CHAPTER 5 : FINANCIAL MATHS. · CHAPTER 5 : FINANCIAL MATHS. © John Wiley and Sons 2013 . © John...

CHAPTER 5 : FINANCIAL MATHS.

© John Wiley and Sons 2013 www.wiley.com/college/Bradley © John Wiley and Sons 2013

Essential Mathematics for Economics and Business, 4th Edition

http://www.wiley.com/college/Bradley

www.wiley.com/college/Bradley © John Wiley and Sons 2013

• Compound interest: formula • Compound interest: Calculations • Present values • Annual percentage rates



How compounding is carried out (annual interest rate i %)

The next slide explains ….how interest is calculated at the end of each year

….interest earned is added to the principal ….principal at the start of next year = (principal + interest) from previous year


The compound interest formula: an explanation The table below will be filled in, row by row… ..to demonstrate the idea of compounding annually at an interest rate i %

Amount at start of year = principal

Interest earned during year

Amount at end of Year = principal + interest

Copyright© 2013 Teresa Bradley and John Wiley & Sons Lt

iP2 P2 + iP2 = P2(1+ i) = P3

iP1 P1 + iP1 = P1(1+ i) = P2

Year t Pt-1 iPt-1 Pt-1 + iPt-1 = Pt-1(1+ i) = Pt

In general, at the end of year t ….

Year 1 P0 iP0 P0 + iP0 = P0(1+ i) = P1

P1 Year 2

Year 3 P2

continued…..


Compound interest formula continued..

principal + interest BUT, write in terms of P0… P0 + iP0 = P0(1+ i) = P1

In general….

Next year P1 + iP1 = P1(1+ i) = P2

Next year P2 + iP2 = P2(1+ i) = P3

But P1(1+ i) = P0(1+ i) (1+ i)

so… P2 = P0(1+ i)2

so… P3 = P0(1+ i)3

so… P1 = P0(1+ i)

But P2(1+ i) = P0(1+ i)2 (1+ i)

P4 = P0(1+ i)4

….Pt = P0(1+ i)t

..and so on…


Worked Example 5.5 (see text) Calculate the amount owed on a loan of £1000 at the end of three years, interest compounded annually, rate of 8%

..the compound interest

formula Method Substitute the values given

into the compound interest formula

• t = 3 years • P0 = 1000 • i =

Calculations

1008

you will need.. P t = P0(1+ i)t

3)08.1(1000=

303 )1( iPP +=

3)08.01(1000 +=

= 0.08

)2597120.1(1000=

712.1259=


Terminology: present value; future value

In the compound interest formula ….

P t = P0(1+ i)t

Pt is called the future value of P0 at the end of t years

when interest at i% is compounded annually.

P0 is called the present value of Pt when discounted at i%

annually.

…see following examples



The present value formula is deduced from the compound interest formula as follows:

tt iPP )1(0 +=

t

t

tt

iiP

iP

)1()1(

)1(0+

+=

+

0)1(

Pi

Pt

t =+

tt

iPP

)1(0 +=


Worked Example 5.6 (a)(i) £5000 is invested at an interest rate of 8% for three years

..the compound interest

formula Method Substitute the values given

into the compound interest formula

• t = 3 years • P0 = 5000 • i = = 0.08

Calculations

1008

You will need P t = P0(1+ i)t

3)08.1(5000=

303 )1( iPP +=

3)08.01(5000 +=

)2597120.1(5000=

5.6298=


Revise terminology. present value and future value

In the compound interest formula

P t = P0(1+ i)t

future value present value

In Worked Example 5.6

Pt = 6298.5 is called the future value of P0 = 5000 at the end of 3 years when invested at 8% interest compounded annually

P0 = 5000 is called the present value of Pt= 6298.5 when discounted at 8% annually for 3 years


Worked Example 5.6(b)(i) Present value calculations (£6298.5 discounted at 8% annually for three years)

..the present value

formula will be required Method Substitute the values given

into the present value formula

• t = 3 years • Pt = 6298.5 • i = = 0.08

Calculations

1008

tt

i

PP

)1(0

+=

33

0)1( i

PP+

=

3)08.01(5.6298

+=

3)08.1(5.6298

=

5000=

Worked Example 5.6 (b)(ii) Present value calculations (£15,000 discounted at 8% annually for three years)

..the present value

formula will be required Method Substitute the values given

into the present value formula

• t = 3 years • Pt = 15,000 • i = = 0.08

Calculations

1008

tt

i

PP

)1(0

+=

33

0)1( i

PP+

=

3)08.01(15000+

=

3)08.1(15000

=

48.11907=


Compound interest compound twice annually (rate = i % pa)

tt iPP )1(0 += ..compounding once annually

t

tiPP

2

0 21

+= ..compounding twice annually

2 x t compoundings necessary in t years

Two compoundings necessary in 1 year At each compounding

use the annual rate, i, divided by 2



How to compound three times annually (rate = i% pa)


t

tiPP

3

0 31

+= ..compounding three times annually

At each compounding

use the annual rate, i, divided by 3 3 x t compoundings

necessary in t years

Three compoundings necessary in 1 year



How to compound m times annually (rate = i% pa)


mt

t miPP

+= 10 ..compounding m times annually

At each compoumding

use the annual rate,i, divided by m

m x t compoundings necessary in t years

m compoundings necessary in 1 year



Compounding continuously t

t iPP )1(0 += …compounding once annually

mt

t miPP

+= 10 …compounding m times annually

tm

t miPP

+= 10 …rearranging

[ ] ittit ePePP 00 ==

∞→→

+ me

mi i

m as 1

itt ePP 0=


Worked Example 5.8 (a). £5000 is invested at an interest rate of 8% for three years compounded semi-annually

Method Substitute the values given

in the question into the compound interest formula above

• m = 2 • t = 3 years • P0 = 5000 • i = = 0.08

Calculations

1008

595.6326)2653190.1(5000

)04.1(5000 6

===

6)04.01(5000 +=

mt

t miPP

+= 10 3

03 1×

+=

m

miPP

32

3 208.015000

×

+=P

you will need the formula..

Worked Example 5.8 (c)(i). £5000 is invested at an interest rate of 8% for three years compounded monthly


into the compound interest formula above

• m = 12 • t = 3 years • P0 = 5000 • i = = 0.08

Calculations

1008

185.6351)270237.1(5000

==

mt

t miPP

+= 10 3

03 1×

+=

m

miPP

312

3 1208.015000

×

+=P

you will need the formula..

Worked Example 5.8 (c)(ii) £5000 is invested at an interest rate of 8% for three years compounded daily (assume 365 days per year)



• m = 365 • t = 3 years • P0 = 5000 • i = = 0.08

Calculations

1008

1095)0002192.1(5000=

mt

t miPP

+= 10 3

03 1×

+=

m

miPP

3365

3 36508.015000

×

+=P

you will need the formula...

)2712157.1(5000=

079.6356=

Worked Example 5.9 £5000 is invested at an interest rate of 8% for three years compounded continuously



• t = 3 years • P0 = 5000 • i = = 0.08

Calculations

1008

24.05000e=

itt ePP 0= 3

03×= iePP

308.03 5000 ×= eP

you will need the formula...

)2712492.1(5000=

246.6356=


How much do you gain when interest is compounded more than once annually? Review results in Worked Examples 5.6, 5.7 and 5.9 £5000 is invested at a nominal interest rate of 8% for three years but compounded at various intervals annually. The future value at the end of 3 years was calculated: • 6298.560 compounded once annually

• 6326.595 compounded twice annually

• 6351.185 compounded monthly

• 6356.079 compounded daily

• 6356.246 compounded continuously



How much do you gain when interest is compounded more than once annually? Review results in Worked Examples 5.6, 5.7 and 5.9 £5000 is invested at a nominal interest rate of 8% for three years but compounded at various intervals annually • 6298.560 one conversion period

• 6326.595 2 conversion periods

• 6351.185 12 conversion periods

• 6356.079 365 conversion periods • 6356.246 infinite conversion periods (continuous)


How much do you gain by compounding more than once annually?

Conversion periods/year

Amount at end of 3 years

Difference over annual compounding

1 6298.560

2 6326.595 6326.595 - 6298.560 = 28.035

12 6351.185 6351.185 - 6298.560 = 52.625

365 6356.079 6356.079 - 6298.560 = 57.519

Infinitely many (continuous)

6356.246

6356.246 - 6298.560 = 57.686

Copyright© 2008 Teresa Bradley and John Wiley & Sons Lt


How do we make comparisons when different conversions periods are used? • Use Annual Percentage Rates: APR • What is the APR? • The APR is the interest rate, compounded annually that

yields an amount Pt • the same amount Pt would be yield when any • other method of compounding is used, for example..



Annual Percentage Rates: APR Pt calculated using the APR rate annually is the same as Pt calculated by the given method

itt ePP 0=

tt APRPP )1(0 +=

mt

t miPP

+= 10



Calculate the APR when interest is compounded m times annually

mt

t miPP

+= 10

compounding m times annually at a nominal rate of i % p.a.

tt APRPP )1(0 += compounding once annually at

APR% p.a.

But Pt is the same whichever method is used, hence mt

tmiPAPRP

+=+ 1)1( 00

Next slide



Calculate the APR when interest is compounded m times annually But Pt is the same which-ever method is used, hence

mtt

miPAPRP

+=+ 1)1( 00

mtt

miAPR

+=+ 1)1(

m

miAPR

+=+ 1)1(

11 −

+=

m

miAPR



Calculate the APR when interest is compounded continuously

But Pt is the same which-ever method is used, hence

itt ePAPRP 00 )1( =+

itt eAPR =+ )1(

ieAPR =+ )1(

1−= ieAPR



Calculate the APR: Progress Exercises 5.4 no 11(a)

Pt is the same which-ever method is used, hence

323

206.015500)1(5500

×

+=+ APR

323

206.01)1(

×

+=+ APR

2

206.01)1(

+=+ APR

( ) 0609.0103.01 2 =−+=APR Correct to 4 decimal places



Calculate the APR: Progress Exercises 5.4 no 11(a)


323

206.015500)1(5500

×

+=+ APR

( ) 0609.0103.01 2 =−+=APR

Nominal interest rate is 6% When interest is compounded twice annually

the APR is 6.09%

Correct to 4 decimal places



Calculate the APR: Progress Exercises 5.4 no 11(b)


3123

1206.015500)1(5500

×

+=+ APR

3123

1206.01)1(

×

+=+ APR

( )12005.01)1( +=+ APR

( ) 0617.01005.1 12 =−=APR Correct to 4 decimal places





3123

1206.015500)1(5500

×

+=+ APR

( )12005.01)1( +=+ APR

( ) 0617.01005.1 12 =−=APR Correct to 4 decimal places

Nominal interest rate is 6% When interest is compounded twelve times

annually the APR is 6.17%





33653

36506.015500)1(5500

×

+=+ APR

33653

36506.01)1(

×

+=+ APR

365

36506.01)1(

+=+ APR

0618.010618.1 =−=APR Correct to 4 decimal places





3123

1206.015500)1(5500

×

+=+ APR

( )12005.01)1( +=+ APR

0618.010618.1 =−=APR Correct to 4 decimal places

Nominal interest rate is 6% When interest is compounded daily

the APR is 6.18%



Calculate the APR Progress Exercises 5.4 no 11(d) But Pt is the same which-ever method is used, hence

306.00

30 )1( ×=+ ePAPRP

306.03)1( ×=+ eAPR

06.0)1( eAPR =+

0618.0106.0 =−= eAPR Correct to 4 decimal places

Nominal rate is 6%

APR is 6.18%


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