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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: +)$$ × 3)/)$4 = +))$
#/ 'fter two years: +))$ × 3)/)$4 = +)#)
,/ 'fter three years: +)#) × 3)/)$4 = +),,/)$
Calculating ( the short< way 3preferred4:
8 F# of a single amount invested today at r 4 for n periods is:
(n = >$3)?r4n
* he e7pression 3) ? r4n is the future value interest factor 3(I(4 for a singlesum/
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Single Sums$
Future Value of % Single Sum
(, = )$$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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+)$$ +),,/)$
Timeline
>$ (,
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Single Sums$
Future Value of % Single Sum
67ample: Future value of a single sum
hat will +)$$$ 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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Future Value of % Single Sum
(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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Present Value of % Single Sum
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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Present Value of % Single Sum
(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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Single Sums$ Pro&lem Variations
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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Single Sums$ Pro&lem Variations
67ample: Solving for the un'nown rate of return (r)
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/ $/$5B5 = r = 5/B59 p/a/
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Single Sums$ Pro&lem Variations
67ample: Solving for the un'nown rate of return (r)
If you sell land for +))B,, 3(4 that you bought fiveyears ago 3n4 for +-$$$ 3>4 what is your annual rate ofreturnF
!sing the same method as in the previous e7ample youwill find that the rate of return 3r4 is e.ual to )B9 p/a/
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Single Sums$ Pro&lem Variations
67ample: Solving for the un'nown rate of return (n)
Kuppose you placed +)$$ 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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Single Sums$ Pro&lem Variations
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*