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Evaluation of Investment

Proposals

1. Time Value of money

2. Net present value (NPV

3. !raphical representation of NPV

". Internal rate of return (I##

$. %n economic rationale for I## rule

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Time Value of &oney Defnition:% rationale human 'ein )oul* not value

the opportunity to receive a speci+c amount of moneyto*ay e,ually )ith the opportunity to have the same

amount at some future *ate. &ost human 'einsvalue the opportunity to receive money no) hiherthan receive one or t)o years from no) the sameamount. The a**itional amount that is re,uire* forreceivin after a certain time perio* in future than theamount receive* to*ay is -no)n as time value of

money. That is this a**itional amount is iven asvalue of time )aitin. %ctually the percentae chanein value of a certain amount of money for a certaintime perio* ap is -no)n as time value of money

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#ationale for Time Value of

&oneyTime value of money is eiste* for the

follo)in reasons: /uture uncertainty 0acri+ce present consumption or preference

for hiher consumption in future perio* %lternative investment opportunities i.e.

opportunity cost. 0acri+ce of cash hol*in preference Ination

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Terminoloies

Present value: The value of to*ay that iso'taine* 'y *iscountin a future cash o)

or a series of cash o)s 'y the opportunitycost of fun* as *iscount rate.

Future value: The amount or value )ill 'eo'taine* at a certain time point in future of

a cash o) or a series of cash o)s 'ycompoun*in at a iven interest rate oropportunity cost over a certain time perio*.

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Terminoloies

Discounting: The process of +n*in thepresent value of a cash o) or a series

of cash o)s 'y usin a iven *iscountrate.

Compounding:The arithmetic process of

*eterminin the +nal value of a casho) or a series of cash o)s 'y usin acertain interest rate

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Terminoloies

Simple interest rate: The interest ratethat chare* only on the principal

amount for a speci+c perio* is calle*simple interest rate.

Compound interest rate: The interest

rate that is chare* 'oth on principalan* interest amount perio* to perio* iscalle* compoun* interest rate.

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Terminoloies

Installment: Perio*ic payments or receipts relate*to any transaction or contract are -no)n asinstallment.

%nnuity: The e,ual amount of cash o) incurre* ate,ual time interval is calle* annuity.

%nnuity *ue: The annuity un*er )hich the casho) is incurre* at the 'einnin of each perio* is

calle* annuity *ue. %nnuity imme*iate: The annuity un*er )hich the

cash o) is incurre* at the en* of each perio* iscalle* annuity imme*iate.

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Terminoloies

Perpetuity: The annuity un*er )hich the cash o)is incurre* for a in+nite perio* of time is calle*Perpetuity..

Nominal interest rate: #ate of interest state* in anareement for transferrin fun* from one party toanother party is -no)n as nominal interest rate.

Eective interest rate: #ate of interest ultimately

pai* 'y the user of fun* to the supplier of fun* 'yta-in into consi*eration of timin fre,uencies an*other chares is -no)n as eective interest rate.

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0olvin for PV:

The arithmetic metho* Pro'lem: o) much shoul* you set

asi*e no) to et T-.144 after 3 years

from no)5

0olve the eneral /V e,uation for PV: PV 6 /Vn7 ( 1 8 i n

PV 6 /V37 ( 1 8 i 3

6 T-.144 7 ( 1.14 3

6 T-.9$.13

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/in*in the interest rate

an* time perio* Pro'lem: hat is the rate of interest 'y )hat

T-.144 'ecomes T-.244 in " years5

2446144(18i"

(18i"62; 18i62 17"62.2$61.1 per annum5

2446144(18.1$n

(1.1$n62; n lo(1.1$6lo(2

n6lo(27lo(1.1$6".=? years

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Compoun*in more than

once in year Pro'lem:@ou li-e to set asi*e an amount of

money so that you et T-.$4;444 after $years from no). Aan- Bne oers you 14>annual interest rate an* Aan- T)o oersyou =.$> interest rate compoun*e*monthly. here shoul* you put the money5

Aan- Bne: PV6$4;4447(1.1$6T-.314"?.49 Aan- T)o: PV6$4;4447

(18.4=$712?46T-.311$2."? Aan- Bne is a 'etter choice

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Classi+cations of interest

rates Eective (or e,uivalent annual rate (E%# 6

E//> the annual rate of interest actually'ein earne*; ta-in into account

compoun*in. E//> for 14> semiannual investment

E//> 6 ( 1 8 iNB&7 m mD 1

6 ( 1 8 4.14 7 2 2 1 6 14.2$>

%n investor )oul* 'e in*ierent 'et)een aninvestment oerin a 14.2$> annual returnan* one oerin a 14> annual return;compoun*e* semiannually.

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Eective %nnual #ateE//>6 ( 1 8 iNB&7 m mD 1

Pro'lem $: % Cre*it car* chares2> interest rate per month. hatis the eective interest rate5

E%#6(18.2"71212D1

6(1.4212D1

62?.

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hy is it important to consi*er

eective rates of return5 %n investment )ith monthly payments is *ierent

from one )ith ,uarterly payments. &ust put eachreturn on an E//> 'asis to compare rates of

return. &ust use E//> for comparisons. 0eefollo)in values of E//> rates at variouscompoun*in levels.

E%#%NN%F14.44>

E%#G%#TE#F@ 14.3

E%#&BNTF@ 14."9>

E%#H%IF@ (3?$ 14.$2>

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Present Value %nnuity %ll -in*s of consumers cre*it schemes follo)

present value annuity. % lump sum amount is

'orro)e* no) aainst )hat payments )oul*'e ma*e in e,ual installments at a reularinterval for a *e+nite perio* of time. /oreample; at 14> interest rate; you can 'orro)

T-.193.$$ in a 2 year annuity of T-.144installment. The amount of T-.193.$$ iscompose* of (the PV of /V1of T-.144 or

T-.=4.=1 an* (/V2of T-.144 or T-.

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PVI/%6

1D

1

(18in

i

/ormulae for Present Value

Interest /actor of %nnuity

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Present Value of %nnuity

Pro'lem:%t 14> interest rate; o) muchcan you 'orro) no) aainst the repayment

3 e,ual annual installments of T-.14445 PV %nnuity6CK(PVI/%

6CLM1D(17(18in7iO

61444LM1D(17(1.137.1O

61444K2."

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Pro'lem:@ou have a plan to *eposit T-.1;444per month in a 'an- for net 24 years. If theinterest rate is per annum then ho) much

can you 'orro) from the 'an- aainst that5

PVI/%6L1D17(18.4

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Pro'lem:/in* the amount of installment ofa loan of T-.$;444 to 'e repai* in " e,ual

monthly installment at 12> interest. &a-ean amortiation sche*ule.

$4446C(PVI/%; i6.12; m612; n6"

6C(3.=41=??

C6$44473.=41=??612

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%mortiation 0che*ule

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Pro'lem:@ou nee* T-.12 la-h no) to'uy a car; un*er the terms an*

con*ition of monthly installments for14 year. Interest rate is 1$> perannum. (a hat )oul* 'e the

amount of installments5 (' o)much )oul* 'e the accumulate*lia'ility of interest5

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0olution:

(a Installment 6PV %nnuity7PVI/%

612;44;4447?1.=

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Pro'lem: In 1==2; a ?4 year ol* nurse'ouht a Q12 *ollar lottery tic-et an*

)on the 'iest Rac-pot to that *ate ofQ=.3 million. Fater it turne* up that she)oul* 'e pai* in 24 annual installments

of Q"?$;444 each. If the interest rate)as ; then )hat )as the amount she)as *eprive* of in present value5

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%ns)er to previouspro'lem

PV 6 Q"?$;444KPVI/% i6.4

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/uture Value of %nnuity

/VI/%6M(18inD17i /V of %nnuity6CK/VI/% 0uppose; there is a 2 year annuity of

Q144 installments at 14> interest. Thefuture value is

/V %nnuity6 CK/VI/%66144KM(1.12D174.16Q214

This is compose* of Q114 an* Q144.

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Pro'lem:@ou li-e to *eposit T-.1444per month for a perio* of 1$ years.

%ssumin an interest of 14> ho) much)oul* you et at the en*5

/V %nnuity6CK(/VI/%

61444KLM(18.17121$K12

D1O7(.171261444K"1"."943

6T-.";1";"94.34

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Pro$lem:@ou nee* to have T-.1 millionafter 24 years from no). %ssumin the

mar-et interest rate of 13> per annum ifyou li-e to *eposit e,ual ,uarterlyinstallments *urin the perio* in a 'an-then ho) much )oul* 'e the amount of

each installment5 hat is the interestaccumulation in the annuity5

!iven; /V6T-.1;444;444; i6.137";

n624K"; C65

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0olution:

C6/V7/VI/%.C61;444;44473??.91?"6T-.2;92?.=4

Interest accumulation6/V %nnuityDTotalpayments

61;444;444D(CKn61;444;444D

(292?.=4K

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%nnuity Hue

Problem:You need to receive Tk.10,000monthly for a period of 2 years to pursue yourM! pro"ram. You make an arran"ement #itha ank that says the interest rate is 1$%.

&a' (o# much #ill you have to return back tothe bank at the end)

&b' (o# much should you deposit to the bankno# to "et the same monthly installmentsthrou"hout the M! pro"ram)

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0olution:

&a' *+ !nnuity-*+/*!

10000&1.1$12'23415&.1$12'

1000026.67707Tk.2,66,770.70

8ince you need the money at the be"innin" of

the month so it is an annuity due.

/n that case,*+ !nnuity

9ue2,66,770.70&1.1$12'Tk.2,71,$3.30

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0olution:

(' This is the present value annuity *ue. PV %nnuity *ue6CKPVI/%K(18i

614;444K24.?2"23K(18.1$712 62;4

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He+nition

1. NPV: NPV is *e+ne* as the summation of thepresent values of cash o)s after ta in eachyear over the proRect or investment perio*

minus the summation of present values of netcash outo)s in each year *urin that perio*.

2% &P' Profle: ()eraphical presentation ofrelationship 'et)een a proRects an* the +rms

cost of capital or *iscount rate is calle* NPVpro+le. % raph that plots a proRects NPVaainst the *iscount rates is *e+ne* as theproRects NPV pro+le.

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He+nition

3. IRR: The discount rate that makes equal the present valueof a projects future cash inflows to the present value ofits total costs. Equivalently the rate that forces the net

present value to equal ero is internal rate of return.

!. "odified IRR: The discount rate at which the present

value of a projects cost is equal to the present value ofits terminal value# where the terminal value is found asthe sum of the future values of cash inflows# compoundat the firms cost of capital.

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He+nition

$. %et Terminal &alue: 'hen terminal amount of anyproject o(tained from reinvestment is discounted intopresent value at a certain required rate of return andthe present value of cash outlays is deducted from thefirst# then the net result is known as net terminal value.

). *rofita(ility Inde+:It measures the present value ofreturns per unit of investment is calle*pro+ta'ility in*e. It is *e+ne* as the ratio that iso'taine* *ivi*in the present value of future cashino)s 'y the present values of cash outo)s.

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Eample

,n asset can (e purchased for Tk.-#$# that will

provide net (enefits Tk./##0 Tk.3##0

Tk.1#/ 2 Tk.1 in years /# 1# 3 2 !respectively. Reinvestment rate is 4 and cost of

capital is /.$4.'ould it (e wise to purchase the

asset under the followin5 techniques67i8 %*& 7ii8 IRR 7iii8 *I 7iv8 "IRR

7v8 %T&

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%n Economic #ationale for NPV

#ule/. It reco5nies time value of money.

1. It considers all cash flows occurrin5 overthe entire life of the project to calculate its

rate of return.

3. It is consistent with the shareholderswealth ma+imiation 5oal.

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%n Economic #ationale for NPV #ule

!. It does not use the concept of required rateof return.

$. It is easily understanda(le to the (usinesse+ecutives and non9technical people.

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%n Economic #ationale for I##

#uleIf the internal rate of return e+ceeds the cost

of capital of the fund used to finance a

project# a surplus remains after payin5 forthe capital and this surplus accrues to thefirms stockholders. Therefore# takin5 on a

project whose IRR e+ceeds its cost ofcapital increases the value of the firmsstock.

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Tra*itional measures of investment)orth

Pa*$ac+ Period ,et)od: Pay'ac- metho* meansho) many years )ill it ta-e for the cash 'ene+tsto pay the oriinal cost of an investment;

normally *isrear*in salvae value. Itmeasures the num'er of years re,uire* for thecash o) after ta to pay'ac- the oriinaloutlay re,uire* in an investment proposal. If the

proRect enerates constant annual cash o)s;the pay'ac- perio* can 'e compute* 'y*ivi*in cash outlay 'y annual cash ino).

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Discounted Pa*$ac+ Period: The lenthof time re,uire* for an investments

*iscounte* cash o)s to e,ual its initialcost is -no)n as *iscounte* pay'ac-perio*. Aase* on *iscounte* pay'ac-rule; an investment is accepta'le if its

*iscounte* pay'ac- is less than someprespeci+e* num'er of years.

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Tra*itional measures of investment )orth.ample o/ pa*$ac+ and discounted pa*$ac+:

@ear

Cash o)s Cum cash o)s Initialcost is

T-.344

444cost ofcapitalis

12.$>

Pay'ac-63years

Hiscounte*pay'ac-6"years

(appr

n*isc Hisc n*isc Hisc

1 144444

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0verage0ccounting ate o/ eturn: Theaccountin rate of return also -no)n as

the return on investment; uses accountininformation as reveale* 'y +nancialstatement to measure the pro+ta'ility ofan investment. It is foun* 'y *ivi*in the

averae after ta pro+t 'y the averaeinvestment. The averae investment)oul* 'e e,ual to half of the oriinalinvestment if it is *epreciate* constantly.

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.ample o/ 0verage0ccounting ateo/ eturn: #e,uire* investment is

T-.$44444 an* proRect life epectancyis $ years. Epecte* net income afterDta are T-.144444; T-.1$4444;

T-.$4444; T-.4 U (T-.$4444 in net $

years. %verae net income is T-.$4444an* averae investment7'oo- value is

T-.2$4444. 0o averae rate of return is

24>.

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ome)or-

uestions pro$lems:

3%5 3%6 3%12 3%14

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