Time Value Questions
-
Upload
jared-opondo -
Category
Documents
-
view
446 -
download
43
description
Transcript of Time Value Questions
0
1
THE UNIVERSITY OF NAIROBI
DFI 515 – CORPORATE VALUATION
TIME VALUE OF MONEY
PRESENTED BY JARED OCHIENG OPONDO
2
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?
3
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
4
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
5
= 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%
6
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%
7
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
8
………………… …………………
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}
9
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
10
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
11
13
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
12
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
13
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
14
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
15
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
16
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
17
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
18