Essenial Mahemaics for Economics and Business, 4 h Ediion CHATER 5 : FINANCIAL MATHS. John Wiley and Sons 201 www.wiley.com/college/bradley John Wiley and Sons 201
Compound ineres: formula Compound ineres: Calculaions resen values Annual percenage raes www.wiley.com/college/bradley John Wiley and Sons 201
How compounding is carried ou (annual ineres rae i %) The nex slide explains.how ineres is calculaed a he end of each year.ineres earned is added o he principal.principal a he sar of nex year (principal + ineres) from previous year www.wiley.com/college/bradley John Wiley and Sons 201
The compound ineres formula: an explanaion The able below will be filled in, row by row..o demonsrae he idea of compounding annually a an ineres rae i % Amoun a sar of year principal Ineres earned during year Amoun a end of Year principal + ineres Year 1 0 i 0 0 + i 0 0 (1+ i) 1 Year 2 1 i 1 1 + i 1 1 (1+ i) 2 Year 2 i 2 2 + i 2 2 (1+ i) In general, a he end of year. Year -1 i -1-1 + i -1-1 (1+ i) Copyrigh 201 Teresa Bradley and John Wiley & Sons L coninued..
Compound ineres formula coninued.. principal + ineres BUT, wrie in erms of 0 0 + i 0 0 (1+ i) 1 so 1 0 (1+ i) Nex year Bu 1 (1+ i) 0 (1+ i) (1+ i) 1 + i 1 1 (1+ i) 2 so 2 0 (1+ i) 2 Nex year 2 + i 2 2 (1+ i) In general. Bu 2 (1+ i) 0 (1+ i) 2 (1+ i) so 0 (1+ i) 4 0 (1+ i) 4..and so on. 0 (1+ i) www.wiley.com/college/bradley John Wiley and Sons 201
Worked Example 5.5 (see ex) Calculae he amoun owed on a loan of 1000 a he end of hree years, ineres compounded annually, rae of 8% you will need.. 0 (1+ i)..he compound ineres formula Mehod Subsiue he values given ino he compound ineres formula years 0 1000 8 i 0.08 100 Calculaions + 0( 1 i) 1000 (1 + 0.08) 1000(1.08) 1000(1. 2597120) 1259.712
Terminology: presen value; fuure value In he compound ineres formula. 0 (1+ i) is called he fuure value of 0 a he end of years when ineres a i% is compounded annually. 0 is called he presen value of when discouned a i% annually. see following examples www.wiley.com/college/bradley John Wiley and Sons 201
The presen value formula is deduced from he compound ineres formula as follows: (1 i) 0 + (1 + i) 0 (1 + (1 + i) i) (1 + i) 0 0 (1 + i) www.wiley.com/college/bradley John Wiley and Sons 201
Worked Example 5.6 (a)(i) 5000 is invesed a an ineres rae of 8% for hree years You will need 0 (1+ i)..he compound ineres formula Mehod Subsiue he values given ino he compound ineres formula years 0 5000 8 i 0.08 100 Calculaions 0 ( 1 i) + 5000 (1 + 0.08) 5000(1.08) 5000(1.2597120) 6298.5
Revise erminology. presen value and fuure value In he compound ineres formula fuure value 0 (1+ i) presen value In Worked Example 5.6 6298.5 is called he fuure value of 0 5000 a he end of years when invesed a 8% ineres compounded annually 0 5000 is called he presen value of 6298.5 when discouned a 8% annually for years www.wiley.com/college/bradley John Wiley and Sons 201
Worked Example 5.6(b)(i) resen value calculaions ( 6298.5 discouned a 8% annually for hree years) 0..he presen value formula will be required Mehod Subsiue he values given ino he presen value formula years 6298.5 8 i 0.08 100 (1 + i) Calculaions 0 ( 1+ i) 6298.5 (1 + 0.08) 6298.5 (1.08) 5000
Worked Example 5.6 (b)(ii) resen value calculaions ( 15,000 discouned a 8% annually for hree years) 0..he presen value formula will be required Mehod Subsiue he values given ino he presen value formula years 15,000 8 i 0.08 100 (1 + i) Calculaions 0 ( 1+ i) 15000 (1 + 0.08) 15000 (1.08) 11907.48
Compound ineres compound wice annually (rae i % pa) (1 i) 0 +..compounding once annually 2 i 0 1 +..compounding wice annually 2 A each compounding use he annual rae, i, divided by 2 Two compoundings necessary in 1 year 2 x compoundings necessary in years www.wiley.com/college/bradley John Wiley and Sons 201
How o compound hree imes annually (rae i% pa) (1 i) 0 +..compounding once annually 0 i 1 +..compounding hree imes annually A each compounding Three compoundings necessary in 1 year use he annual rae, i, divided by x compoundings necessary in years www.wiley.com/college/bradley John Wiley and Sons 201
How o compound m imes annually (rae i% pa) (1 i) 0 +..compounding once annually 0 1+ i m m..compounding m imes annually A each compoumding use he annual rae,i, divided by m m compoundings necessary in 1 year m x compoundings necessary in years www.wiley.com/college/bradley John Wiley and Sons 201
Compounding coninuously (1 i) 0 + compounding once annually 0 1+ i m 1 + m i m m compounding m imes annually 0 rearranging [ i ] i e e 0 0 0 e i 1 + i m m e i as m www.wiley.com/college/bradley John Wiley and Sons 201
Worked Example 5.8 (a). 5000 is invesed a an ineres rae of 8% for hree years compounded semi-annually you will need he formula.. Mehod Subsiue he values given in he quesion ino he compound ineres formula above m 2 years 0 5000 8 i 0.08 100 1+ i m Calculaions 0 m m i 0 1 + m 2 0.08 5000 1 + 2 6 5000 (1 + 0.04) 6 5000(1.04) 5000(1.265190) 626.595
Worked Example 5.8 (c)(i). 5000 is invesed a an ineres rae of 8% for hree years compounded monhly you will need he formula.. m 1+ Mehod Subsiue he values given ino he compound ineres formula above m 12 years 0 5000 8 i 0.08 100 i m Calculaions 0 m i 0 1 + m 5000 1 + 0.08 12 5000(1.27027) 651.185 12
Worked Example 5.8 (c)(ii) 5000 is invesed a an ineres rae of 8% for hree years compounded daily (assume 65 days per year) you will need he formula... Mehod 0 1+ Subsiue he values given ino he compound ineres formula above m 65 years 0 5000 8 i 0.08 100 i m m Calculaions m i 0 1 + m 5000 1 + 1095 5000(1.0002192) 0.08 65 5000(1. 2712157) 656.079 65
Worked Example 5.9 5000 is invesed a an ineres rae of 8% for hree years compounded coninuously Mehod Subsiue he values given ino he compound ineres formula above years 0 5000 8 i 0.08 100 Calculaions you will need he formula... i 0 e i 0 e 5000e 0.24 5000e 0.08 5000(1.2712492) 656.246
How much do you gain when ineres is compounded more han once annually? Review resuls in Worked Examples 5.6, 5.7 and 5.9 5000 is invesed a a nominal ineres rae of 8% for hree years bu compounded a various inervals annually. The fuure value a he end of years was calculaed: 6298.560 compounded once annually 626.595 compounded wice annually 651.185 compounded monhly 656.079 compounded daily 656.246 compounded coninuously www.wiley.com/college/bradley John Wiley and Sons 201
How much do you gain when ineres is compounded more han once annually? Review resuls in Worked Examples 5.6, 5.7 and 5.9 5000 is invesed a a nominal ineres rae of 8% for hree years bu compounded a various inervals annually 6298.560 one conversion period 626.595 2 conversion periods 651.185 12 conversion periods 656.079 65 conversion periods 656.246 infinie conversion periods (coninuous) www.wiley.com/college/bradley John Wiley and Sons 201
How much do you gain by compounding more han once annually? Conversion periods/year Amoun a end of years 1 6298.560 Difference over annual compounding 2 626.595 626.595-6298.560 28.05 12 651.185 651.185-6298.560 52.625 65 656.079 656.079-6298.560 57.519 Infiniely many 656.246 656.246-6298.560 57.686 (coninuous) Copyrigh 2008 Teresa Bradley and John Wiley & Sons L
How do we make comparisons when differen conversions periods are used? Use Annual ercenage Raes: AR Wha is he AR? The AR is he ineres rae, compounded annually ha yields an amoun he same amoun would be yield when any oher mehod of compounding is used, for example.. www.wiley.com/college/bradley John Wiley and Sons 201
Annual ercenage Raes: AR calculaed using he AR rae annually is he same as calculaed by he given mehod 1+ 0 i m m (1 AR) 0 + 0 e i www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR when ineres is compounded m imes annually 0 1 + i m m compounding m imes annually a a nominal rae of i % p.a. (1 AR) 0 + compounding once annually a AR% p.a. Bu is he same whichever mehod is used, hence 0( 1+ AR ) 0 1+ i m m Nex slide www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR when ineres is compounded m imes annually Bu is he same which-ever mehod is used, hence 0( 1+ AR ) 0 1+ www.wiley.com/college/bradley John Wiley and Sons 201 i m m i ( 1+ AR ) 1+ m m (1 + AR ) 1+ AR i m m i 1 + 1 m m
Calculae he AR when ineres is compounded coninuously Bu is he same which-ever mehod is used, hence 0 ( 1+ AR) 0 e i (1+ AR ) e i i (1+ AR ) e AR e i 1 www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR: rogress Exercises 5.4 no 11(a) is he same which-ever mehod is used, hence 0.06 5500(1 + AR ) 5500 1 + 2 2 0.06 (1 + AR ) 1 + 2 2 0.06 ( 1 + AR ) 1 + 2 2 AR 2 ( 1+ 0.0) 1 0. 0609 Correc o 4 decimal places www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR: rogress Exercises 5.4 no 11(a) is he same which-ever mehod is used, hence 5500(1 + AR ) 5500 1 + 0.06 2 2 Nominal ineres rae is 6% When ineres is compounded wice annually he AR is 6.09% AR 2 ( 1+ 0.0) 1 0. 0609 Correc o 4 decimal places www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500(1 + 0.06 + AR ) 5500 1 12 12 (1 + AR ) 1 + ( 1+ AR) + 0.06 12 ( 1 0.005) 12 12 AR 12 ( 1.005) 1 0. 0617 Correc o 4 decimal places www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500(1 + AR ) 5500 1 + 0.06 12 12 Nominal ineres rae is 6% When ineres is compounded welve imes ( 1+ AR annually ) ( 1he + 0.005 AR is ) 12 6.17% AR 12 ( 1.005) 1 0. 0617 Correc o 4 decimal places www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500(1 + 0.06 + AR ) 5500 1 65 65 0.06 (1 + AR ) 1 + 65 65 0.06 ( 1 + AR ) 1 + 65 65 AR 1.0618 1 0.0618 Correc o 4 decimal places www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500(1 + AR ) 5500 1 + 0.06 12 12 Nominal ineres rae is 6% When ineres is compounded daily ( 1+ AR) he AR + is 6.18% ( 1 0.005) 12 AR 1.0618 1 0.0618 Correc o 4 decimal places www.wiley.com/college/bradley John Wiley and Sons 201
Calculae he AR rogress Exercises 5.4 no 11(d) Bu is he same which-ever mehod is used, hence 0 (1 AR) + e 0 0.06 (1 + AR) e 0.06 ( 1+ AR ) e 0.06 AR e 0.06 1 0.0618 Nominal rae Correc o 4 decimal is 6% places AR is 6.18% www.wiley.com/college/bradley John Wiley and Sons 201