Essential Mathematics for Economics and Business, 4 th Edition CHAPTER 5 : FINANCIAL MATHS.

Transcription

1 Essenial Mahemaics for Economics and Business, 4 h Ediion CHATER 5 : FINANCIAL MATHS. John Wiley and Sons John Wiley and Sons 201

2 Compound ineres: formula Compound ineres: Calculaions resen values Annual percenage raes John Wiley and Sons 201

3 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 John Wiley and Sons 201

4 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 i 0 0 (1+ i) 1 Year 2 1 i i 1 1 (1+ i) 2 Year 2 i i 2 2 (1+ i) In general, a he end of year. Year -1 i i -1-1 (1+ i) Copyrigh 201 Teresa Bradley and John Wiley & Sons L coninued..

5 Compound ineres formula coninued.. principal + ineres BUT, wrie in erms of 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) John Wiley and Sons 201

6 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 i Calculaions + 0( 1 i) 1000 ( ) 1000(1.08) 1000( )

7 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 John Wiley and Sons 201

8 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) John Wiley and Sons 201

9 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 i Calculaions 0 ( 1 i) ( ) 5000(1.08) 5000( )

10 Revise erminology. presen value and fuure value In he compound ineres formula fuure value 0 (1+ i) presen value In Worked Example is called he fuure value of a he end of years when invesed a 8% ineres compounded annually is called he presen value of when discouned a 8% annually for years John Wiley and Sons 201

11 Worked Example 5.6(b)(i) resen value calculaions ( 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 i (1 + i) Calculaions 0 ( 1+ i) ( ) (1.08) 5000

12 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 (1 + i) Calculaions 0 ( 1+ i) ( ) (1.08)

13 Compound ineres compound wice annually (rae i % pa) (1 i) 0 +..compounding once annually 2 i 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 John Wiley and Sons 201

14 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 John Wiley and Sons 201

15 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 John Wiley and Sons 201

16 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 e i 1 + i m m e i as m John Wiley and Sons 201

17 Worked Example 5.8 (a) 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 i i m Calculaions 0 m m i m ( ) (1.04) 5000( )

18 Worked Example 5.8 (c)(i) 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 i i m Calculaions 0 m i m ( )

19 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 i i m m Calculaions m i m ( ) ( )

20 Worked Example 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 i Calculaions you will need he formula... i 0 e i 0 e 5000e e ( )

21 How much do you gain when ineres is compounded more han once annually? Review resuls in Worked Examples 5.6, 5.7 and 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: compounded once annually compounded wice annually compounded monhly compounded daily compounded coninuously John Wiley and Sons 201

22 How much do you gain when ineres is compounded more han once annually? Review resuls in Worked Examples 5.6, 5.7 and is invesed a a nominal ineres rae of 8% for hree years bu compounded a various inervals annually one conversion period conversion periods conversion periods conversion periods infinie conversion periods (coninuous) John Wiley and Sons 201

23 How much do you gain by compounding more han once annually? Conversion periods/year Amoun a end of years Difference over annual compounding Infiniely many (coninuous) Copyrigh 2008 Teresa Bradley and John Wiley & Sons L

24 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.. John Wiley and Sons 201

25 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) e i John Wiley and Sons 201

26 Calculae he AR when ineres is compounded m imes annually 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 John Wiley and Sons 201

27 Calculae he AR when ineres is compounded m imes annually Bu is he same which-ever mehod is used, hence 0( 1+ AR ) John Wiley and Sons 201 i m m i ( 1+ AR ) 1+ m m (1 + AR ) 1+ AR i m m i m m

28 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 John Wiley and Sons 201

29 Calculae he AR: rogress Exercises 5.4 no 11(a) is he same which-ever mehod is used, hence (1 + AR ) (1 + AR ) ( 1 + AR ) AR 2 ( ) Correc o 4 decimal places John Wiley and Sons 201

30 Calculae he AR: rogress Exercises 5.4 no 11(a) is he same which-ever mehod is used, hence 5500(1 + AR ) Nominal ineres rae is 6% When ineres is compounded wice annually he AR is 6.09% AR 2 ( ) Correc o 4 decimal places John Wiley and Sons 201

31 Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500( AR ) (1 + AR ) 1 + ( 1+ AR) ( ) AR 12 ( 1.005) Correc o 4 decimal places John Wiley and Sons 201

32 Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500(1 + AR ) Nominal ineres rae is 6% When ineres is compounded welve imes ( 1+ AR annually ) ( 1he AR is ) % AR 12 ( 1.005) Correc o 4 decimal places John Wiley and Sons 201

33 Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500( AR ) (1 + AR ) ( 1 + AR ) AR Correc o 4 decimal places John Wiley and Sons 201

34 Calculae he AR: rogress Exercises 5.4 no 11(b) is he same which-ever mehod is used, hence 5500(1 + AR ) Nominal ineres rae is 6% When ineres is compounded daily ( 1+ AR) he AR + is 6.18% ( ) 12 AR Correc o 4 decimal places John Wiley and Sons 201

35 Calculae he AR rogress Exercises 5.4 no 11(d) Bu is he same which-ever mehod is used, hence 0 (1 AR) + e (1 + AR) e 0.06 ( 1+ AR ) e 0.06 AR e Nominal rae Correc o 4 decimal is 6% places AR is 6.18% John Wiley and Sons 201