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1.
Time
Value of Money numerical with solutions
1. Future value (FV) of a single cash
flow
= PV × (1+r) n = PV × FVF r,
n
2. Future value of Annuity
= Annuity amount × [(1+r) n + 1] ÷ r
=
Annuity amount × FVAF r, n
3. Future value of Annuity
due = [Annuity amount × [(1+r) n + 1]
÷ r] × (1+r)
= Annuity amount × FVAF r, n × (1+r)
4. Present value (PV) of a single cash
flow = FV ÷ (1+r) n = FV × PVF r,
n
5. Present value (PV) of
Annuity = Annuity amount × [1 – {1 ÷ (1 + r)
n}] ÷ r
= Annuity amount × PVAF r, n
6. Present value (PV) of
Annuity due = [Annuity amount × [1 – {1 ÷ (1 + r)
n}] ÷ r] × (1+r)
= Annuity amount × PVAF r, n × (1+r)
7. Present value of
perpetuity = Cash flow ÷ r
8. Present value of growing
perpetuity = Cash flow ÷ (r – g)
9. Present value of growing
annuity = [Cash flow 1 ÷ (r-g) ] ×
[ 1- { (1+g) ÷(1+r)}n ]
10. Present value of growing annuity due
= [Cash flow 1 ÷ (r-g) ] × [ 1- { (1+g) ÷ (1+r)}n ] × (1
+ r)
11. Future value of growing annuity
= Present value of growing annuity × (1+r) n
= [Cash flow 1 ÷ (r-g) ] × [ (1+r)n – (1+g)n]
12. Future value of growing annuity due
= Present value of growing annuity × (1+r) n × (1 +r)
= [Cash flow 1 ÷ (r-g) ] × [ (1+r)n – (1+g)n]
× (1 +r)
In case, r = g
13. Present value of growing
annuity = CF1 × [n ÷ (1 + r)]
14. Present value of growing
annuity due = CF1 × [n ÷ (1 + r)] ×(1
+ r)
15. Future value of growing
annuity = CF1 × n ×
(1+r) n−1
16. Future value of growing
annuity due = CF1 × n ×
(1+r) n−1 × (1 + r)
Finding growth rates
FV = PV (1 +g) n
g (growth rate) = (FV/PV)
PVFr, n = PVIFr, n
=
Present value interest factor at rate of interest r after n periods.
PVFAr, n = PVIFAr, n
=
Present value interest factor for an annuity at rate of interest r after n
periods.
FVFr, n = FVIFr,n
= CVIPr,n = CVPr,n = Future or compound value interest
factor at rate of interest r after n periods.
FVFAr, n = FVIFAr, n
= CVIPAr, n = CVPAr, n = Future or compound value interest
factor for an annuity at rate of interest r after n periods.
1. Mr. X invested Rs. 7, 10,000 at
the beginning of the year one at the rate of 10% compounded annually. Calculate
how much he will receive after the end of 1st, 2nd and 3rd year of his
investment?
Future value at the end of year 1 FV1 = PV (1+r)
1 = 710000 * (1+0.1)1 = 710000 * 1.1= 781000
Or FV1 = PV × FVFr, n = 710000 × PVF.1,
1 = 710000 × 1.1 = 781000
|
A |
B |
A × B |
|
Present
value at year 0 |
FVFr,n ( r= 0.1, n=1,2,3) taken from PVF table |
Future Value
at the end of |
|
Rs. 7,10,000 |
FVF0.1,
1 = 1.1 |
Year 1 - FV1 =
PV × FVF0.1, 1 = Rs. 7,10,000 × 1.1 = Rs. 7,81,000 |
|
Rs. 7,10,000 |
FVF0.1,
2 = 1.21 |
Year 2 - FV2 = PV × FVF0.1, 2
= Rs. 7,10,000 × 1.21 = Rs. 8,59,100 |
|
Rs. 7,10,000 |
FVF0.1,
3 = 1.33 |
Year 3 - FV3
= PV × FVF0.1, 3 = Rs. 7,10,000 × 1.33 = Rs. 9,65,600 |
2. Mr. X invested Rs. 1,000, Rs. 2,000
and Rs. 5,000 at the starting of 1st, 2nd and 3rd year. What will be compounded
value of his investment at the end of 3rd year when interest is provided at the
rate of 12%.
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Solution:
|
Method 1 |
Method 2 |
||
|
A |
B |
C = A × B |
FV = PV × (1+
r)n |
|
Money invested at the beginning of year |
FVF r,n |
|
|
|
1- Rs. 1000 |
FVF 0.12, 3 = 1.405 |
Rs. 1000×1.405= Rs. 1,405 |
= Rs. 1000× (1+ .12)3=
Rs. 1,405 |
|
2 - Rs. 2000 |
FVF 0.12, 2 = 1.254 |
Rs. 2000×1.254= Rs. 2,508 |
= Rs. 2000× (1+ .12)2=2,508 |
|
3 - Rs. 5000 |
FVF 0.12, 1 = 1.120 |
Rs. 5000×1.120= Rs. 5,600 |
= Rs. 5000× (1+ .12)1
= 5,600 |
|
compound value
of his investment at the end of 3rd year when interest is provided at the
rate of 12% |
1405 + 2508 + 5600 = Rs. 9513 |
1405 + 2508 + 5600 = Rs. 9,513 |
|
3. Mr. X has invested an amount of Rs.
15,000 each at the end of 1st, 2nd and 3rd year. Calculate the compound value
of his investment at the end of 3rd year if interest is provided at a rate of 9
% compounded annually.
Solution:
FV at the end of year 3 = Annuity × FVAF 0.09, 3 =
Rs. 15,000 × 3.278 = Rs. 49,170
4. A has invested Rs. 7,000 for 3
years at an interest rate of 12 % per annum compounded semiannually. What
amount he will get after 3 years?
Solution:
FV = PV ×
FVF 0.06, 6 = Rs. 7,000 × 1.419 = Rs. 9,933
(In case of
semiannual compounding divide r by 2 and multiply n by 2)
5. Vitthal has invested Rs. 25, 000
now for 3 years at the rate of 8 % per annum compounded quarterly. What amount
he will get after 3 years?
Solution: FV = PV × FVF 0.02, 12 = Rs. 25,000 ×
1.268 = Rs. 31,700
(In case of quarterly compounding divide r by 4 and multiply
n by 4)
6.
Ravi wants to deposit Rs. 10, 00,000 in a bank for a year. He has received
following offers of rate of interest from different banks
SBI-10.75%
p.a. compounded weekly
PNB-11%
p.a. compounded monthly
HSBC-11.25%
p.a. compounded quarterly
ICICI
- 11.2% p.a. compounded half yearly
HDFC-
11.5% p.a. compounded yearly.
In
which bank should he deposit his money?
Solution:
|
Bank |
Nominal /
stated / normal Rate of interest (r) |
Period of
compounding |
No. of
compounding period in a year (m) |
Effective
rate of interest re
= (1+r/m)m -1 |
|
SBI |
0.1075 |
Weekly |
52 |
(1 + |
|
PNB |
0.11 |
Monthly |
12 |
(1 + |
|
HSBC |
0.1125 |
Quarterly |
4 |
(1 + |
|
ICICI |
0.112 |
Half yearly |
2 |
(1 + |
|
HDFC |
0.115 |
yearly |
1 |
(1 + |
|
Ravi should invest in HSBC as effective rate of
interest is highest for HSBC = 0.1173 or 11.73 % |
||||
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7.
Mr. Ajai will receive Rs. 30,000 after 3 years. How much he has invested now if
rate of interest is 10%.
Solution:
Money invested today = Present value = FV × PVF .1, 3
= Rs. 30,000 × 0.751 = 22,530
= FV / (1+r)n = Rs. 30000/ (1.1)3 = 22530
8.
Shyam will receive Rs. 6,000, Rs. 4,000 and Rs. 2,000 at the end of 1st, 2nd
and 3rd year. Find present value of these cash flows considering discounting
rate be 15 %.
Solution:
|
|
A |
B |
C= A×B |
|
Year |
FV |
PVFr,n |
PV = FV ×
PVFr,n |
|
1 |
Rs. 6,000 |
PVF.15,
1 = 0.870 |
Rs. 5,220 |
|
2 |
Rs. 4,000 |
PVF.15,
2 = 0.756 |
Rs. 3,024 |
|
3 |
Rs. 2,000 |
PVF.15,
3 = 0.658 |
Rs. 1,316 |
|
Present value of future cash flows |
=5220+3024+1316 = Rs. 9560 |
||
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9.
Find the present value of the annuity consisting of a cash inflow of 17,000 per
year for 3 years discounting rate being 18 %
Solution:
Present value of annuity = Annuity × PVAF .18, 3 =
Rs. 17,000 × 2.174 = Rs. 36, 958
10.
How much Rina has to invest to yield Rs. 10,000 p.a. in perpetuity if
opportunity cost of capital (r) is 11 %.
Solution:
Amount to be invested by Rina = Present value of perpetuity =
Cash flow ÷ r = Rs. 10,000 ÷.11
= Rs. 90,909
11.
Rajni makes recurring deposit of Rs. 13,000 in the beginning of each of 5 years
starting now at 12% p.a. how much she will get after 5 years?
Solution:
Future value of Annuity due = Annuity × FVAF0.12, 5
× (1+.12) = Rs. 13,000 × 6.353 × 1.12
= Rs. 92,500
12.
What amount should be invested now to get an amount of Rs. 25,000 in the
beginning of next 5 years at 8 % p.a. rate of interest?
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Solution:
Present value of Annuity due = Annuity × PVAF0.09, 5 ×
(1+.09) = 25000 × 3.890 × 1.09
= Rs. 1,06, 002.5
13.
TCS wants to offer scholarship of Rs. 35,000 per year to 100 disabled sports
persons starting from one year now and
it will increase at a constant rate of 5% every year Find the present value of
this scholarship if rate of interest is 7%.
Solution:
Present value of perpetuity with constant growth rate = CF ÷
(r-g) = Rs. 35,000 ÷ (0.07-0.05)
= Rs. 17, 50, 000
Present value of scholarship given to 100 disabled sports
persons = Rs. 17,50, 000 × 100
= Rs. 17, 50, 00, 000
14.
TCS wants to offer scholarship of 35000 per year to 100 disabled sports
persons starting from one year now for
next 10 years and it will increase at a constant rate of 5% every year Find the
present value of this scholarship if rate of interest is 7%.
Solution:
Present value of Annuity with constant growth rate = [Cash flow 1 ÷ (r-g) ] × [
1- { (1+g) ÷(1+r)}n ]
=
[Cash flow 1 / (r-g) ] × [ 1- { (1+g) /(1+r)}n ]
=
Rs. 17,
50, 000 × (1- 0.981310)
=
Rs. 17,
50, 000 × (1- 0.828) = Rs. 3,00, 917
Present value of scholarship given to 100 disabled sports persons for 10 years= Rs.3, 00,917 × 10 = Rs. 3, 00, 91, 700
15.
A machinery is to be replaced after 7 years which will cost Rs. 10, 00, 000 at the
end of 7 years. However, Rs. 75,000 can be realized by selling the existing
machinery at that time. Calculate the amount to be deposited every year to get
the amount required to purchase new machinery after 7 years considering deposit
yields a rate of 6 %.
Solution:
Replacement cost of machinery at the end of 7 year after
deducting realized value of existing machinery = Rs. 10, 00, 000 – Rs. 75, 000
= Rs. 9, 25, 000
Future value of Annuity = Annuity × FVAF0.06, 7
→ Annuity = FV/ FVAF0.06, 7 = Rs. 9, 25, 000 /
8.394 = Rs. 1, 10, 198
Amount to be deposited every year to get Rs. 9, 75, 000 at
the end of year 7 Rs. 1, 10, 198.
16.
Mahesh took a loan of Rs. 15, 00,000 to be paid in ten equal annual
installments at rate of 15 %. Calculate annual installment payable by him.
Solution:
PV of annuity = Annuity × PVAF 0.15, 10 =>
Annuity = PV of annuity / × PVAF 0.15, 10
Annuity / Annual installment = Rs. 15, 00,000 /5.019 = Rs. 2,
98,864
17.
A financial institution offered that it will pay a lump sum amount of Rs.
12,000 at the end of 10 years to investors who pay 1,000 annually for 10 years.
Find out implicit rate of interest in the offer.
Solution: FV = Annuity × FVAF r, 10
=> FVAF r, 10 = FV/Annuity = Rs. 12,000 /Rs. 1,000
= 12
Looking into FVAF table at 10 years, implicit rate of
interest is 4 %.
18.
SBI has issued deep discount bonds of Rs. 15,000 each which will mature after 7
years for 35,000. What is implicit rate of interest of these DDRBs.
Solution:
FV = PV × (1+r) 7 => (1+r) 7 = FV /
PV = Rs. 35,000 / Rs. 15,000 = 2.333
=> (1+r) 7 = FVPr, 7 = 2.333
1+r = (2.333)
1 + r = 1.1284
S0, r = 1.1284 – 1 =0.1284 = 12.84 %
19. Profit of Tesla has grown from Rs. 10
crores to Rs. 100 crores in 5 years’ time. Calculate compound rate of growth
company has maintained in profit during this period. Assume that profit has
grown evenly throughout this period.
Solution:
Growth rate = (FV/PV) 1/5 - 1 = (100/10)
1/5 - 1 = 1.5849 – 1 = 0.5849 = 58.49 %
20. Find the number of years required to
accumulate a sum of Rs. 5,000 with an initial investment of 1,200 at 10% rate
of interest.
Solution:
FV = PV × (1+r) n PV × FVF0.1, n
FVF0.1,
n or (1+r)
n = Rs. 5000 / Rs. 1200 = 4.17
(1.1) n = 4.17
Looking at FVF table at 10 % no. of years for value
of 4.17 is close to 15 years
So, n = 15 years
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