FINANCE FOR PREMASTER
CHAPTER 3
Valuation
What is the value of an asset (a stock, a firm, a project, …)?
In a competitive market, the value today (present value) of all the cash flows that accrue to its
investors in the future
Two main components:
1. (Expected) Cash flows → timing! (paying 1 mln in one year or paying 1 mln in 100 years gives different
value)
2. Discount rate → appropriate rate to bring future cash flows backward in time given risk or uncertainty
in the cash flows!
Cash flows can be pre-determined → only timing matters
• Coupons on government bonds
• Taking out a loan
Cash flows can be risky → Both timing and uncertainty matter: when uncertain cash flows correlate with the
state of the economy, this will impact the discount rate
• Dividends on stocks (you don’t know when it’s going to be paid or what the dividend is going to be →
depends on how the firm is performing)
• Weather derivatives (contracts where the payoff depends on the weather → weather is
unpredictable)
TWO MAIN CONCEPTS: 1. TIMING
You can invest in two projects:
1) earns $100 in one year
2) earns $100 in 10 years
Which do you prefer? Or, how much are you willing to pay for each investment?
P1 > P2, the difference reflects time preference
• Quantified using: P1 = 100/ (1 + r) vs P2 = 100 / (1 + r)^10
• R = interest (or more generally: discount) rate that allows you to calculate value of an asset at any
point in time (past, present, future)
TWO MAIN CONCEPTS: 2A. RISK
You can invest in two projects:
1) earns $100 with certainty in one year
2) earns $0 or $200 with equal probability in one year
Which do you prefer? Or, how much are you willing to pay for each investment?
For most people: P1 > P2, which reflects risk preference
, • Quantified as: P1 = 100 / (1 + R risk free) vs P2 = E(CF) / (1 + R risky) where
R risky = R risk free + risk premium
• The risk premium is lower for government bonds (which are on average likely to repay) than for
corporate bonds (which are on average less likely to repay as the firm may end up bankrupt)
TWO MAIN CONCEPTS: 2B. RISK
You can invest in two projects and each projects gives 0 or 200 with equal probability in one year. However:
1) Earns 0 when the economy is doing worse than average and 200 when the economy is doing better than
average
2) Earns 200 when the economy is doing worse than average and 0 when the economy is doing better than
average
Which do you prefer? Or, how much are you willing to pay for each investment?
For most people, P1 < P2, which reflects risk preference
• Quantified as P1 = E(CF) / (1 + R risky-1) vs P2 = E(CF) / (1 + R risky-2)
[R risky-1 = R risk-free + risk premium (1)] > [ R risky-2 = R risk-free + risk premium (2)]
• The appropriate risk premium is higher the more the project’s cash flows correlate with the overall
economy
o People like project (2) as it pays off exactly when times are bad
o Later, we introduce models to estimate:
Risk premium = amount of risk x price of risk
The second asset has a higher price, but means it has a lower risk premium (investors are willing to pay a high
price for this investment because it pays of exactly when they need the money) → provides additional money
exactly when investors need it (bad state of economy)
THE THREE RULES OF TIME TRAVEL
1. Only cash flows at the same point in time can be compared or combined
2. To move a cash flow backward in time, you must discount it:
PV = Cn / (1 + r)^n
3. To move cash flow forward in time, you must compound it:
FVn = C0 x (1 + r)^n
Notation:
Cn = Cash flow at date n
r = discount rate
PV = present value (today)
FVn = future value on date n
EXAMPLE: COMPARING CF’S AT SAME POINT IN TIME
Which alternative do you prefer if the one year interest rate r = 4%?
1) €100 today
2) €103 in 1 year time
You prefer option 1, because you can get €104 with the interest rate in 1 year time. Even if you plan to use
money only next year, use the bank to move money forward in time
, Strategies Cash flow today Cash flow in one year
1A. Take alternative 1 100 0
1B. take alternative 1 and put money in bank 100 – 100 = 0 0 + 100 * (1 + r) = 104
account
2A. Take alternative 2 0 103
2B. Take alternative 2 and borrow PV of 103 today 0 + 103 / (1+ r) = 99 103 -103 = 0
→ Comparing Present Value and Future Value is equivalent (1A dominates 2B and 1B dominates 2A)
Thus, comparing cash flows at the same point in time (rule #1) is not just a paper trick, we can use the bank by
saving or borrowing to actually do it
EXAMPLE: COMPOUNDING AND DISCOUNTING
What is the future value of putting €100 in a bank account for 2 years at 4%?
C1 = C0 x (1 + r) = 100 x (1 + r) = 104
C2 = C1 x (1 + r) = C0 x (1 + r) x (1 + r) = C0 x (1 +r)^2
C2 = 100 x 1.04^2 = 108.16
Thus, FVn = C0 x (1 + r)^n
What is the present value (or today’s worth) of a cash-flow of €110 received in 2 years?
Put an amount C0 in a bank account so that it accumulates to C2 = 110 in two years.
Hence 110 = C0 x 1.04^2 → C0 = .04^2 = 101.70
Thus, PV = Cn / (1 + r)^n
PRESENT VALUE AND NPV DECISION RULES
Discounting is used by financial managers to decide which project(s) to invest in. The Net Present Value (NPV)
of a project or investment:
NPV = PV(benefits) – PV(costs)
NPV measures increase in today’s firm value from taking a project that will provide some cash flows in the
future.
NPV decision rules:
• Good projects have NPV > 0
• When choosing among positive NPV projects, choose the largest NPV
• With a budget, choose the combination of projects that maximizes the sum of the NPV’s
EXAMPLE
Consider the following investment opportunity:
Invest €100 today
Receive €30 at the end of year 1; Receive €75 at the end of year 2
The interest rate is 4%
Should we make this investment?
, Not simply a €5 profit; cash flows don’t accrue on same date
PV (benefits) = .04 + .04^2 = 98.1
PV (costs) = 100
NPV = -100 + 98.1 = -1.9 → do not invest!
A related concept that will come back in this course: Internal Rate of Return:
IRR is the discount rate that sets the NPV of an investment equal to zero
In many cases, but not always, higher IRR is better
Example: -100 + 30 / (1 + IRR) + 75 / (1 + IRR)^2 = 0 → IRR = 2.89% (→ with Excel solver)
EXAMPLE: CHOOSING BETWEEN INVESTMENTS
Problem: You have $10,000 to invest and are considering three one-year risk-free investment options
1) Invest up to $10,000 in a Treasury-Bill (a one-year US government bond) paying 2%
Note, 2% is the risk-free interest rate
2) Invest in a project that cost $6,000 and returns $6,100 in one year
3) Invest in a project that costs $4,000 and returns $4,100 in one year
How should the $10,000 investment be allocated?
1) Investing $10,000 in the T-Bill
NPV = $10,000 (1.02) / 1.02 – $10,000 = $0.00
2) Investing $6,000 and receiving $6,100
NPV = $6,.02 - $6,000 = -$19.61
3) Investing $4,000 and receiving $4,100
NPV = $4,.02 - $4,000 = $19.61
Which one to choose? → Option 3. Option 1 creates no value, option 2 destroys value
ARBITRAGE AND THE LAW OF ONE PRICE
Arbitrage – the practice of buying and selling equivalent goods in different markets to take advantage of a price
difference. An arbitrage opportunity occurs when it is possible to make a profit without taking any risk or
making any investment → an investment that doesn’t cost anything today, but you have a positive probability
of making some money (positive cash flow) in the future
1. When the Law of One Price (two assets that give exactly the same future pay off should have the same
price) is violated there is an arbitrage opportunity:
a. The same asset’s price is different in two markets (Royal Dutch Shell shares are traded on the
Amsterdam, London and New York stock exchanges)
2. When an asset’s price ≠ PV (future cash flows)
a. If you have a risk free investment opportunity that gives you 3% return even though the risk
free interest rate is 2%, you have an arbitrage opportunity. A risk free investment should give
the return that is exactly compensating for the risk free interest rate of 2%
Normal market – a competitive market in which there are no arbitrage opportunities (and the Law of One Price
Holds) → assumed throughout the course and not too far form reality
CHAPTER 3
Valuation
What is the value of an asset (a stock, a firm, a project, …)?
In a competitive market, the value today (present value) of all the cash flows that accrue to its
investors in the future
Two main components:
1. (Expected) Cash flows → timing! (paying 1 mln in one year or paying 1 mln in 100 years gives different
value)
2. Discount rate → appropriate rate to bring future cash flows backward in time given risk or uncertainty
in the cash flows!
Cash flows can be pre-determined → only timing matters
• Coupons on government bonds
• Taking out a loan
Cash flows can be risky → Both timing and uncertainty matter: when uncertain cash flows correlate with the
state of the economy, this will impact the discount rate
• Dividends on stocks (you don’t know when it’s going to be paid or what the dividend is going to be →
depends on how the firm is performing)
• Weather derivatives (contracts where the payoff depends on the weather → weather is
unpredictable)
TWO MAIN CONCEPTS: 1. TIMING
You can invest in two projects:
1) earns $100 in one year
2) earns $100 in 10 years
Which do you prefer? Or, how much are you willing to pay for each investment?
P1 > P2, the difference reflects time preference
• Quantified using: P1 = 100/ (1 + r) vs P2 = 100 / (1 + r)^10
• R = interest (or more generally: discount) rate that allows you to calculate value of an asset at any
point in time (past, present, future)
TWO MAIN CONCEPTS: 2A. RISK
You can invest in two projects:
1) earns $100 with certainty in one year
2) earns $0 or $200 with equal probability in one year
Which do you prefer? Or, how much are you willing to pay for each investment?
For most people: P1 > P2, which reflects risk preference
, • Quantified as: P1 = 100 / (1 + R risk free) vs P2 = E(CF) / (1 + R risky) where
R risky = R risk free + risk premium
• The risk premium is lower for government bonds (which are on average likely to repay) than for
corporate bonds (which are on average less likely to repay as the firm may end up bankrupt)
TWO MAIN CONCEPTS: 2B. RISK
You can invest in two projects and each projects gives 0 or 200 with equal probability in one year. However:
1) Earns 0 when the economy is doing worse than average and 200 when the economy is doing better than
average
2) Earns 200 when the economy is doing worse than average and 0 when the economy is doing better than
average
Which do you prefer? Or, how much are you willing to pay for each investment?
For most people, P1 < P2, which reflects risk preference
• Quantified as P1 = E(CF) / (1 + R risky-1) vs P2 = E(CF) / (1 + R risky-2)
[R risky-1 = R risk-free + risk premium (1)] > [ R risky-2 = R risk-free + risk premium (2)]
• The appropriate risk premium is higher the more the project’s cash flows correlate with the overall
economy
o People like project (2) as it pays off exactly when times are bad
o Later, we introduce models to estimate:
Risk premium = amount of risk x price of risk
The second asset has a higher price, but means it has a lower risk premium (investors are willing to pay a high
price for this investment because it pays of exactly when they need the money) → provides additional money
exactly when investors need it (bad state of economy)
THE THREE RULES OF TIME TRAVEL
1. Only cash flows at the same point in time can be compared or combined
2. To move a cash flow backward in time, you must discount it:
PV = Cn / (1 + r)^n
3. To move cash flow forward in time, you must compound it:
FVn = C0 x (1 + r)^n
Notation:
Cn = Cash flow at date n
r = discount rate
PV = present value (today)
FVn = future value on date n
EXAMPLE: COMPARING CF’S AT SAME POINT IN TIME
Which alternative do you prefer if the one year interest rate r = 4%?
1) €100 today
2) €103 in 1 year time
You prefer option 1, because you can get €104 with the interest rate in 1 year time. Even if you plan to use
money only next year, use the bank to move money forward in time
, Strategies Cash flow today Cash flow in one year
1A. Take alternative 1 100 0
1B. take alternative 1 and put money in bank 100 – 100 = 0 0 + 100 * (1 + r) = 104
account
2A. Take alternative 2 0 103
2B. Take alternative 2 and borrow PV of 103 today 0 + 103 / (1+ r) = 99 103 -103 = 0
→ Comparing Present Value and Future Value is equivalent (1A dominates 2B and 1B dominates 2A)
Thus, comparing cash flows at the same point in time (rule #1) is not just a paper trick, we can use the bank by
saving or borrowing to actually do it
EXAMPLE: COMPOUNDING AND DISCOUNTING
What is the future value of putting €100 in a bank account for 2 years at 4%?
C1 = C0 x (1 + r) = 100 x (1 + r) = 104
C2 = C1 x (1 + r) = C0 x (1 + r) x (1 + r) = C0 x (1 +r)^2
C2 = 100 x 1.04^2 = 108.16
Thus, FVn = C0 x (1 + r)^n
What is the present value (or today’s worth) of a cash-flow of €110 received in 2 years?
Put an amount C0 in a bank account so that it accumulates to C2 = 110 in two years.
Hence 110 = C0 x 1.04^2 → C0 = .04^2 = 101.70
Thus, PV = Cn / (1 + r)^n
PRESENT VALUE AND NPV DECISION RULES
Discounting is used by financial managers to decide which project(s) to invest in. The Net Present Value (NPV)
of a project or investment:
NPV = PV(benefits) – PV(costs)
NPV measures increase in today’s firm value from taking a project that will provide some cash flows in the
future.
NPV decision rules:
• Good projects have NPV > 0
• When choosing among positive NPV projects, choose the largest NPV
• With a budget, choose the combination of projects that maximizes the sum of the NPV’s
EXAMPLE
Consider the following investment opportunity:
Invest €100 today
Receive €30 at the end of year 1; Receive €75 at the end of year 2
The interest rate is 4%
Should we make this investment?
, Not simply a €5 profit; cash flows don’t accrue on same date
PV (benefits) = .04 + .04^2 = 98.1
PV (costs) = 100
NPV = -100 + 98.1 = -1.9 → do not invest!
A related concept that will come back in this course: Internal Rate of Return:
IRR is the discount rate that sets the NPV of an investment equal to zero
In many cases, but not always, higher IRR is better
Example: -100 + 30 / (1 + IRR) + 75 / (1 + IRR)^2 = 0 → IRR = 2.89% (→ with Excel solver)
EXAMPLE: CHOOSING BETWEEN INVESTMENTS
Problem: You have $10,000 to invest and are considering three one-year risk-free investment options
1) Invest up to $10,000 in a Treasury-Bill (a one-year US government bond) paying 2%
Note, 2% is the risk-free interest rate
2) Invest in a project that cost $6,000 and returns $6,100 in one year
3) Invest in a project that costs $4,000 and returns $4,100 in one year
How should the $10,000 investment be allocated?
1) Investing $10,000 in the T-Bill
NPV = $10,000 (1.02) / 1.02 – $10,000 = $0.00
2) Investing $6,000 and receiving $6,100
NPV = $6,.02 - $6,000 = -$19.61
3) Investing $4,000 and receiving $4,100
NPV = $4,.02 - $4,000 = $19.61
Which one to choose? → Option 3. Option 1 creates no value, option 2 destroys value
ARBITRAGE AND THE LAW OF ONE PRICE
Arbitrage – the practice of buying and selling equivalent goods in different markets to take advantage of a price
difference. An arbitrage opportunity occurs when it is possible to make a profit without taking any risk or
making any investment → an investment that doesn’t cost anything today, but you have a positive probability
of making some money (positive cash flow) in the future
1. When the Law of One Price (two assets that give exactly the same future pay off should have the same
price) is violated there is an arbitrage opportunity:
a. The same asset’s price is different in two markets (Royal Dutch Shell shares are traded on the
Amsterdam, London and New York stock exchanges)
2. When an asset’s price ≠ PV (future cash flows)
a. If you have a risk free investment opportunity that gives you 3% return even though the risk
free interest rate is 2%, you have an arbitrage opportunity. A risk free investment should give
the return that is exactly compensating for the risk free interest rate of 2%
Normal market – a competitive market in which there are no arbitrage opportunities (and the Law of One Price
Holds) → assumed throughout the course and not too far form reality
