4 QT

 

Rajiv Srivastava & Anil Misra

Financial Management, 2e

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 Fi nanc ncier ier Ltd(a Ltd(an n NBFC NBFC ) is offe offerring pers p ersona ona l loa loa n within within 24 hours of a pp li lic c a ti tion, on, without any a ny do c umenta umentati tion. on. The The interest interest ra ra te c ha harged rged by the Fi Fina nanc ncier ier Ltd Ltd is 25% 25% with monthly c ompo unding. The b orr orrowe owerr, Ana nd has another opti op tion, on, to b orr orrow ow the required required money from from Banko (a national bank) at 27% p.a with annual compounding. The borrower feels that the op ti tion on o f bo row owing ing from from NBFC NBFC is not only o nly convenient co nvenient but b ut is is a lso c hea pe r. Is the firs first opti op tion on actually cheaper? Solution:  The  The iint nter eres estt rate giv given en iin n the the ffiirst option is the the nomi nominal nal rate whil while iin n tthe he sec sec ond option sin inc ce it iis s a c ase of a nnual c compo ompo unding the interes interestt offered rate is the effec ti tive ve rate.  To  To co mpare the the ttwo wo opti options ons the the eff effec ec tive tive iint nter eres estt rate for th the e fir first option needs to be computed. For the first option: No mina l intere st ra te (i) = 0.25 o r 25% Numb er o f c om omp o und ing p e rio d s p e r yea r (n) = 12 ⎡ i ⎤ Effective interest rate (r) =   − 1 ⎢1 + n

⎢ ⎣

⎡

r = ⎢1 + ⎢⎣

. 25 12

⎤ ⎥  ⎦

n

⎥ ⎦

12

−

r=

1

or

0.2807 28.07%

No, the bo rrow ower er is not c orrec orrec t as the effe effec c ti tive ve rate of interes interestt for the fir firs st opti op tion on is more mor e than that for tthe he s sec ec ond opti op tion. on. Hence borrowing from Banko @ 27% p.a is a cheaper option than borrowing from Fi Fina nanc ncier ier L Ltd td (in which whic h effe effec c tive rra a te is 28. 28.07% 07%).

4-2: Future Value of a Single Sum Alpha Alp ha Lt Ltd d ha s a sur urplus plus of Rs. Rs. 20 lac ks a vailab le for the next 5 yea yearrs. At the end e nd of the pe perriod the firm firm will be nee needing ding the a mount for fo r invest investing ing in its its busi business ness expa expans nsion. ion. If the firm firm dec d ec ided to invest invest the sur surplus plus in a scheme sche me that tha t offers a 13 13% % pe r annum, how muc h would wo uld the a mount grow to a t the end of 5 years? years? Solution: Future Futur eV Va a lue (FV) = Present Va Value lue (PV (PV)) x Future Future Va V a lue Interest F Fa a c tor (FVIF n FV = (1+r) Where; r =ra te o r re turn 13% n = time p e rio d o f investme nt (ye a rs) 5 Pre sent Va lue (PV)= Rs 20.00 la khs Futur Fut ure eV Va a lue (FV)  = Rs 36.85 36.85 lakhs

 5

© Oxford University Press 2011

 

Rajiv Srivastava & Anil Misra

Financial Management, 2e

Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT 2e Rajiv Srivastava - Dr. Anil Misra

4-3: Future Value of Annuity An inves investor tor is thi thinki nking ng of inves investi ting ng in a rec ur urri ring ng de po pos sit sc sc heme that offers o ffers a n interest interest ra ra te of of 12% 12 % pe r annum. The The inves investment tment that he is pla planning nning is for the higher highe r educ a tion of his son who is jus justt two yea ye a rs no now. w. In ca c a se the investor d ec ide ides s to inves inve st Rs Rs. 20,00 20,000 0 every year, yea r, wha t will b e the total money available to him when his son reaches the age of 20 years?

Solution: Future Futu re V a lue A nnu nnuiity Fac tor  FV A (r, (r, n)=

Interest rate offered off ered (r ) = No . o f ye a rs (n) = A mo unt investe d (yea rly)

(1+ r)n r

−

1

12 12% % 18 Rs 20,000

So future va value lue of a nnuit nnuity y at the end of 18 year yea rs @ 12 12% % p.a wil willl be Future Futur e value va lue Annuity An nuity Fac tor 55 55.74 .7497 97 Future Futur eV Va a lue = A moun mountt of a annuity nnuity x FV Future Va lue = 20000 x 55.7497 = Rs 11,14,994

4-4: Determining the Implied Rate of Interest  Two  T wo inves investm tment ent options are available available to Pr Prateek. The fi first option prom promiises to pa y Rs. 75,0 ,00 00 at the end e nd of 8 yea rs if the an a n investor d ep osit osits s Rs Rs.. 7,000 7,000 a nnua lly for 8 yea rs. The sec on ond d sc heme promis promises es a lumps lump sum of Rs. Rs. 50,00 50,000 0 at the end of five yea ye a rs on a n annua a nnuall depo de po sit of Rs.  7,000 7,0 00 for 5 yea yea rs. Pra Pra teek is c onfo onfounde unded d a s to whic h of the two sche mes give him better returns? Please suggest. Solution: Sch chem eme eA Rs 7,000 8 Rs 75,0 ,00 00

A mo unt to b e inve sted a nnua lly  Time ime peri period od (years (years) Lum umps psum um rec ei eived ved at the end

Sch chem eme eB Rs 7,000 5 Rs 50,00 ,000

Decision can be taken based on the interest rates implied in the offer.  The  The iimpli mplied ed inter interes estt rates for the the ttwo wo s sc c hemes c an be c computed omputed as fol folllows: ows: Sch chem eme e A Sch chem eme eB Future Futur e Va lue Interest Fac tor for A nnuity (FVIFA) 75000 75000// 700 7000 0 50 50000 000// 700 7000 0 = 10.71 7.14 Now looking at the FVIFA table to locate the FVIFA of 10.71 against 8 years we can find out the rate of retur return. n. Retur Return n for op option tion A lies betwee be tween n 8 and 9%. Now looki looking ng a t the FVIFA FVIFA tab table le to loc ate the FVI FVIF FA of 7.1 7.14 4 ag agains ainstt 5 yea yea rs we c an fi find nd out the rate of return. Return for option B is around 18%. Hence opti op tion on B iis s be bett tter er than than option o ption A.

 5

© Oxford University Press 2011

 

Rajiv Srivastava & Anil Misra

Financial Management, 2e

Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT  Rajiv Srivastava - Dr. Anil Misra

4-5: Present Value of a Single Sum Assuming Ass uming a dis disc c ount rate of 12% find out o ut which whic h one of the following g ives the highes highe st returns returns: a . Rs 1, 1,60 60,00 ,000 0 ava il ila a ble tod toda ay b. Rs 1,7 1,75, 5,00 000 0 to be rec received eived a fter 8 yea yearr c. Rs 25,000 p.a. in perpetuity d. Rs 10 10,00 ,000 0 per pe r month for a yea yearr and Rs 1,0 1,00,0 0,000 00 at the end of the yea e. Rs 25, 25,000 000 p er yea r for the next n ext 10 years yea rs. Solution: Since the the options have varied maturity they are incomparable. To make them c ompa rab le we need nee d to find out tthe he pres present ent value of all the above ab ove options o ptions.. 1,60,000 is a ) Since it is is a vailab le tod toda a y, thi this s is the p prresent value o only nly i.e. i.e. Rs 1,60,000  is the present value. ⎛ 

b) PVIF = ⎜

⎝ 

 ⎞ ⎟

1 1

+

n

o r PVIF a t 12%=

 

0.4039

r 

 ⎠

Presen Pres entt Va lue = Future Future Va lue *(PVIF)

Rs 70 70,68 ,680 0 1

is know known, n, Present Value Va lue o off a Pe Perpetuity rpetuity = c ) A s it is Present ent Valu alue Fact actor or for Per erpe pettuity at 12% Pr Pres esen entt val alu ue of Rs 25,000 in pe perrpet petui uitty =

r

  8.3333 Rs 2,08,333

Presen t Va V a lue =1000 10000 0 x(PV x(PVIFA IFA 12%/ 12,12 12,12)) + 1 10000 00000 0 x (PV (PVIF12 IF12% %,1yea r d) Present   Pres Present ent Value Interes Interestt Fac tor for Annuity (PVIFA   PV PVIFA IFA 12%/ 12, 12 mo months nths 11.2551 11.25 51   Pres Presen entt Va lue Interest Fac tor (PVIF (PV IF)) = 0.892 0.8929 9 Presentt Va lue = 300 Presen 30000 00 x x(PV (PVIFA IFA .12/ .12/12,12 12,12 mo months) nths) +5000 500000 00 x (PV (PVIF.12 IF.12,1yea ,1yea r = (10000 x 11.25508) + (100000 x 0.8929) = Rs 2,01,840.80 Presentt Va Value lue = 2500 25000 0 x (PVIFA.12,10 (PV IFA.12,10 e) Presen PV PVIFA IFA =

⎡ ⎢1 − ⎢ ⎢ ⎢ ⎣

1

+

(1

i

 

)

n

i

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

1 ⎡ ⎤ 1− (1 + . 12 ) 10 ⎥ = ⎢ ⎥ ⎢

⎢ ⎢ ⎣

. 12

PVIFA = 5.65022 Presentt Va lue =2500 Presen 25000 0 x 5.65 5.65022 022 = Rs 1,41,255.50

 5

© Oxford University Press 2011

⎥ ⎥ ⎦

 

Rajiv Srivastava & Anil Misra

Financial Management, 2e

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 Shruti Enterpri Enterpris ses is is c on ons side ideri ring ng ma king investment for expa nd nding ing its op era erations tions to the no nort rth h zone of the country. Such an expansion is likely to entail a capital expenditure of 120 lacks.  The  T he increment incremental al infl inflows ows fr from om such an inves investm tment ent are li likel kely y to be avail ava ilab ab le for th the e next next 10 years. The cash flows on account of such an expansion plan are as follows:  Year  Y ears Cash Cas h Flow ows s 0 -12,00,000 1 2,10,000 2 3,05,085 3 4,25,060 4 4,05,002 5 3,24,545 6 3,05,689 7 2,45,065 8 1,55,400 9 1,35,600 10 1,25,000 Assuming Ass uming the dis d isc c ount rate to be 14 14% % find out the p resent value of o f the c a sh outflows. Should Shruti Ltd go ahead with the expansion plans? Solution:  The  The pr problem oblem can be solv solved ed in thr three s steps teps as fol follows lows::  Step 1 : Fi Finding nding the PV fac tor. As iitt is known tha thatt PVIF

1 ⎛  ⎜ ⎝  1 +

r 

 ⎞ ⎟  ⎠

n

 Year wise PV fac factor tors s are given given below in Col 3 3.. They have been co mput mputed ed usi using tthe he abo ve model model..  Step 2:  Fi  Finding nding out the Pres Present ent value. va lue. Pres Present ent Va lue = C a sh flows x PV Fa Fac c to Step 3: 3: Fi  Finding nding the Net Present Present Va lue (NPV). NPV = PV o off C Ca a sh Inflow Inflows s - PV of of C Ca a sh outflows. NPV is show hown n in the las lastt cell c ell of the T Ta a ble below: be low: PV C ash Flows PV Fac to PV (Rs) C ol.1 C ol.2 C ol.3 C ol. 4 0 -120,00,000 1 -120,00,000 1 18,00,000 0.8772 15,78,947 2 30,50,850 0.7695 23,47,530 3 4 5 6 7 8 9 10

42,50,600 40,50,020 32,45,450 30,56,890 2,45,065 1,55,400 1,35,600 1,25,000

0.6750 0.5921 0.5194 0.4556 0.3996 0.3506 0.3075 0.2697

28,69,034 23,97,937 16,85,585 13,92,678 97,937 54,477 41,698 33,718 4,99,541

Acceptance of the Project is likely to create value for the business to the tune of 4.99 lakhs approx. hence Shruti Ltd is advised to go ahead with the expansion plans.

 5

© Oxford University Press 2011

 

Rajiv Srivastava & Anil Misra

Financial Management, 2e

Solutions to Numerical Problems Chapter 4

FINANCIAL MANAGEMENT 2e Rajiv Srivastava - Dr. Anil Misra

 4-7: Co Comparing mparing the Yield Rahull Sha Rahu harma rma is to retire retire in 15 yea rs time. He is c on ons side ideri ring ng two tw o sche sc hemes mes ava ilab ilable le for fo r investment.T inves tment.The fir firs st sc sc heme pa ys him an inter interest est ra ra te of o f 8% pe perr annum whil w hile e the second sec ond sc heme will pa y him Rs Rs.20 .2000 00 pe r a nnum perpetua pe rpetuall lly y beginning be ginning from the end of 16 years, years, if he de po sit its s at the e nd of e very year Rs. Rs. 2500 2500 til tilll he retir etires. es. Whic Whic h sc sc heme should Mr. Sha Sharrma op t for?  Solution: Sc heme I 8.00% 11

Inte re st ra te (p e r a nnum)  Ti  Time per period iod ((year years s)

Sc heme II ? 11

Assuming Ass uming tha thatt the interest interest rate inherent in the sec ond sc heme is 'r' 'r' Future  Value  Annuity  Factor  FVA(r, n) =

=

(1+ r)n − 1 r

1.0815 − 1

=

0.08

27.1521

 The  The v value alue of Rs 2,00 ,000 c ontr ontriibuted for 15 year years s wi with th an int inter eres estt rate of 8 = Rs 2,500 x 27.1521 = Rs 67880.25  The  The pr pres esent ent valu value e of an annui annuity ty =1/  The  The v value alue of Rs 2,00 ,000 to be rrec ec eiv eived ed in perpetuity perpetuity at the the start tart of Y Yea earr 16 =Rs 2,0 2,000/ 00/ If this this is eq uate uated d to Sc Sc heme I then the implied rate in Sc heme II iis s given b y: 2,000 r

give s r =

= 67,880.25 2.95%

4-8: Calc ulating g the Gr Growth Rate Calculatin An investor investor bought bo ught a share of o f a Blue Blue C hip 20 yea yearrs ba c k at a p rice of Rs 50 50.. If the pr p rice of the share has gone up to Rs 350, what is the compound rate of growth in the price of shares of the firm? firm? Solution: 1

⎡ EndingValue ⎤ n ⎢BeginningValue⎥ − 1 ⎦ ⎣

CAGR

C AG R

10.2 10.22% 2%

 5

© Oxford University Press 2011