BF Lecture Notes Topic 2 Part 1(1).ppt

8/19/2019 BF Lecture Notes Topic 2 Part 1(1).ppt

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Topic 2

Financial Mathematics/Time Value of MoneyPart 1


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Overview

In this lecture we will:

Discuss the time value of money concept;

Learn about simple interest;

Learn about compounding and discounting;

Learn about compound interest;

Calculate the present value and future value of a singleamount for both one period and multiple periods;

Calculate the present value and future value of multiplecash flows; &

Calculate the present value and future value of annuities;

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Time Value of Money

* Receiving +) today is worth more than +) in the

future* he opportunity cost of +) in the future is theinterest we could have earned on +) if receivedearlier

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Today Future


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Time Value Terminology

For a single sum time value problem there are four variables that have to be

taken into account:

n - The number of interest paying time periods between a present value and

a future value;

R - The rate of interest for discounting or compounding;

Note - n and r need to be consistent - if interest (r is paid monthly the

number of periods n has to be worked out in terms of months (we will seee!ample of this later on which will help to e!plain it;

"#$ % "resent value % the price&value of the asset&investment now (at time

period 'ero (T$

F#n % Future value % the price&value of the asset&investment at some futurespecified time (Tn

ll single sum time value )uestions involve four values: "#* F#* r and n -

given three of the values it is always possible to calculate the unknown

fourth value+


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Future Sum ith Simple !nterest

,f the ank pays you simple interest on a deposit the interest payment each periodwill be the same and will be the interest rate times the initial amount+

.imple interest refers to interest earned only on the original capital investment

amount+

The formula for the future value of a single sum calculated with simple interest is:

F#n / "#(0 1 (r ! n

2!ample: 30$$ invested at 0$4 p+a+ simple interest for three years

0+F#5 / 30$$(01($+0$ ! 5

6+F#5 / 30$$(0+5$

5+F#5 / 305$+$$

Therefore* interest earned / 35$+$$

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"ompounding and #iscounting

"ompounding ranslating +) today into its e.uivalent future value/

#iscountingranslating a future +) into its e.uivalent present value today/

imeline

$ 0 6 5 7T$ T0 T6 T5 T7

"#$ F#7

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"ompound !nterest

If the ban1 pays you compound interest you

will receive interest payments not 2ust on theinitial amount but also on previous interestpayments/

Compound interest refers to interest earnedon both the initial capital investment and onthe interest reinvested from prior periods 3i/e/

earning interest on interest4/In finance compound interest is usually used/

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Single Sums$

Future Value of % Single Sum

67ample: Future value of a single sum

8ou invest +)$$ in a savings account that earns )$9 p/a/ interest3compounded4 for three years/

Calculating ( the long< way:

)/ 'fter one year: +)$$ × 3)/)$4 = +))$

#/ 'fter two years: +))$ × 3)/)$4 = +)#)

,/ 'fter three years: +)#) × 3)/)$4 = +),,/)$

Calculating ( the short< way 3preferred4:

8 F# of a single amount invested today at r 4 for n periods is:

(n = >$3)?r4n

* he e7pression 3) ? r4n is the future value interest factor 3(I(4 for a singlesum/

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Single Sums$


(, = )$$3)/)$4,

)/(, = )$$3)/,,)4

#/(, = +),,/)$ $ ) # ,

Interest earned = +,,/)$

@otice:

* Interest earned with compounding +,,/)$;

* Interest earned with simple interest +,$/$$;

Difference +,/)$ A due to compounding

3i/e/ interest on interest4/

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+)$$ +),,/)$

Timeline

>$ (,


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Single Sums$


67ample: Future value of a single sum

hat will +)$$$ amount to in five yearsE time if interest is )#9 p/a/ compounded annually F

n = - 3interest is calculated - times4 r = $/)#:

)/ (- = +)$$$3)/)#4-

#/ (- = +)$$$3)/50#,4

,/ (- = +)50#/,$

@ow assume interest is )#9 per annum compounded monthly /

'lways remember that n is the number of compounding periods not the number of years/

* n = -yrs 7 )# months per year = 0$ 3i/e/ interest is calculated 0$ times4/

* r = $/)# p/a/G)# months per year = $/$) 3i/e/ interest rate is )9 per month4/

* (0$ = +)$$$3)/$)40$

* (0$

= +)$$$3)/%)054

* (0$ = +)%)0/5$

Difference 3+)%)0/5$ A +)50#/,$ = +-H/H$4 is due to compounding more often over entireinvestment period 3i/e/ 0$ times )9 v/ - times )#94/

(uture values also depend critically on the assumed interest rate A the higher the interest rate thegreater the future value/

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(or a given number of periods the higher the interest rate the higher thefuture value.

(or a given interest rate the more compounding periods the greater thefuture value.

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Present Value of % Single Sum

67ample: Present value of a single sum

If you will receive +)$$$ in three yearsE time what is its > if your opportunity

costGdiscount rateGinterest rate is )$9 p/a/F

Calculating > the long< way:

8r ,: +)$$$ 3)/)$4A) = +B$B/$B

8r #: +B$B/$B 3)/)$4A) = +%#0/H-

8r ): +%#0/H- 3)/)$4A)

= +5-)/,#

Calculating ( the short< way:

8 "# of a single future amount

discounted back to today at r 4 for n periods is:

>$ = (n3)?r4An

>$ = +)$$$3)/)$4A,

>$ = +)$$$3$/5-),4

>$ = +5-)/,$

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Timeline

$ ) # ,

>$ (,

+5-)/,# +%#0/H- +B$B/$B +)$$$/$$


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8our rich grandmother promises to give you

+)$$$$ in )$ yearsE time/ If interest rates are)#9 per annum how much is this gift worthtodayF

>$ = (n3)?r4

An

>$ = +)$$$$3)/)#4A)$

>$ = +)$$$$3$/,##$4A,

>$ = +,##$/$$

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(or a given number of periods the higher the interest rate the lower the present value.

(or a given interest rate the greater the number of discounting periodsthe lower the present value.

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Single Sums$ Pro&lem Variations

In general the problems that students will confront in this

course will either involve wor1ing out present values orfuture values/

Jowever it is of course possible to also want to wor1 outn if given > ( and r/ It is .uite a common problem to

want to 1now how long it will ta1e an investment to growfrom its > to its ( at a given interest rate/

It is also possible to wor1 our r given n > and (/ It is.uite a common problem to want to 1now what the rate of

return on an asset is when it grows from > to ( over agiven period of time/

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67ample: Solving for the un'nown rate of return (r)

8ou currently have +)$$ available for investment for a #)year period/ 't what annual interest rate must you investthis amount in order for it to be worth +-$$ at maturityF

Remember given any three factors in the present value orfuture value of a single sum formula the fourth factor canbe solved/

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Kince we 1now both the > and ( 3and n4 we can use either the > or the( of a single sum formula to find the un1nown interest rate 3r4/

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* %* PV of a single sum

* >$ = (n3)?r4An

* )/ )$$ = -$$3)?r4A#)

* #/ )$$G-$$ = 3)?r4A#)

* ,/ $/#$ = 3)?r4A#)

* H/ 3$/#$4) = 3)?r4A#)

* -/ 3$/#$4)GA#) = 3)?r4A#)GA#)

* 0/ 3$/#$4A$/$H50# = )?r

* 5/ )/$5B5 = )?r

* %/ )/$5B5A) = )?rA)

* B/ $/$5B5 = r = 5/B59 p/a/

* +* FV of a single sum

* (n = >$3)?r4n

* )/ -$$ = )$$3)?r4#)

* #/ -$$G)$$ = 3)?r4#)

* ,/ - = 3)?r4#)

* H/ 3-4) = 3)?r4#)

* -/ 3-4)G#) = 3)?r4#)G#)

* 0/ 3-4$/$H50# = )?r

* 5/ )/$5B5 = )?r

* %/ )/$5B5A) = )?rA)

* B/ $/$5B5 = r = 5/B59 p/a/


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If you sell land for +))B,, 3(4 that you bought fiveyears ago 3n4 for +-$$$ 3>4 what is your annual rate ofreturnF

!sing the same method as in the previous e7ample youwill find that the rate of return 3r4 is e.ual to )B9 p/a/

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67ample: Solving for the un'nown rate of return (n)

Kuppose you placed +)$$ in an account that pays interest of B/09 p/a/compounded monthly/ Jow long will it ta1e for your account to grow to +-$$F

note: r = $/$B0G)# = $/$$% 3i/e/ $/%9 per month4

Kince we 1now both the > and ( 3and r4 we can use either the > or the( of a single sum formula to find the un1nown number of investment periods3n4/ o get the answer we must use natural logs 3the ln button on your

calculator4/

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* %* PV of a single sum

* >$ = (n3)?r4An

* )/ )$$ = -$$3)/$$%4An

* #/ )$$G-$$ = 3)/$$%4An

* ,/ $/#$ = 3)/$$%4An

* H/ ln3$/#$4 = Anln3)/$$%4

* -/ A)/0$BH = An3$/$$5B0%4

* 0/ A)/0$BHG$/$$5B0% = An

* 5/ A#$# = An = #$# months

* +* FV of a single sum

* (n = >$3)?r4n

* )/ -$$ = )$$3)/$$%4n

* #/ -$$G)$$ = 3)/$$%4n

* ,/ - = 3)/$$%4n

* H/ ln3-4 = nln3)/$$%4

* -/ )/0$BH = n3$/$$5B0%4

* 0/ )/0$BHG$/$$5B0% = n

* 5/ #$# = n = #$# months


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Jint (or Kingle Kum >roblems

here are only H variables: FV PV r and n.

8ou will always be given three variables andas1ed to solve for the fourth/

his hint ma1es solving single sum timeAvalueproblems much easier/

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Multiple ,neven "ash-Flows

67ample: Future Value of Multiple ,neven "ash-Flows

8ou deposit +)$$$ now +)-$$ in one year +#$$$ in two years and +#-$$ in three years in an

account paying interest of )$9 p/a/ Jow much will you have in the account at the end of the thirdyearF

's each of the cashAflows is of a different value you must first calculate the future value of eachcash flow individually as a single sum and then total the future values/

(n = >$3)?r4n

+)$$$3)/)$4, = +)$$$3)/,,)4 = +) ,,)

+)-$$3)/)$4# = +)-$$3)/#)4 = +) %)-

+#$$$3)/)$4) = +#$$$3)/)$4 = +# #$$

+# -$$3)/$$4 = = +# -$$ $ ) # ,

otal = +5 %H0 +)$$$ +)-$$ +#$$$ +#-$$

+),,)

+)%)-

+##$$

+#-$$

+5%H0

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Timeline


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Multiple ,neven "ash-Flows

67ample: Present Value of Multiple ,neven "ash-Flows

8ou deposit +)-$$ in one year +#$$$ in two years and +#-$$ in three years in an account paying

interest of )$9 p/a/ hat is the present value of these cash flowsF 's each of the cashAflows is of a different value you must first calculate the present value of each

cash-flow individually as a single sum and then total the present values/

>$ = (n3)?r4An

+)-$$3)/)$4A) = +)-$$3$/B$B)4 = +),0H

+#$$$3)/)$4A# = +#$$$3$/%#0H4 = +)0-,

+#-$$3)/)$4A, = +#-$$3$/5-),4 = +) %5%

otal = +H %B- $ ) # ,

+)-$$ +#$$$ +#-$$

+),0H

+)0-,

+)%5%

+H%B-

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Timeline


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%nnuities

hat is an 'nnuityF ' series of constantGfi7ed cashAflows

3payments or receipts4 ocurring at regular intervals e/g/ asuperannuationGpension payment/

ypes of 'nnuities:

rdinary annuity;

'nnuity due;

Deferred annuity;

>erpetuity; &

rowing perpetuity/

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%nnuities

Ordinary annuity A ' series of constant cashAflows occurring at the end of each period forsome fi7ed number of periods and commencingat the end of the first period (i.e. commencing

at T 1).

Timeline

$ ) # ,

* +)$$ +)$$ +)$$

* 67amples include mortgage repayments 3payment

annuity4 and superannuationGpension payments3receipt annuity4/

%nnuity due A ' series of constant cashAflowsoccurring at the start of each period for somefi7ed number of periods and commencing at thebeginning of the first period (i.e. commencing

at T 0 ).

Timeline

$ ) # ,

+)$$ +)$$ +)$$

* 67amples include paying rent or uni/ fees inadvance/

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%nnuities

#eferred annuity A ' series of constant cashAflows occurring at the end of each period forsome fi7ed number of periods and commencing

some future period after period one (e.gcommencing at T 3 (the end of the third period)).

Timeline

$ ) # , H -

+)$$ +)$$ +)$$

* 67amples include a lump sum pension plan/

* Perpetuity - ' series of constant cashAflowsoccurring at the end of each period indefinitely(i.e. forever).

Timeline

$ ) # , /////NNN O

+)$$ +)$$ +)$$ +)$$

* 67amples include a scholarship fund availableeach year forever 3e/g/ Rhodes Kcholarship4/

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%nnuities

(uture value of an ordinary annuity:

* he compounding term 3s.aure brac1eted term4 is calledthe future value interest factor of the annuity 3(I('4/

he formula gives the F at the time the last payment!receipt is made.

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( )

+=

r

0-r 0 "9TF#

n

n


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%nnuities

67ample: Future Value of %n Ordinary %nnuity

If you invest +)$$$ at the end of each of the ne7t , years at %9 p/a/ howmuch will you have after , yearsF

Timeline

$ ) # , +)$$$ +)$$$ +)$$$

(, = +,#H0/H$

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( )

[ ]

7$+67:*53

67:7+5$$$*03

$;+$

0$;+0($$$*03

r

0-r 0 "9TF#

5

5

5

5

n

n

=

=

−=

+=

FV

FV

FV


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%nnuities

>resent value of an ordinary annuity:

* >M = the annuity payment

* he discounting term 3value in the big s.uare brac1et4 iscalled the present value interest factor of the annuity 3>I('4/

@ote the formula always assumes that it is an ordinaryannuity and it provides the " one period before thefirst payment or receipt ta#es place$ i.e. it provides "at T 0 .

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( )

+−=

−

r

r PMT PV

n

00$


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%nnuities

67ample: Present Value of %n Ordinary %nnuity

hat is the > of receiving +)$$$ at the end of each of the ne7t , years if theopportunity cost is %9 p/a/F

Timeline

$ ) # , +)$$$ +)$$$ +)$$$ >$ = +#-55/)$

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( )

( )

[ ]

0$+


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%nnuities

(inding an un1nown >M

* In the previous problems we were given:n the number of investment periods;

r the discountG interest rate per investment period; &

>M the regular periodic annuity paymentGreceipt

and as1ed to calculate the > of the ordinary annuity/

Jowever it is common to want to 1now >M if given n r and >/

his is particularly so in instances of trying to wor1 out the regularperiodic payments on a loan/

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%nnuities

(inding an un1nown >M

* !sing the previous numerical e7ample:

>$= +#-55/)$ r = %9 p/a/ n = , years PMT .

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[ ]

$$$*03


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Future Values 0 Present Values$

Single Sums Multiple ,neven "ash-Flows 0

%nnuities

)/ Draw a timeline

#/ Determine what un1nown the problem involves:

r$ n$ "$ F$ "%T&

,/ Identify the class of problem:

single sum multiple uneven cash-flow annuity

H/ Recognise any PtrapsE in the problem:

%nnual interest rate and more than one

compounding period per year %d3ust r and n*

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BF Lecture Notes Topic 2 Part 1(1).ppt