Solved problems on time value of money. For executive MBA course

1. E-MBA 1 NileshParab
Time Value of Money (TVM)
Present Value (PV) = Future value (FV) x PVIF
Present Value Interest Factor (PVIF) = 1 / (1+r) ^n OR 1 / FVIF
Present Value of Annuity (PVA) = Annuity Value (A) x PVIFA
Present Value Interest Factor Annuity (PVIFA) = [1 – 1/(1+r) ^n] / r OR (1 – PVIF)/ r
Future Value (PV) = Present value (PV) x FVIF
Future Value Interest Factor (FVIF) = (1+r) ^n
Future Value of Annuity (FVA) = Annuity Value (A) x FVIFA
Future Value Interest Factor Annuity (FVIFA) = [(1+r) ^n – 1]/ r OR (FVIF -1)/r
Present Value (PV)
Q1. Find the present value of a $100 cash flow that is to be received 5 years from now if the interest
rate equals 10%.
PVIF (10%, 5) = 0.620921
A1. Information available:
Future Value (FV) = $100
Tenure (n) = 5 Years
Interest Rate (r) = 10% p.a.
PVIF (10%, 5) = 0.620921 (Present Value Interest Factor), [1/(1+r) ^n]
Present Value (PV) = Future value (FV) x PVIF
∴ PV = 100 x 0.620921
∴ PV = 62.0921
Q2. If you wish to accumulate $140,000 in 13 years, how much you must deposit today in an account
that pays an annual interest rate of 14%?
PVIF (14%, 13) = 0.1821
A2. Information available:
Future Value (FV) = $140,000
Tenure (n) = 13 Years
Interest Rate (r) = 14% p.a.
PVIF (14%, 13) = 0.1821 (Present Value Interest Factor)
Present Value (PV) = Future value (FV) x PVIF
∴ PV = 140,000 x 0.1821
∴ PV = 25,494
Q3. If you wish to accumulate $197,000 in 5 years, how much deposit today in an accountthat pays
a quoted annual interest rate of 13% with semi-annual compounding of interest?
PVIF (6.5%, 10) = 0.53335
A3. Information available:
Future Value (FV) = $197,000
Tenure (n) = 5 Years,
Interest Rate (r) = 13% p.a.
For Semi-annual compounding interest (interest accrued twice in a year)
N = n*2 = 5*2 = 10 times
R=r/2 = 13/2 = 6.5%
PVIF (6.5%, 10) = 0.53335 (Present Value Interest Factor)
Present Value (PV) = Future value (FV) x PVIF
∴ PV = 197,000 x 0.53335
∴ PV = 105,069.95
Q4. Supposed I want to withdraw $5000 at the end of five years and withdraw $6000 at the end of
six years leaving the zero balance in the account after the last withdrawal. If I can earn 5% on
balance how much I have to deposit today to satisfy my withdrawals need?
A4. Information available:

2. E-MBA 2 NileshParab
Future Value (FV1) = $5,000
Future Value (FV2) = $6,000
Tenure (n1) = 5 Years
Tenure (n2) = 6 Years
Interest Rate (r) = 5% p.a.
PVIF1 (5%, 5) = 0.7835 (Present Value Interest Factor)
PVIF2 (5%, 6) = 0.7462 (Present Value Interest Factor)
Present Value (PV1) = Future value (FV1) x PVIF1
∴ PV1 = 5,000 x 0.7835
∴ PV1 = 3,917.5
Present Value (PV2) = Future value (FV2) x PVIF2
∴ PV2 = 6,000 x 0.7462
∴ PV2 = 4,477.2
Total amount to be deposited today (PV) = PV1 + PV2
∴ PV = 3,917.5 + 4,477.2
∴ PV = 8,394.7
Q5. Given the uneven streams of cash flows shown in the following table, answer parts (a) and (b):
Year (n) Project A Project B PVIF (15%, n)
1 50,000 10,000 0.869565
2 40,000 20,000 0.756144
3 30,000 30,000 0.657516
4 20,000 40,000 0.571753
5 10,000 50,000 0.497177
Total
(Undiscounted)
150,000 150,000
a. Find the present value of each stream, using a 15% discount rate.
b. Compare the calculated present values and discuss them in light of the fact that the
undiscounted total cash flows amount to $150,000 in each case.
A5. Here we need to calculate present values of future cash flows, hence we use to formula
Present Value (PV) = Future value (FV) x PVIF
For Project A
PV1 = 50,000 x 0.869565 = 43,479
PV2 = 40,000 x 0.756144 = 30,246
PV3 = 30,000 x 0.657516 = 19,725
PV4 = 20,000 x 0.571753 = 11,435
PV5 = 10,000 x 0.497177 = 4,972
Total for Project A = PV1 + PV2 + PV3 + PV4 + PV5 = 109,857
For Project B
PV1 = 10,000 x 0.869565 = 8,696
PV2 = 20,000 x 0.756144 = 15,123
PV3 = 30,000 x 0.657516 =19,725
PV4 = 40,000 x 0.571753 = 22,870
PV5 = 50,000 x 0.497177 = 24,859
Total for Project B = PV1 + PV2 + PV3 + PV4 + PV5 = 91,273
Cash flows for Project A has a higher present value (109,857) than cash flows for Project B
(91,273) because Project A has larger cash flows in the early years and hence gets more of the
150,000 sooner. Although both the Projects have total 150,000 on an undiscounted basis, the
large early-year cash flow of Project A result in its higher present value.
Q6. Using the interest of 5% per year, what is the value today of the following cash flows?
Year (n) Cash flows PVIF (5%, n)

3. E-MBA 3 NileshParab
1 2,000 0.9524
2 5,000 0.907
3 10,000 0.8638
4 10,000 0.8227
A6. Here we need to calculate present values of future cash flows, hence we use to formula
Present Value (PV) = Future value (FV) x PVIF
PV1 = 2,000 x 0.9524 = 1,905
PV2 = 5,000 x 0.907 = 4,535
PV3 = 10,000 x 0.8638 = 8,638
PV4 = 10,000 x 0.8227 = 8,227
Present Value (PV) = PV1 + PV2 + PV3 + PV4 = 23,305
Present Value of Annuity (PVA)
Q1. Find the present value of a $100 annuity that is to be received annually over the next 5 years if
the interest rate equals 10%.
PVIFA (10%, 5) = 3.7908
A1. Information available:
Annuity Value (A) = $100 p.a.
Tenure (n) = 5 Years,
Interest Rate (r) = 10% p.a.
PVIFA (10%, 5) = 3.7908 (Present Value Interest Factor Annuity)
Present Value of Annuity (PVA) = Annuity Value (A) x PVIFA
∴ PVA = 100 x 3.7908
∴ PVA = 379.08
Q2. Youplan to borrow $ 389000 now and repay in 25 equal annual instalments (END)if the annual
rate of interest is 14% how much your payment will be?
PVIFA (14%, 25) = 6.8729
A2. Information available:
Present Value of Annuity (PVA) = $389,000
Tenure (n) = 25 Years
Interest Rate (r) = 14% p.a.
PVIFA (14%, 25) = 6.8729 (Present Value Interest Factor Annuity)
Present Value of Annuity (PVA) = Annuity Value (A) x PVIFA
∴ A = PVA/PFIFA
∴ A = 389,000/6.8729
∴ A = $56,599
∴ Total Payment = 56,599 x 25
= $14,14,975
Q3. $10,000, 5 year loan at 10% per year, what will the yearly installment?
PVIFA (10%, 5) = 3.7908
A3. Information available:
Present Value of Annuity (PVA) = $10,000
Tenure (n) = 5 Years
Interest Rate (r) = 10% p.a.
PVIFA (10%, 5) = 3.7908 (Present Value Interest Factor Annuity)
Present Value of Annuity (PVA) = Annuity Value (A) x PVIFA
∴ A = PVA/PFIFA
∴ A = 10,000/3.7908
∴ A = $2,637.97

4. E-MBA 4 NileshParab
Q4. A company has made an investment in government bonds. The bonds will generate an interest
incomeof $25,000 eachyearfor5 years.The interest rate is 10%compounded annually. Compute
present value of the stream of interest income for 5 years.
PVIFA (10%, 5) = 3.7908
A4. Information available:
Annuity Value (A) = $25,000
Tenure (n) = 5 Years
Interest Rate (r) = 10% p.a.
PVIFA (10%, 5) = 3.7908 (Present Value Interest Factor Annuity)
Present Value of Annuity (PVA) = Annuity Value (A) x PVIFA
∴ PVA = 25,000 x 3.7908
∴ PVA = $ 94,770

5. E-MBA 5 NileshParab
Future Value (FV)
Q1. What will $247,000 grow to be in 9 years if it is invested today in an account with an annual
interest rate of 11%?
FVIF (11%, 9) = 2.5580
A1. Information available:
Present Value (PV) = $247,000
Tenure (n) = 9 Years
Interest Rate (r) = 11% p.a.
FVIF (11%, 9) = 2.5580 (Future Value Interest Factor), [(1+r) ^n]
Future Value (FV) = Present Value (PV) x FVIF
∴ FV = 247,000 x 2.5580
∴ FV = 631,826
Q2. Find the future value in 5 years of a $100 cash flow if the interest rate equals 10%.
FVIF (10%, 5) = 1.6105
A2. Information available:
Present Value (PV) = $100
Tenure (n) = 5 Years
Interest Rate (r) = 10% p.a.
FVIF (10%, 5) = 1.6105 (Future Value Interest Factor)
Future Value (FV) = Present Value (PV) x FVIF
∴ FV = 100 x 1.6105
∴ FV = 161.05
Q3. How much interest on interest is earned in an account by the end of 5 years if $100,000 is
deposited and interest in 4% per year compounded semi-annually.
FVIF (2%, 10) = 1.2190; FVIF (4%, 5) = 1.2167
A3. Information available:
Present Value (PV) = $100,000
Tenure (n) = 5 Years
Interest Rate (r) = 4% p.a.
For Semi-annual compounding interest (interest accrued twice in a year)
N = n*2 = 5*2 = 10 times
R=r/2 = 4/2 = 2%
FVIF (2%, 10) = 1.2190 (Future Value Interest Factor)
Future Value (FV) = Present Value (PV) x FVIF
∴ FV1 = 100,000 x 1.2190
∴ FV1 = 121,900
If Interest was accrued annually, FVIF (4%, 5) = 1.2167
∴ FV2 = 100,000 x 1.2167
∴ FV2 = 121,670
Additional Interest earned = FV1 - FV2
= 121,900 – 121,670
= $230

6. E-MBA 6 NileshParab
Q4. How much will be in an account at the end of five years the amount deposited today is $10,000
and interest is 8% per year compounded semi-annually.
A4. Information available:
Present Value (PV) = $10,000
Tenure (n) = 5 Years
Interest Rate (r) = 8% p.a.
For Semi-annual compounding interest (interest accrued twice in a year)
N = n*2 = 5*2 = 10 times
R=r/2 = 8/2 = 4% = 0.04
FVIF = (1 + R) ^N = (1+.04) ^10 = 1.04 ^10 = 1.4802
Future Value (FV) = Present Value (PV) x FVIF
∴ FV = 10,000 x 1.4802
∴ FV = 14,802