Corporate Financial Management
Week 1: Time Value Of Money And Investment Rules
Q: Perpetuities and growth
Consider the formula for a perpetuity with a growing cash flow:
Which problem might arise with high growth rates? And which companies would be most affected
by this problem?
If the growth rate g exceeds the discount rate (say r = 3%, g = 5%) then your present value is negative
(if C is positive). Of course, this isn’t correct, you don’t destroy value with a growing set of positive
cash flows. In fact, the value explodes, and it becomes infinite. The formula breaks down, it’s not
applicable anymore. This is most likely to affect companies that are very safe, and therefore have a
low discount rate, so that even a modest growth rate can cause trouble. Or companies that grow
very fast, such as start-ups.
NPV calculation (Q1)
You are considering opening a new plant. The plant will cost $100 million upfront. After that, it is
expected to produce profits of $30 million at the end of every year. The cash flows are expected to
last forever. Calculate the NPV of this investment opportunity if your cost of capital is 8%. Should
you make the investment?
Since the cash flows are expected to last forever, this will be a perpetuity calculation.
𝐶 𝐶 30
𝑃𝑉 = 𝑟 → 𝑃𝑉 = 𝑟 = 0.08 = 375
NPV = PV – Investment = 375 – 100 = 275
NPV > 0, so yes make the investment!
NB: this is a very (perhaps unrealistically) nice project. Even with an investment 3x as big, or a cost
of capital twice as high, it would still have a positive NPV.
NPV calculation (Q2)
FastTrack Bikes, Inc. considers developing a new composite road bike. Development will take two
years and the cost is $600,000 per year. After those two years, the bike enters production and is
expected to make $300,000 per year for 10 years. Assume the cost of capital is 10%. Calculate the
NPV of this investment opportunity, assuming all cash flows occur at the end of each year. Should
the company make the investment?
𝐶 1
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 = 𝑟 ∗ [1 − (1+𝑟)𝑛]
600.000 1
𝑃𝑉𝑐𝑜𝑠𝑡𝑠 = ∗ [1 − (1+0,10)2] = 1.041.322
0,10
For the revenues, the process is the same, except for the fact that the revenues start only 2 years
later… (hint: make a timeline)
1 300.000 1 1
𝑃𝑉𝑟𝑒𝑣𝑒𝑛𝑢𝑒𝑠 = (1+0,10)2 ∗ ∗ [1 − (1+0,10)10 ] = (1+0,10)2 ∗ 1.843.370 = 1.523.446
0,10
NPV = PVrevenues – PVcosts = 1,523,446 – 1,041,322 = 482,123
NPV > 0, so yes, the company should make the investment.
You also can solve this problem in another way, but this is slower than the previous one:
−600 −600 −600
𝑁𝑃𝑉 = (1+0.10)1 + (1+0.10)2 + ⋯ + (1+0.10)12 = 482123
1
,NPV calculation (Q3)
You are deciding between two mutually exclusive investment opportunities. Both require the same
initial investment of $10 million. Investment A will generate $2 million per year (starting at the end
of the first year) in perpetuity (so constant perpetuity). Investment B will generate $1.5 million at
the end of the first year and its revenues will grow at 2% per year for every year after that (growing
perpetuity). Which option should you choose? (hint: keep the discount rate as an unknown, your
answer depends on it).
If you have to choose between mutually exclusive investments, choose the investment with the
highest NPV. But this is also dependent on what the rate is, which is not given.
𝐶 2 𝐶 1.5
𝑃𝑉𝐴 = 𝑟 = 𝑟 𝑎𝑛𝑑 𝑃𝑉𝐵 = 𝑟 = 𝑟−0.02
2 1.5
So 𝑟 = 𝑟−0.02
2𝑟 − 0.02 = 𝑟 ∗ 1.5
2𝑟 − 0.04 = 1.5r
0.5𝑟 = 0.04
0.04
𝑟= = 0.08 → This means that you should choose investment A if r > 0.08, if not choose B.
0.5
Investment 1 pays more initially, while investment 2 pays more after 15 years. Which one you choose
depends on how important early cash flows are compared to later cash flows (which is captured by
the discount factor!)
Q: Real and nominal interest rates
In the year 2008, the average 1-year Treasury Constant Maturity rate was about 1.82% and the rate
of inflation was about 0.28%. What was the real interest rate in 2008?
Use the formula of the interest rate with inflation (only good approximation for low inflation rates,
not for high rates). With I = nominal rate and π = inflation.
𝑖−𝜋 0.0182−0.0028
r = 1−𝜋 = = 0.015
1+0.0028
2
, APRs and effective interest rates (Q1)
Suppose your bank pays interest monthly with the interest rate quoted as an effective annual rate
(EAR) of 6%. What amount of interest will you earn each month? If you have no money in the bank
today, how much will you need to save at the end of each month to accumulate €120.000 in 8 years
(FV annuity)?
1
EMR = (1 + 0.06)12 – 1 = 0.0049 (Effective Monthly Rate)
120000
𝐹𝑉 = 𝑃𝑉 ∗ (1 + 𝑟)𝑛 → 120000 = 𝑃𝑉 ∗ (1 + 0.0049)96 → 𝑃𝑉 = (1+0.0049)96 = 75289
𝑐 1
𝑃𝑉 = 𝑟 ∗ [1 − (1+𝑟)𝑛]
𝑐 1
75289 = 0.0049 ∗ [1 − (1+0.0049)96]
C = 984
APRs and effective interest rates (Q2)
You have found three investment choices for a one-year deposit: 10% APR compounded monthly,
10% APR compounded annually, and 9% APR compounded daily. Compute the EAR for each
investment choice. (Assume that there are 365 days in the year.)
r n
EAR = (1 + 𝑛) − 1
0.1 1
Annual EAR = (1 + ) − 1 = 0.1 = 10%
1
0.1 12
Monthly EAR = (1 + 12 ) − 1 = 0.105 = 10.5%
0.09 365
Daily EAR = (1 + 365 ) − 1 = 0.094 = 9.4%
Payback period (Q1)
Projects A, B, and C each have an expected life of 5 years. Given the initial cost and annual cash flow
information below, what is the payback period for each project?
Assume CFs are the same every year throughout the life of the project.
80
𝑃𝑎𝑦𝑏𝑎𝑐𝑘𝐴 = 25 = 3.2 𝑦𝑒𝑎𝑟𝑠
120
𝑃𝑎𝑦𝑏𝑎𝑐𝑘𝐵 = = 4.0 𝑦𝑒𝑎𝑟𝑠
30
150
𝑃𝑎𝑦𝑏𝑎𝑐𝑘𝑐 = = 4.3 𝑦𝑒𝑎𝑟𝑠
35
3
Week 1: Time Value Of Money And Investment Rules
Q: Perpetuities and growth
Consider the formula for a perpetuity with a growing cash flow:
Which problem might arise with high growth rates? And which companies would be most affected
by this problem?
If the growth rate g exceeds the discount rate (say r = 3%, g = 5%) then your present value is negative
(if C is positive). Of course, this isn’t correct, you don’t destroy value with a growing set of positive
cash flows. In fact, the value explodes, and it becomes infinite. The formula breaks down, it’s not
applicable anymore. This is most likely to affect companies that are very safe, and therefore have a
low discount rate, so that even a modest growth rate can cause trouble. Or companies that grow
very fast, such as start-ups.
NPV calculation (Q1)
You are considering opening a new plant. The plant will cost $100 million upfront. After that, it is
expected to produce profits of $30 million at the end of every year. The cash flows are expected to
last forever. Calculate the NPV of this investment opportunity if your cost of capital is 8%. Should
you make the investment?
Since the cash flows are expected to last forever, this will be a perpetuity calculation.
𝐶 𝐶 30
𝑃𝑉 = 𝑟 → 𝑃𝑉 = 𝑟 = 0.08 = 375
NPV = PV – Investment = 375 – 100 = 275
NPV > 0, so yes make the investment!
NB: this is a very (perhaps unrealistically) nice project. Even with an investment 3x as big, or a cost
of capital twice as high, it would still have a positive NPV.
NPV calculation (Q2)
FastTrack Bikes, Inc. considers developing a new composite road bike. Development will take two
years and the cost is $600,000 per year. After those two years, the bike enters production and is
expected to make $300,000 per year for 10 years. Assume the cost of capital is 10%. Calculate the
NPV of this investment opportunity, assuming all cash flows occur at the end of each year. Should
the company make the investment?
𝐶 1
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 = 𝑟 ∗ [1 − (1+𝑟)𝑛]
600.000 1
𝑃𝑉𝑐𝑜𝑠𝑡𝑠 = ∗ [1 − (1+0,10)2] = 1.041.322
0,10
For the revenues, the process is the same, except for the fact that the revenues start only 2 years
later… (hint: make a timeline)
1 300.000 1 1
𝑃𝑉𝑟𝑒𝑣𝑒𝑛𝑢𝑒𝑠 = (1+0,10)2 ∗ ∗ [1 − (1+0,10)10 ] = (1+0,10)2 ∗ 1.843.370 = 1.523.446
0,10
NPV = PVrevenues – PVcosts = 1,523,446 – 1,041,322 = 482,123
NPV > 0, so yes, the company should make the investment.
You also can solve this problem in another way, but this is slower than the previous one:
−600 −600 −600
𝑁𝑃𝑉 = (1+0.10)1 + (1+0.10)2 + ⋯ + (1+0.10)12 = 482123
1
,NPV calculation (Q3)
You are deciding between two mutually exclusive investment opportunities. Both require the same
initial investment of $10 million. Investment A will generate $2 million per year (starting at the end
of the first year) in perpetuity (so constant perpetuity). Investment B will generate $1.5 million at
the end of the first year and its revenues will grow at 2% per year for every year after that (growing
perpetuity). Which option should you choose? (hint: keep the discount rate as an unknown, your
answer depends on it).
If you have to choose between mutually exclusive investments, choose the investment with the
highest NPV. But this is also dependent on what the rate is, which is not given.
𝐶 2 𝐶 1.5
𝑃𝑉𝐴 = 𝑟 = 𝑟 𝑎𝑛𝑑 𝑃𝑉𝐵 = 𝑟 = 𝑟−0.02
2 1.5
So 𝑟 = 𝑟−0.02
2𝑟 − 0.02 = 𝑟 ∗ 1.5
2𝑟 − 0.04 = 1.5r
0.5𝑟 = 0.04
0.04
𝑟= = 0.08 → This means that you should choose investment A if r > 0.08, if not choose B.
0.5
Investment 1 pays more initially, while investment 2 pays more after 15 years. Which one you choose
depends on how important early cash flows are compared to later cash flows (which is captured by
the discount factor!)
Q: Real and nominal interest rates
In the year 2008, the average 1-year Treasury Constant Maturity rate was about 1.82% and the rate
of inflation was about 0.28%. What was the real interest rate in 2008?
Use the formula of the interest rate with inflation (only good approximation for low inflation rates,
not for high rates). With I = nominal rate and π = inflation.
𝑖−𝜋 0.0182−0.0028
r = 1−𝜋 = = 0.015
1+0.0028
2
, APRs and effective interest rates (Q1)
Suppose your bank pays interest monthly with the interest rate quoted as an effective annual rate
(EAR) of 6%. What amount of interest will you earn each month? If you have no money in the bank
today, how much will you need to save at the end of each month to accumulate €120.000 in 8 years
(FV annuity)?
1
EMR = (1 + 0.06)12 – 1 = 0.0049 (Effective Monthly Rate)
120000
𝐹𝑉 = 𝑃𝑉 ∗ (1 + 𝑟)𝑛 → 120000 = 𝑃𝑉 ∗ (1 + 0.0049)96 → 𝑃𝑉 = (1+0.0049)96 = 75289
𝑐 1
𝑃𝑉 = 𝑟 ∗ [1 − (1+𝑟)𝑛]
𝑐 1
75289 = 0.0049 ∗ [1 − (1+0.0049)96]
C = 984
APRs and effective interest rates (Q2)
You have found three investment choices for a one-year deposit: 10% APR compounded monthly,
10% APR compounded annually, and 9% APR compounded daily. Compute the EAR for each
investment choice. (Assume that there are 365 days in the year.)
r n
EAR = (1 + 𝑛) − 1
0.1 1
Annual EAR = (1 + ) − 1 = 0.1 = 10%
1
0.1 12
Monthly EAR = (1 + 12 ) − 1 = 0.105 = 10.5%
0.09 365
Daily EAR = (1 + 365 ) − 1 = 0.094 = 9.4%
Payback period (Q1)
Projects A, B, and C each have an expected life of 5 years. Given the initial cost and annual cash flow
information below, what is the payback period for each project?
Assume CFs are the same every year throughout the life of the project.
80
𝑃𝑎𝑦𝑏𝑎𝑐𝑘𝐴 = 25 = 3.2 𝑦𝑒𝑎𝑟𝑠
120
𝑃𝑎𝑦𝑏𝑎𝑐𝑘𝐵 = = 4.0 𝑦𝑒𝑎𝑟𝑠
30
150
𝑃𝑎𝑦𝑏𝑎𝑐𝑘𝑐 = = 4.3 𝑦𝑒𝑎𝑟𝑠
35
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