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THE UNIVERSITY OF NAIROBI

DFI 515 – CORPORATE VALUATION

TIME VALUE OF MONEY

PRESENTED BY JARED OCHIENG OPONDO

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QUESTION ONE

If you invest Sh.20, 000 today, how much will you have in 25 years at 12 percent?

SOLUTION

PV= Shs 20000

Time= 25 years

Rate = 12%

FV=PV (1+k )n

FV = 20000 (1.1225)

= Shs 340,001.2881

QUESTION FOUR

You are making the following deposits into an investment that earns annual rate of 10percent. What will be the value of the investment at the end of the five year period?

Year 1 2 3 4 5Cash flow 5,000 6,000 3,000 2,000 1,500

0 1 2 3 4 5

C1 C2 C3 C4 C5

Future value of uneven stream of cash flows

FV = C1((1+k )4 + C2(1+k )3+ C3¿ + C4(1+k) + C5

FV = 5,000(1.1)4 +6,000(1.1)3 +3,000(1.1)2 +2,000(1.1) + 1,500

= Ksh 22,636.5

QUESTION 5

Assume you are making Sh. 5,000 deposit each year into an investment for 15 years earning an annual rate of 10 percent. What will be the value of the investment at the end of the fifteenyear period?

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Annutiy = Ksh 5,000

Time = 15 years

Rate = 10%

FV annuity= PMT {(1+k )n−1

k}

FV annuity= 5,000{(1+0.1)15−1

0.1}

FV annuity = Shs 158,862.4085

QUESTION 11

CAT ltd is contemplating acquiring RAT Ltd. On successful acquisition, incremental cashflows will as follows

Year 1-10 11-20 21-40 41-60Cash flows in millions

3 2 1.5 0.5

If the appropriate discounting rate is 10 percent, how much should CAT Ltd pay in order toacquire RAT Limited?

Rate = 10%

PV annuity = {1−(1+K )−n

k} for period between 1-10

PV annuity = A[{1−(1+k )−n1

k} - {

1−(1+k )−n

k}] For periods 11-20, 21-40, 41-60

Substituting the values;

PV 1−10= 3{1−(1+0.1)−10

0.1} = Shs 18.433million

PV 11−10= 2[{1−(1+0.1)−20

0.1} - {

1−(1+0.1)−10

0.1}] = 4.737993228 million

PV 21−40= 1.5[{1−(1+0.1)−40

0.1} - {

1−(1+0.1)−20

0.1}] = 1.898230498 million

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PV 41−60= 0.5[{1−(1+0.1)−60

0.1} - {

1−(1+0.1)−40

0.1}] = 0.094053289 million

Total PV = 18.433+4.737993228+1.898230498+0.094053289

PV = 25.16327702 million

QUESTION 12

CAT ltd is contemplating acquiring RAT Ltd. On successful acquisition, incremental cashflows will as follows

YEAR 1-5CASH FLOW (Sh’Millions)

2.5

After the fifth year cash flows are expected to grow at an annual rate of 3 percent forever. Ifthe appropriate discounting rate is 10 percent, how much should CAT Ltd pay in order toacquire RAT LTD.?

0 5

∞

Annuity1−5 = 2.5 million

Annuity period n = 5 years

k= 10%

Perpetuity5−∞ growth rate g = 3%

PV = C{1−(1+K )−n

k} +(

Ck−g )*(1+k)-n

2.5{1−(1+0.1)−5

0.1} + {(

2.50.1−0.03

)* (1+k ¿¿−5

= 9.476966923+ 22.17576154

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= Ksh 31.6527846 million

QUESTION 13

The estimated free cash flow for Savvy solutions ltd at the end of the current period is Sh.1.5 million. After the end of current period, cash flows are expected to grow an annual rateof 8 percent for 25 years. After the 8 percent growth period, annual growth rate in cash flowsis expected to be 3 percent forever. The weighted average cost of capital during the highgrowth period is 10 percent and 5 percent in the stable growth period. Compute the enterprisevalue of Savvy solutions ltd.

Solution

C1 = 1.5 million K1 = 10%

g1 = 8% K2 = 5%

n1 =26 years

g2 = 3%

0 26

g1 = 8% K1 =10% g2= 3% K2 = 5% ∞

At the end of year one T1, Cash flow C1 =1.5 million

At the end of year 2 T2, cash flow C2 = 1.5(1.08)

After 25 years i.e. at T26 cash flow C26 =1.5(1.08)25 = Ksh 10.27271279 million

25 26 27

g1 = 8% g1 =10% g2 = 3%

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Cash flow at T27 = C26( 1.03)

= 10.27271279 * 1.03

C27 = Ksh 10.58089418 million

PVSavvy ltd = PV growing Annuity + PVgrowing perpetuity

PVSavvy ltd = C1

k 1−g1 * [1- (

1+g11+k 1

) n1] + [C27

k 2−g2* (1+k1)-n

1]

PV = 1.5

0.1−0.08 * [1-(

1+0.081.1

) 26] + [ 10.58

0.05−0.03* (1+0.1)-26]

PVSAVVY = Kshs (28.45542255+44.38598458) million

= Kshs 72.84140713 million

QUESTION 14

Madam Grace is celebrating her 35th birthday today and wants to start saving for her anticipated retirement at age of 65. She wants to be able to withdraw khs. 120,000 from her savings account on each birthday for 15 years following her retirement; the first withdrawal will be on her 66th birthday. She is interested in investing in her local commercial bank, which offers 8% interest per year. She wants to make equal annual payments on each birthday into account established at the local commercial bank for her retirement fund.

a) If she starts making these deposits on her 36th birthday and continues to make deposits until she is 65, (the last deposit will be on her 65th birthday), what amount she must deposit annually to be able to make the desired withdrawals at retirement.

b) Suppose Madam Grace has just inherited a large sum of money. Rather than making equal annual payments, she has decided to make one lump sum payment on her 35 birthday to cover her retirement needs. What amount does she have to deposit?

c) Suppose Madam Grace’s employer will contribute Ksh. 1,500 to the account every year as part of the company’s profit sharing plan. In addition she expects a ksh. 25,000 distributions from family trust fund on her 55th birthday, which she will also put into the retirement account. What amount must she deposit annually now to be able to make the desired withdrawals at retirement?

Solution

Rate of interest = 8%

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C1 =amount saved to retirement

C2 = Amount withdrawn after retirement = Ksh 120000

Time to retirement n1 = 30 years

Time after retirement n2 = 15 years

35 36 37 65 66 80

0 1 2 30 31 45

C1 C1 C1 C2 C2 C2 C2

a)

PV65 Annuity = C2 {1−(1+k )−n2

k}

PV65 = 120000{1−(1+0.08 )−15

0.08}

PV65 = Ksh 1027137.443

FVA65 = C{ (1+k )n−1

k}

1027137.443 = C{ (1+0.08)30−1

0.08}

1027137.443 = C (113.2832111)

C1 = Ksh 9,066.987353 (Amount to be deposited annually)

b) To be able to receive Ksh 120,000.00 per month for 15 years, the accumulated amount at

T65 = Ksh 1,027,137.44

35 65 80

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………………… …………………

0 30 45

Discounting amount at T65 to T35 = C65 (1+k ¿¿−n

N =31

= 1,027,137.443(1+0.08¿¿−31

= Ksh 94,513.12893

c)

Let Yearly contribution to be C1

Employer’s contribution yearly C2 = Ksh 1,500

Distribution at 55th birthday i.e. at T20 = Kshs 25,000

Accumulated Amount at 65th Birthday i.e. T30 to be to be able to withdraw Ksh 120,000 yearly = Ksh 1,027,137.443

FV of Ksh 25,000 at T30 = C (1+k ¿¿n

Where C = Ksh 25000 n = 10 years k = 8%

= 2,500(1+0.08¿¿10

= Ksh 53,973.12493

35 55 65 80

0 20 30 45

The balance to be accumulated = Ksh (1027137.443 - 53973.12493)

= KSh 973,164.3181

FVannuity = C{(1+k )n−1

k}

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Where C = C1 + C2

K = 8%

n =30 years

C1 + 1500{(1+0.08)30−1

0.08} = Ksh 973,164.3181

(C1 + 1500)*113.2832111 =973,164.3181

C1 + 1500 = Ksh 8,590.643193 million

C1 = (8,590.543193-1,500) million

C1 = 7,090.543193 million

QUESTION 15

Assume that your guardian plans to borrow Sh. 1.5 Million from a local commercial bank to meet cost of education of your sibling who is travelling abroad for post-graduate studies. The annual interest rate on the loan is 15.9 percent and the loan term is ten years with equal annual installments. Prepare a loan amortization schedule for your guardian.

Solution

Loan amount = Ksh 1.5 million Interest Rate = 15.9% Period = 10 years

PVIFA = Present value interest factor of annuity; Annual instalment = Loan AmountPVIFA

PVIFA = 1−(1+k )−n

kPVIFA =

1−(1+0.159)−10

0.159 = 4.85127629

Annual instalment = 1.5million

4.85127629 = Ksh 309,196.9846

Amortization table

Loan Data      Original Principal  Ksh 1,500,000   Loan Term (Years)                     10   Annual Interest Rate 15.90%  Payments per Year                       1   Payment Ksh 309,196.98            

Year Payment Interest Principal Balance0          1,500,000.00 

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1    309,196.98     238,500.00       70,696.98     1,429,303.02 2    309,196.98     227,259.18       81,937.81     1,347,365.21 3    309,196.98     214,231.07       94,965.92     1,252,399.29 4    309,196.98     199,131.49     110,065.50     1,142,333.80 5    309,196.98     181,631.07     127,565.91     1,014,767.89 6    309,196.98     161,348.09     147,848.89         866,919.00 7    309,196.98     137,840.12     171,356.86         695,562.13 8    309,196.98     110,594.38     198,602.61         496,959.53 9    309,196.98       79,016.56     230,180.42         266,779.11 10    309,196.98       42,417.88     266,779.11                      0.00 

QUESTION 16

Your brother, James, will Join University in seven years, for his higher education. His ambition is to pursue medicine at the University of Nairobi. The Cost of education will be Sh. 1.5 Million per year for five years. Anticipating James’s ambitions, your parents started investing Sh. 100,000 per year five years ago and will continue to do so each year for the next seven years. How much more will your parents have to invest each year for the next seven years to have the necessary funds for the education of your brother? Use 12 percent as the appropriate interest rate throughout this problem

(a) The cost assumed to come at the end of each year

(b) The cost assumed to come at the beginning of each year

today start

0 1 2 3 4 5 6 7 8 9 10 11 12

If John is going to pay 1.5 million for years starting at T12

Annuity C2 = Ksh 1.5 million

Rate = 12%

Time n2 = 5 years

a) Ordinary Annuity

PVA12 = C2 {1−(1+k )−n2

k}

PVA13 = 1.5{1−(1+0.12)−5

0.12} million

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Ksh 5.407164303 million {Amount to be accumulated at the start of School.}

Amount accumulated from 5 years ago to now at C1 = Ksh 100,000 annually

FVannuity = C1{(1+K )n−1

k}

FVA5 = 100,000{(1+0.12)5−1

0.12} = Ksh 635,284.7364

FVA12 of 635,284.7364 = (1.12)7*635,284.7364

Ksh 1,404,412.155

Balance = Ksh 4,002,752.148

Let the additional money to deposited from T5 to T12 be C

FVA12 = C+C1 {(1+K )n−1

k}

Where n = 7 years C1= 100,000 C = Amount required n = 7years

4,002,752.148 = (C + 100,000) {(1+0.12)7−1

0.12}

4,002,752.148 = (C + 100,000)10.08901173

(C + 100,000) = 4002752.14810.08901173

(C + 100,000) = 396,743.7303

C = Ksh 296,743.7303

b) Annuity Due

PVA12 = PVA12 (ordinary annuity) * (1+k)

PVA12 = Ksh 5.407164303 *(1.12) million

PVA12 = 6.056024019 million {amount to be accumulated at the start of School)

FVA5 of Shs 100,000 deposited until T5

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FVA5 = 100,000{(1+0.12)5−1

0.12} = Ksh 635,284.736

FVA12 of 635,284.7364 = (1.12)7*635,284.7364

= Ksh 1,404,412.155

Balance at T12 = 6,056,024.019 - 1,404,412.155

= Ksh 4,651,611.865

FVA12 = C+C1 {(1+K )n−1

k}

4,651,611.865= C+ 100,000 {(1+0.12)7−1

0.12}

4,651,611.865 = C+ 100,000 (10.08901173)

461,057.2362 = C+ 100,000

C = Ksh 361,057.2362

QUESTION 17

Solution

1 5 10 15

Parents investment to year 5 (current year)

A = Ksh 100,00 per month

n = 5 years = 60 months

k = 12% pa =1% per month

FVn5 =A{ (1+k )n−1

k}

FVn5 = 10,000{ (1+0.01)60−1

0.01}

FVn5 = Ksh 816,696.6986

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PV of the entire fee at n10 = ?

N = 3 *5 = 15 periods

Rate = 12%/3 = 4% per semester

Annuity A = Ksh 600,000

PVn10 = { 1−(1+k )−n

k}A

PVn10 = { 1−(1+0.04)−15

0.04}600,000

PVn10 = 6,671,032.459

The FV of the amount accumulated the past 5 years = Ksh 816,696.6986

So the FV of sum already accumulated at n10 = (1+k )n

= 816,696.6986(1.01)60

= Ksh 1,483,690.196

Balanace at n10 : Total fee due= 6,671,032.459

Amount accumulated at n10= Ksh 1,483,690.196

Balanace at n10 = (6,671,032.459- 1,483,690.196)

Ksh 5,187,342.263

Let the additional contribution be C

Annuity from n5-n10 = (10,000+C)

FVAn10 = A{ (1+k )n−1

k}

5,187,342.263 = (10,000 +C){(1+0.01)60−1

0.01}

5,187,342.263 = (10,000 +C){81.66966986}

63,516.141 = Ksh (10,000 +C)

C = Ksh 53,516.141

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QUESTION 18

Solution

45 60 75

0 15 30

Average rate of return = (0.5*7) + (0.5*13)

k = 10%

a) Between 60th -75th years;

Terminal amount = 1 million

Annuity C = Ksh 500,000

K = 10%

n = 15 years

PVAage 60 = C {1−(1+k )−n

k}

PVA = 500,000 {1−(1+0.1)−15

0.1}

PVAage 60 = Ksh 3,803,039.753

PVage 60 of terminal value 1 million

PVage 60 = C (1+k )−n

PV = 1 (1+0.1)−15

= Ksh 239,392.04

Total PVage 60 = Ksh 4,042,431.75

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b)

Amount at T15 = Ksh 4,042,431.75

Balance in bank at T0 = Ksh 600,000.00

FV of 600,000 at T15 = A ((1+k )n

= 600,000 ((1+0.1)15

= Ksh 2,506,348.90

Amount due from annuities = Ksh (4,042,431.75 -2,506,348.90)

= Ksh 1,536,082.85

FVAT15= A{ (1+k )n−1

k}

1,536,082.85 = A{ (1+0.1)15−1

0.1}

1,536,082.85 = A(31.77248169)

A = Ksh 48,346.3289

c)

Donation 200,000 200,000 200,000

age 72 73 74 75

FVA = A{ (1+k )n−1

k}*(1+k)

A = Ksh 200,000

K =10%

n = 3 years

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FVAage 75 = 200,000{ (1+0.1)3−1

0.1}*(1+0.1)

FVA age 75 = Ksh 728,200.00

Discounting to T15 (age 60) = C (1+k )−n

728,200.00 (1+0.1)−15

= Ksh 174,325.29

d)

Salary = Ksh 400,000

Salary growth = 12%

Discount rate = 8%

Age   Amount Growth Rate PVIF PV

46 1 400,000 1.12 1.08 0.925925926       370,370.37 

47 2 448,000.00 1.12 1.08 0.85733882       384,087.79 

48 3 501,760.00 1.12 1.08 0.793832241       398,313.27 

49 4 561,971.20 1.12 1.08 0.735029853       413,065.61 

50 5       629,407.74  1.12 1.08 0.680583197

       428,364.33 

51 6       704,936.67  1.12 1.08 0.630169627

       444,229.68 

52 7       789,529.07  1.12 1.08 0.583490395

       460,682.63 

53 8       884,272.56  1.12 1.08 0.540268885

       477,744.95 

54 9       990,385.27  1.12 1.08 0.500248967

       495,439.21 

55 10    1,109,231.50  1.12 1.08 0.463193488       513,788.81 

56 11    1,242,339.28  1.12 1.08 0.428882859       532,818.02 

57 12    1,391,420.00  1.12 1.08 0.397113759       552,552.02 

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58 13    1,558,390.40  1.12 1.08 0.367697925       573,016.91 

59 14    1,745,397.24  1.12 1.08 0.340461041       594,239.76 

60 15    1,954,844.91  1.12 1.08 0.315241705       616,248.64 

    14,911,886       7,254,962.02

PV = Ksh 7,254,962.02

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