments= 36 =

3.

( )

? 11% 0.11 5 $7800

1

7800 1 0.11 5

1 7800 1.55

7800

1 1.55

7800 $5032.26

1 0.11 5

P r t years Amount

S P Prt P

P rt P

S P P

rt

P P

= = = = =

= + = +

= + = + ×

= + = ×

= =

+

= ∴ ≈

+ ×

4. (i)

( )

(

^0.082

)

¹⁰

8% 0.08

$2000 5 2 10

2 2

1

2000 1 $2960.49

2960.49 2000 $960.49

n

P n i

S P i

I S P

= = × = = =

= +

= + ≈

= −

= − =

(ii)

( )

(

^0.05612

)

³⁰

5.6% 0.056

$35 000 30

12 12

1

35000 1 $40 246.47

40 246.47 35000 $5 246.47

n

P n i

S P i

I S P

= = = =

= +

= + ≈

= −

= − =

(23)

FINM-G continued

5.

( )

(

^0.0754

)

¹²

7.5% 0.075

$600 3 4 12

4 4

1

600 1 $749.83

749.83 600 $149.83

n

P n i

S P i

I S P

= = × = = =

= +

= + ≈

= −

= − =

The compound interest is $149.83.

6.

( )

1

1 48 48

$5000 $6500 4 12 48 ?

1

1 1

6500 6500

1 1

5000 5000

0.00548 0.548% . . 0.548% 12 6.576%

n

n n

P S n i

S P i

S S

i OR i

P P

i p m

annual rate

= = = × = =

= +

= − =⎛ ⎞⎜ ⎟ −

⎝ ⎠

⎛ ⎞

= − =⎜ ⎟ −

⎝ ⎠

∴ ≈ =

= × =

7.

4 365

8.2% 0.082 4 8.1% 0.081 365

1 1 1 1

0.082 0.081

1 1 1 1

4 365

0.0846 8.46% 0.0844 8.44%

A A B B

N N

eA eB

r N r N

r r

N N

= = = = = =

⎛ ⎞ ⎛ ⎞

= +⎜ ⎟ − = +⎜ ⎟ −

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞

= +⎜⎝ ⎟⎠ − = +⎜⎝ ⎟⎠ −

≈ = ≈ =

Company B offers the best rate for a loan.

8.

The final repayment would be $2077.90

( ) ( ) ( )

1 2 1 2

48 12 12

0.1 0.1 0.1

12 12 12

10% 0.1 12 12

repayments =debts

1000 1 2000 1 1500 1

1489.35 2209.43 1357.82

$2077.90 i

repay repay debt debt

x x x

−

= =

+ = +

+ + = + + +

+ = +

=

(24)