Financial Management 2E: Rajiv Srivastava - Dr. Anil Misra Solutions To Numerical Problems

 

Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT 2e Rajiv Srivastava - Dr. Anil Misra

4-1: Rela elationsh tionship ip betw be twee een n Effec tive and Nominal Nominal Rates of Interest Fina nc ie r Ltd (a n NBF NBFC) is is o ffe ring p e rsona l lo a n withi within n 24 hou rs o f a p p lic a ti tio o n, witho witho ut a ny d oc um e nta ti tion on . The inte re st ra te c ha rg e d b y the the Fina nc ie r Ltd is is 25% wi with th mo nthl nthly y c o m p o u n d in in g . Th Th e b o rro w e r, An An a n d h a s a n o t h e r o p t io io n , t o b o rr rro w t h e re q u ir ire d m o n e y fr fro m Ba nk nko o ( a na ti tiona ona l b a nk) a t 27% 27% p .a with with a nnua l c om p ound ing . The b or rowe rowe r fe e ls tha t the op ti tion on of b orowi orowing ng from NB NBF FC is not o nl nly y c on ve ni niee nt b ut is is a lso c he a p e r. Is Is the first op ti tion on a c t ua u a lly c h e a p e r?

Solution: The inte re st ra te g ive n iin n the first op tion tion is tthe he no mi mina na l ra te whi whille in the sec on d op ti tion on si sinc nc e it is is a c a se of an nua l c om p oun d ing the inte re st of offfe re d r a te iiss the e ffe c ti tive ve r ate . To c om p a re the two o p ti tions ons the e ffe c ti tive ve iinte nte re st ra ra te for tthe he first op ti tion on n e e d s to b e computed. Fo r the fir first op ti tio o n:  No m in a l iin n te re st ra te (i) = 0.2 5 o r 25%  Nu m b e r o f c o m p o u n d ing p e rio rio d s p e r ye a r (n ) = 12 ⎡ i ⎤ Effe c ti tive ve inte re st ra ra te (r) (r) =   − 1 ⎢1 + n

⎢ ⎣

r=

⎡ ⎢1 + ⎢⎣

. 25 12

⎤ ⎥  ⎦

n

⎥ ⎦

12

−

r=

1

o r 

0 .2 8 0 7 28.07%

 No , t h e b o rro rro w e r iiss n o t c o rre rre c t a s th e e ffe c tive ra te o f iin n te re st fo r th e fi first rst o p tio n is m o re t h a n t h a t ffo o r t h e s e c o n d o p t iio on. He nc e b or rrowi owing ng from Ba nko @ 27% 27% p .a is a c he a pe r op ti tion on tha n b or rrowi owing ng from Fina nc ie r Ltd (in (in w hic hic h e ffe c ti tive ve ra te is 28.07% 28.07%). ).

4-2: Future Value of a Single Sum Alp ha Ltd h a s a surp Alp surp lus of Rs. 20 la la c ks a va ila b le for the ne xt 5 ye ye a rs. At the e nd of the p e riod the firm wi willl b e ne e d ing the a mo unt for inve inve st stiing in its its b usi usine ne ss e xp a nsi nsion on . If If the firm d e c id e d to inve inve st the the sur surpl plus us in a sc he me tha t off offe rs a 13% p e r a nnum , how muc h would would the a mo unt g r o w to to a t th th e e n d o f 5 ye ye a rs?

Solution: Future Va lue (FV (FV) = Pr Pree se nt Va lue (PV (PV)) x F Future uture Va lue Inte re st F Faa c tor (F (FV VIF n

FV = (1+r) Wher e; r = ra t e o r re t u rn n = t im e p e rio d o f in v e st m e n t ( y e a rs) P re se n t Va lu e ( P V) = Futu re Va lue (F (FV) V)   =

 5

13% 5 Rs 2 0 . 0 0 la kh s Rs 36.85 36.85 la khs

 

Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT 2e Rajiv Srivastava - Dr. Anil Misra

4-3: Future Value of Annuity An inve inve st stor or is is thinki thinking ng of inve inve st stiing in a re c ur urrring d e p osi ositt sc sc he me tha t offe offe rs a n inte inte re st ra ra te of 12% p e r a nnum . The inve stme nt tha t he is pla pla nni nning ng is for for the hi high gh e r e d uc a ti tion on of his his son son who is just just two ye a rs now. In In c a se th e inve st stor or de c id e s to inve inve st Rs. 20, 20,000 000 eve ry ye ye a r, wha t wil will b e t h e t o t a l m o n e y a v a ila b le le t o h im im w h e n h is is so n r e a c h e s th th e a g e o f 20 20 ye ye a rs?

Solution: Fut ure  V a lu e  A nnu it y Fa c t o r  FVA(r, n ) =

Inte re st ra te off offee re d (r ) =  No . o f ye a rs (n ) = Am o u n t in v e sstt e d (y e a rly ) =

( 1+ r)n

−

1

r 

12% 18 Rs 2 0 , 0 0 0

So futur futuree va lue of a nnu ity a t the e nd of 18 y yee a rs @ 12% p .a wi willl b e Fu t u re v a lu e An n u it y Fa c t o r = 5 5 .7 4 9 7 Futur uturee Va lue = Amo unt o f a nnuit nnuity yxF FV V Fu t u re Va lu e = 2 0 0 0 0 x 5 5 .7 4 9 7 = Rs 1 1 , 1 4 , 9 9 4

4-4: Determining the Implied Rate of Interest Two inve st stme me nt o p ti tion on s a re a va ila b le to Pra Pra te e k. The first op ti tion on p rom ises to p a y Rs. Rs. 75, 75,000 000 a t the e nd of 8 ye ye a rs ifif the a n inve inve st stor or de p osi osits ts Rs. 7,0 7,000 00 a nnu a lly for for 8 ye a rs. The sec on d sc he me p rom ise s a lump lump sum o f Rs. 50, 50,000 000 a t the e nd of fi five yea rs on a n a nnua l d e p osi ositt of Rs Rs.   7,000 7, 000 for 5 yea rs. Pr Pra te e k is c onf ound e d a s to whic whic h of the the two sche me s gi give ve him him b e tter re tur turns? ns? Ple Ple a se sug ge st.

Solution: Sch chem eme eA Am o u n t t o b e in v e st e d a n n u a lly Tim e p e rio d ( y e a rs) Lu m p ssu u m re c e iiv ve d a t th e e n d

Rs 7 , 0 0 0 8 Rs 7 5 , 0 0 0

Sch chem eme eB Rs 7 , 0 0 0 5 Rs 5 0 , 0 0 0

Dec ision c a n b e ta ke n b a sed on the inte re st ra ra te s imp lie d iin n the o ffe r. The imp lie d iinte nte re st rraa tes for for the two sche me s c a n be c om p ute d a s fol olllows:

Sch chem eme eA Future V Vaa lue IInte nte re st Fa c to r fo r Annu ity (F (FV VIFA)

Sch chem eme eB

75000/7000 50000/7000 = 1 0 .7 1 7 .1 4

 No w loo kin kin g a t t h e FVI FVIFA ta b le to lo c a te th e FVIF VIFA o f 10 10.71 .71 a g a in st 8 y e a rs w wee c a n fi fin nd ou t the ra te of rree tur turn. n. R Ree tur turn n ffor or op tion tion A lliie s b e twe e n 8 a nd 9% 9%..  No w loo kin kin g a t t h e FVI FVIFA ta b le to lo c a te th e FVIF VIFA o f 7.1 4 a g a in st 5 ye a rs w wee c a n fi fin nd ou t the ra te of rree tur turn. n. R Ree tur turn n for op tion tion B is a rou nd 18% 18%.. He nc e op ti tion on B is be tter ttha ha n op ti tion on A.

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Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT  Rajiv Srivastava - Dr. Anil Misra

4-5: Present Value of a Single Sum Assumi As suming ng a d isco unt ra ra te of 12% find ou t whic whic h o ne of the foll ollow ow ing g ive s the the hi hig g he st re tur turns: ns: a . Rs 1, 1,60, 60,000 000 a va ila b le tod a y  b . Rs 1,7 5,0 00 to b e re c e ive d a fte r 8 ye y e a rs c . Rs Rs 25,0 25,000 00 p.a . in in p e rp e tui tuity ty d . Rs Rs 10, 10,00 000 0 pe r mo nth for for a yea r a nd Rs 1,0 1,00, 0,00 000 0 a t the e nd of the the yea r   e . Rs Rs 25, 25,000 000 pe r ye a r for the ne xt 10 ye ye a rs.

Solution: Sin c e t h e t h e o p t iio o n s h a v e v a rie d m a t u rriit y t h e y a re in c o m p a ra b llee . To m a ke t h e m c o m p a ra b le le w e n e e d t o ffiin d o u t tth h e p re se n t v vaa lu e o f a ll t h e a b o v e o p t io io n s . a ) Sinc e it iiss a va ila b le tod a y, thi thiss iiss tthe he p re sent va lue on ly ii.e. .e. R Rss 1,60,000 i  iss tthe he p re se nt v a lu lu e .  b ) P VIF =

⎛  ⎜

 ⎞ ⎟

1 1

+

n

o r PVIF a t 1 2 %=

 

0 .4 0 3 9

r 

 ⎠ ⎝  Pre s e n t V Vaa lu e = Fu tu t u re re Va lu e * *(( P VIF) = is k kno no wn , Pr Pre se nt Va lue o f a Pe rp e tui tuity ty = c ) As it is P re se se n t Va llu ue F Faa c t o r fo r P e rp rp e ttu u it y a t 1 12 2 %= P re se s e n t v a llu u e o f Rs 2 5 , 0 0 0 in p e rrp petu uiit y =

Rs 7 0 0,, 6 8 80 0 1 r 

  8 .3 3 3 3 Rs 2 , 0 8 , 3 3 3

Pre se nt Va lue = 10000 x(PV x(PVIIFA 12% 12%// 12,12) + 100000 x (PVI (PVIF12% 12%,1ye ,1ye a r  d ) Pre   Pre Pre se nt V Vaa lue Inte re st Fa c to r fo r Annuity (PV (PVIIFA   PVIFA 1 2 %/ 1 2 , 1 2 m o n t h s = 1 1 .2 5 5 1   Pre Pre se nt Va lue Inte re st Fa c to r (PV (PVIIF) = 0.892 0.8929 9 Pre Pre se nt Va lue = 30000 x(P x(PVI VIF FA .12/12,12 m o nth s) + 500000 x (PVI (PVIF.12,1ye a r  = (10000 x 11.25508) + (100000 x 0.8929) = Rs 2,01,84 0.80 Pre se nt Va lue = 25000 x (PVI (PVIFA.12,10) e ) Pre PVIFA PVIF A=

⎡ ⎢1 − ⎢ ⎢ ⎢ ⎣

1

+

(1

i

 

)

n

i

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

=

PVIFA = 5.65 022 PVIF Pre Pre se nt Va lue = 25000 x 5.650 5.65022 22 = Rs 1,41,255 .50

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1 ⎡ ⎢ 1 − (1 + . 12 ) 10 ⎢ . 12 ⎢ ⎢ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

 

Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT 2e Rajiv Srivastava - Dr. Anil Misra

4-6: Present Value Value of of a Ca C ash Strea eam m with Varying Varying Sums Shr hruti uti Ente rp rises is is co nsi nsid d e ring ma king inve st stme me nt for e xp a nd ing its op e ra ti tion on s to the no rth zon e of the c ou ntr ntry. y. Suc h a n e xp a nsi nsion on is likel kely y to e nta il a c a p ita l e xp e nd itur turee of 120 120 la la c ks ks.. The inc re me nta l infl nflow ow s from such a n inve inve st stme me nt a re likel kely y to to b e a va ila b le for the the ne xt 10 yea rs. The c a sh fl flows on a c c oun t of suc h a n e xp a nsi nsion on p la n a re a s fol olllows:

 Year  Y ears

Cash Cas h Flow ows s

0 1 2 3 4 5 6 7 8 9 10

-1 2 ,0 0 ,0 0 0 2 ,1 0 ,0 0 0 3 ,0 5 ,0 8 5 4 ,2 5 ,0 6 0 4 ,0 5 ,0 0 2 3 ,2 4 ,5 4 5 3 ,0 5 ,6 8 9 2 ,4 5 ,0 6 5 1 ,5 5 ,4 0 0 1 ,3 5 ,6 0 0 1 ,2 5 ,0 0 0

Assumi Ass uming ng the d isc oun t ra ra te to b e 14% find out the p re se nt va lue of the c a sh o utf utfllows. Shoul Should d Shr hruti uti Ltd g o a he a d wi with th the e xp a nsi nsion on p la ns?

Solution: The p rob le m c a n b e solved olved in thr three e ste ps a s ffol olllows:

 Step 1 : Fi Find ing the PV fa c to r. As it iiss k kno no wn tha t PV PVIIF=

1 ⎛  ⎜ ⎝  1 +

r 

 ⎞ ⎟  ⎠

n

Ye a r wi wisse PV fa c tor torss a re given given b e low iin n Co l 3. The y ha ve b e e n c om p ute d usi using ng the a b o v e m o d e ll..  Step 2:  F  Fiind ing ou t the Pre Pre sen t va lue . P Prre sen t Va Va lue = Ca sh ffllow s x PV Fa c tor 

Step 3: F  Fiind ing the Ne t Pr Pree se nt Va lue (NP (NPV V). N NPV PV = PV o f Ca sh Infl Inflo o ws - PV PV of Ca sh outflows.  NPV is sho sh o w n in t h e la st c e ll o f th e Ta b le b e low :

PV C ol.1

C ash Flows PV Fac to C ol.2 C ol.3

PV (Rs) C ol. 4

0 1 2

-1 2 0 ,0 0 ,0 0 0 1 8 ,0 0 ,0 0 0 3 0 ,5 0 ,8 5 0

1 0 .8 7 7 2 0 .7 6 9 5

-1 2 0 ,0 0 ,0 0 0 1 5 ,7 8 ,9 4 7 2 3 ,4 7 ,5 3 0

3 4 5 6 7 8 9 10

4 2 ,5 0 ,6 0 0 4 0 ,5 0 ,0 2 0 3 2 ,4 5 ,4 5 0 3 0 ,5 6 ,8 9 0 2 ,4 5 ,0 6 5 1 ,5 5 ,4 0 0 1 ,3 5 ,6 0 0 1 ,2 5 ,0 0 0

0 .6 7 5 0 0 .5 9 2 1 0 .5 1 9 4 0 .4 5 5 6 0 .3 9 9 6 0 .3 5 0 6 0 .3 0 7 5 0 .2 6 9 7

2 8 ,6 9 ,0 3 4 2 3 ,9 7 ,9 3 7 1 6 ,8 5 ,5 8 5 1 3 ,9 2 ,6 7 8 9 7 ,9 3 7 5 4 ,4 7 7 4 1 ,6 9 8 3 3 ,7 1 8 4,99,541

Acc e p ta nc e of tthe he Pr Proj ojee c t iiss lliike ly tto o c re a te va lue for the the b usi usine ne ss tto o the tune of 4 4.9 .99 9 la khs ap pr ox ox.. henc e Shruti hruti Ltd iiss a d vis vise d to g o a he a d with with the e xp a nsi nsion on p la ns.

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Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT 2e Rajiv Srivastava - Dr. Anil Misra

Comparing mparing the Yield  4-7: Co Ra hul Sha rma is to re re ti tirre in 15 yea rs time time . He He is co nsi nsid d e ring two sche me s a va ila b le for inve stme nt.The first sc sc he me p a ys hi him m a n inte inte re st ra ra te of 8% pe r a nnum whi whille the sec ond sc he me wi willl pa y him him Rs.2 .200 000 0 pe r a nnum p e rpe tua lly be gi ginni nning ng from the e nd of 16 yea rs, if if he d e p osi osits ts a t the e nd of eve ry ye ye a r Rs. 2500 ti tilll he re ti tirre s. Whi Whicc h sch e me shoul should d Mr Mr.. Sha Sha rma op t for?

 Solution: Sc heme I

Sc heme II

8 .0 0 % 11

? 11

In t e re st ra t e ( p e r a n n u m ) Tim e p e rio d ( y e a rs)

As Asssuming uming tha t the inte re st ra ra te inhe re nt iin n the sec ond sc he me is ''rr' Future  Va lue  Annuity  Fa c t o r   FVA(r, n) =

=

(1 + r)n − 1 r 

1 .08

15

−1

0. 08

=

2 7 .1 5 2 1

The va lue of R Rss 2, 2,000 000 co ntr ntriib ute d for 15 y yee a rs wi with th a n iinte nte re st rraa te of 8% = Rs 2,500 x 27.1521 = R Rss 67880.25 The p re se nt va lue of a n a nnui nnuity ty = 1/r   The va lue of R Rss 2 2,00 ,000 0 to b e re c e ive d in p e rp e tui tuity ty a t the st staa rt of Y Yee a r 16 = R Rss 2 2,0 ,000/r  00/r  If thi thiss iiss eq ua te d to S Scc he me I the n the imp lie d ra te in S Scc he me II is give give n b y: 2,000 r 

g iv e s r =

= 67,880 .25 2 .9 5 %

4-8: Calc ulating g the Gr Growth Rate Calculatin An inve inve stor boug ht a sha re of a Blue Chi Chip p 20 ye ye a rs ba c k at a p ric e of Rs Rs 50. 50. If the p ric e of the sha re ha s g one up to Rs 350 350,, wha t is is the c om p oun d r ate of gr owth in in the p ric e of sha sha re s of the firm ?

Solution: 1

C AGR =

⎡ En d in g V a lu e ⎤ n ⎥ −1 ⎢ ⎣Be g in n in gV a lu e ⎦

C AG R =

1 0 .2 2 %

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