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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
www.wiley.com/college/Bradley © John Wiley and Sons 2013
• Compound interest: formula • Compound interest: Calculations • Present values • Annual percentage rates
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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…..
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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=
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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
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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=
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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=
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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
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How to compound three times annually (rate = i% pa)
tt iPP )1(0 += ..compounding once annually
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
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How to compound m times annually (rate = i% pa)
tt iPP )1(0 += ..compounding once annually
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
www.wiley.com/college/Bradley © John Wiley and Sons 2013
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
Method Substitute the values given
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)
Method Substitute the values given
into the compound interest formula above
• 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
Method Substitute the values given
into the compound interest formula above
• 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=
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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
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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
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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..
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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
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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
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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
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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
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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
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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
( ) 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
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Calculate the APR: Progress Exercises 5.4 no 11(b)
Pt is the same which-ever method is used, hence
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
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Calculate the APR: Progress Exercises 5.4 no 11(b)
Pt is the same which-ever method is used, hence
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%
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Calculate the APR: Progress Exercises 5.4 no 11(b)
Pt is the same which-ever method is used, hence
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
www.wiley.com/college/Bradley © John Wiley and Sons 2013
Calculate the APR: Progress Exercises 5.4 no 11(b)
Pt is the same which-ever method is used, hence
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%
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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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