Investeren en Beleggen
Hoofdstuk 3 t/m 13.6
Hoofdstuk 3
A financial manager’s job is to make decisions on behalf of the firm’s investors. They must compare
the costs and benefits and determine the best decision to make for the value of the firm.
Whenever a good trades in a competitive market (a market in which it can be bought and sold at the
same price), that price determines the cash value of the good.
Valuation Principle: the value of an asset to the firm or its investors is determined by its competitive
market price. The benefits and costs of a decision should be evaluated using these market prices, and
when the value of the benefits exceeds the value of the costs, the decision will increase the market
value of the firm.
For most financial decisions, costs and benefits occur at different points in time. For example, a
typical investment project incurs costs up front and provides benefits in the future.
In general, a dollar today is worth more than a dollar in one year. If u have €1 today, you can invest it
(for example in a bank account paying 7% interest, then after one year you will have €1,07).
Time Value of Money: The difference in value between money today and the money in the future.
By depositing money into a savings account, we can convert money today into money in the future
with no risk. Similarly, by borrowing money from the bank, we can exchange money in the future for
money today. The rate at which we can exchange money today for money in the future is determined
by the current interest rate. An interest rate is like an exchange rate across time. It tells us the market
price today of money in the future.
Risk-free interest rate (rf): the interest rate at which money can be borrowed or lent without risk over
that period.
- The supply of savings = the demand of borrowing
Interest rate factor (1 + rf): for risk free cashflows, has units of “€ in one year/€ today)
Present value (PV): when we express the value of the investment in terms of dollars today
Future value (FV): when we express the value of the investment in terms of dollars in the future.
1
Discount Factor: Provides the discount at which we can purchase money in the future –> 1 + 𝑟
Discount rate: also known as the risk-free interest rate for a risk-free investment
Net Present Value (NPV): the difference between the present value of the benefits and the present
value of the costs (of a project/investment).
- NPV = PV(Benefits) – PV(Costs) = PV(All project cashflows)
- The NPV expresses the value of an investment decision ass an amount of cash received today.
- Als long as the NPV is positive, the decision increases the value of the firm and is a good
decision regardless of your current cash needs or preferences regarding when to spend the
money.
,NPV Decision Rule: When making an investment decision, take the alternative with the highest NPV.
Choosing this alternative is equivalent to receiving its NPV in cash today.
- Accept projects with positive NPV because accepting them is equivalent to receiving their
NPV in cash today
- Reject projects with negative NPV because accepting them would reduce the wealth of
investors, whereas not doing them has no cost (NPV = 0).
- If the NPV is exactly zero, you will neither gain nor lose by accepting the project rather than
rejecting it.
- Regardless of our preferences for cash today versus cash in the future, we should always
maximize NPV first. We can then borrow or lend to shift cash flows through time and find our
most preferred pattern of cash flows.
Arbitrage: The practice of buying and selling equivalent goods in different markets to take advantage
of a price difference.
Arbitrage opportunity: a situation in which it is possible to make a profit without taking any risk or
making any investment.
Normal market: a competitive market in which there are no arbitrage opportunities.
Law of One Price: If equivalent investment opportunities trade simultaneously in different
competitive markets, then they must trade for the same price in all markets.
- If the prices in two markets differ, investors will profit immediately by buying in the market
where it is cheap and selling in the market where it is expensive. In doing so, they will
equalize the prices.
- Because of this law, we can use any competitive price to determine a cash value, without
checking the price in all possible markets.
(Financial) security: an investment opportunity that trades in a financial market.
- Bond: a security sold by governments and corporations to raise money form investors today
in exchange for the promised future payment.
In financial markets, it is possible to sell a security you do not own by doing a short sale.
- Short sale: the person who intends to sell the security first borrows it from someone who
already owns it. Later that person must either return the security by buying it back or pay the
owner the cash flows he or she would have received.
No-arbitrage price: the price in a normal market.
Pricing securities:
1. Identify the cash flows that will be paid by the security
2. Determine the “do-it-yourself” cost of replicating those cash flows on our own
a. That is, the present value of the security’s cash flows.
➔ No-Arbitrage Price of a Security: Price (Security) = PV (All cash flows paid by the security).
Risk-free interest rate → 1 + r = (Cashflow in one year)/(Cashflow today)
Return = Gain at End of Year/Initial Cost
,Risk-free interest rate = Return → When there is no arbitrage → All risk-free investments should offer
investors the same return.
Buying securities at no-arbitrage prices: NPV (Buy security) = PV (All cash flows paid by the security) –
Price (Security) = 0
Selling securities at no-arbitrage prices: NPV (Buy security) = Price (Security) - PV (All cash flows paid
by the security) = 0
Because arbitrage opportunities do not exist in normal markets, the NPV of all security trades must
be zero.
Separation Principle: We can evaluate a decision by focusing on its real components, rather than its
financial ones. So we can separate the firm’s investment decision from its financing choice.
- Security transactions in a normal market neither create nor destroy value on their own.
Therefore, we can evaluate the NPV of an investment decision separately from the decision
the firm makes regarding how to finance the investment or any other security transactions
the firm is considering.
Value Additivity: Price (C) = Price (A+B) = Price (A) + Price (B)
- Because security C is equivalent to the portfolio of A and B, by the Law of One Price, they
must have the same price.
- Value additivity implies that the value of a portfolio is equal to the sum of the values of its
parts.
- The price or the value of the entire firm is equal to the sum of the values of all projects and
investments within it.
To maximize the value of the entire firm, managers should make decisions that maximize NPV. The
NPV of the decision represents its contribution to the overall value of the firm.
Hoofdstuk 4
Stream of cash flows: A series of cash flows lasting several periods.
Timeline: A linear representation of the timing of the expected cash flows
- Inflows = positive cash flows
- Outflows = negative cash flows
Three rules of Time Travel
1. Comparing and combining values: It is only possible to compare or combine values at the
same point in time.
a. To compare or combine cash flows that occur at different points in time, you first
need to convert the cash flows into the same units of move them to the same point
in time.
2. Moving cash flows forward in time: to move the cash flow forward in time, you must
compound it
a. Compounding = the process of moving a value or cash flow forward in time.
b. Compound interest: earning “interest on interest”
c. Future value of a cash flow: 𝐹𝑉𝑛 = 𝐶 × (1 + 𝑟)𝑛
i. C = cashflow
, ii. n = periods
iii. r = interest rate (constant)
3. Moving cash flows back in time: to move the cash flow back in time, you must discount it.
a. Discounting = the process of moving a value or cash flow backward in time; finding
the equivalent value today of a future cash flow.
b. Present value of a cash flow: 𝑃𝑉 = 𝐶 ÷ (1 + 𝑟)𝑛
i. C = cashflow
ii. n = periods
iii. r = interest rate (constant)
𝐶
Present value of a Cash Flow Stream: 𝑃𝑉 = ∑𝑁 𝑁 𝑛
𝑛=0 𝑃𝑉(𝐶𝑛 ) = ∑𝑛=0 (1+𝑟)𝑛
Future value of a Cash Flow with a Present Value: 𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟)𝑛
Net present value (NPV): 𝑁𝑃𝑉 = 𝑃𝑉(𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠) − 𝑃𝑉(𝑐𝑜𝑠𝑡𝑠) = 𝑃𝑉(𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠 − 𝑐𝑜𝑠𝑡𝑠)
Perpetuity: a stream of equal cash flows that occur at regular intervals and last forever.
- The first cash flow does not occur immediately, it arrives at the end of the first period →
payment in arrears
𝐶
- 𝑃𝑉 = ∑∞
𝑛=1 (1+𝑟)𝑛
o Cn = C because the cash flow for a perpetuity is constant
𝐶
- 𝑃𝑉 (𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) = 𝑟
Annuity: a stream of N equal cash flows that is paid at regular intervals. The annuity ends after some
fixed number of payments.
𝐶
- 𝑃𝑉 = ∑𝑁
𝑛=1 (1+𝑟)𝑛
1 1
- 𝑃𝑉 (𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑜𝑓 𝐶 𝑓𝑜𝑟 𝑁 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑤𝑖𝑡ℎ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑟 = 𝐶 × 𝑟 (1 − (1+𝑟)𝑁)
o C = periodic payment = r x P
o P = initial investment
o N = periods
- 𝐹𝑉 (𝑎𝑛𝑛𝑢𝑖𝑡𝑦) = 𝑃𝑉 × (1 + 𝑟)𝑁
1
- 𝐹𝑉 = 𝐶 × 𝑟 ((1 + 𝑟)𝑁 − 1)
Growing perpetuity: a stream of cash flows that occur at regular intervals and grow at a constant rate
forever.
- The first payment occurs at date 1
- The first payment does not include growth
- g<r
𝐶(1+𝑔)𝑛−1
- 𝑃𝑉 = ∑∞
𝑛=1 (1+𝑟)𝑛
o g = growth
𝐶
- 𝑃𝑉 (𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) = 𝑟−𝑔
- The present value of the perpetuity is the first cash flow divided by the difference between
the interest rate and the growth rate
Growing annuity: a stream of N growing cash flows, paid at regular intervals. It is a growing
perpetuity that eventually comes to an end.
Hoofdstuk 3 t/m 13.6
Hoofdstuk 3
A financial manager’s job is to make decisions on behalf of the firm’s investors. They must compare
the costs and benefits and determine the best decision to make for the value of the firm.
Whenever a good trades in a competitive market (a market in which it can be bought and sold at the
same price), that price determines the cash value of the good.
Valuation Principle: the value of an asset to the firm or its investors is determined by its competitive
market price. The benefits and costs of a decision should be evaluated using these market prices, and
when the value of the benefits exceeds the value of the costs, the decision will increase the market
value of the firm.
For most financial decisions, costs and benefits occur at different points in time. For example, a
typical investment project incurs costs up front and provides benefits in the future.
In general, a dollar today is worth more than a dollar in one year. If u have €1 today, you can invest it
(for example in a bank account paying 7% interest, then after one year you will have €1,07).
Time Value of Money: The difference in value between money today and the money in the future.
By depositing money into a savings account, we can convert money today into money in the future
with no risk. Similarly, by borrowing money from the bank, we can exchange money in the future for
money today. The rate at which we can exchange money today for money in the future is determined
by the current interest rate. An interest rate is like an exchange rate across time. It tells us the market
price today of money in the future.
Risk-free interest rate (rf): the interest rate at which money can be borrowed or lent without risk over
that period.
- The supply of savings = the demand of borrowing
Interest rate factor (1 + rf): for risk free cashflows, has units of “€ in one year/€ today)
Present value (PV): when we express the value of the investment in terms of dollars today
Future value (FV): when we express the value of the investment in terms of dollars in the future.
1
Discount Factor: Provides the discount at which we can purchase money in the future –> 1 + 𝑟
Discount rate: also known as the risk-free interest rate for a risk-free investment
Net Present Value (NPV): the difference between the present value of the benefits and the present
value of the costs (of a project/investment).
- NPV = PV(Benefits) – PV(Costs) = PV(All project cashflows)
- The NPV expresses the value of an investment decision ass an amount of cash received today.
- Als long as the NPV is positive, the decision increases the value of the firm and is a good
decision regardless of your current cash needs or preferences regarding when to spend the
money.
,NPV Decision Rule: When making an investment decision, take the alternative with the highest NPV.
Choosing this alternative is equivalent to receiving its NPV in cash today.
- Accept projects with positive NPV because accepting them is equivalent to receiving their
NPV in cash today
- Reject projects with negative NPV because accepting them would reduce the wealth of
investors, whereas not doing them has no cost (NPV = 0).
- If the NPV is exactly zero, you will neither gain nor lose by accepting the project rather than
rejecting it.
- Regardless of our preferences for cash today versus cash in the future, we should always
maximize NPV first. We can then borrow or lend to shift cash flows through time and find our
most preferred pattern of cash flows.
Arbitrage: The practice of buying and selling equivalent goods in different markets to take advantage
of a price difference.
Arbitrage opportunity: a situation in which it is possible to make a profit without taking any risk or
making any investment.
Normal market: a competitive market in which there are no arbitrage opportunities.
Law of One Price: If equivalent investment opportunities trade simultaneously in different
competitive markets, then they must trade for the same price in all markets.
- If the prices in two markets differ, investors will profit immediately by buying in the market
where it is cheap and selling in the market where it is expensive. In doing so, they will
equalize the prices.
- Because of this law, we can use any competitive price to determine a cash value, without
checking the price in all possible markets.
(Financial) security: an investment opportunity that trades in a financial market.
- Bond: a security sold by governments and corporations to raise money form investors today
in exchange for the promised future payment.
In financial markets, it is possible to sell a security you do not own by doing a short sale.
- Short sale: the person who intends to sell the security first borrows it from someone who
already owns it. Later that person must either return the security by buying it back or pay the
owner the cash flows he or she would have received.
No-arbitrage price: the price in a normal market.
Pricing securities:
1. Identify the cash flows that will be paid by the security
2. Determine the “do-it-yourself” cost of replicating those cash flows on our own
a. That is, the present value of the security’s cash flows.
➔ No-Arbitrage Price of a Security: Price (Security) = PV (All cash flows paid by the security).
Risk-free interest rate → 1 + r = (Cashflow in one year)/(Cashflow today)
Return = Gain at End of Year/Initial Cost
,Risk-free interest rate = Return → When there is no arbitrage → All risk-free investments should offer
investors the same return.
Buying securities at no-arbitrage prices: NPV (Buy security) = PV (All cash flows paid by the security) –
Price (Security) = 0
Selling securities at no-arbitrage prices: NPV (Buy security) = Price (Security) - PV (All cash flows paid
by the security) = 0
Because arbitrage opportunities do not exist in normal markets, the NPV of all security trades must
be zero.
Separation Principle: We can evaluate a decision by focusing on its real components, rather than its
financial ones. So we can separate the firm’s investment decision from its financing choice.
- Security transactions in a normal market neither create nor destroy value on their own.
Therefore, we can evaluate the NPV of an investment decision separately from the decision
the firm makes regarding how to finance the investment or any other security transactions
the firm is considering.
Value Additivity: Price (C) = Price (A+B) = Price (A) + Price (B)
- Because security C is equivalent to the portfolio of A and B, by the Law of One Price, they
must have the same price.
- Value additivity implies that the value of a portfolio is equal to the sum of the values of its
parts.
- The price or the value of the entire firm is equal to the sum of the values of all projects and
investments within it.
To maximize the value of the entire firm, managers should make decisions that maximize NPV. The
NPV of the decision represents its contribution to the overall value of the firm.
Hoofdstuk 4
Stream of cash flows: A series of cash flows lasting several periods.
Timeline: A linear representation of the timing of the expected cash flows
- Inflows = positive cash flows
- Outflows = negative cash flows
Three rules of Time Travel
1. Comparing and combining values: It is only possible to compare or combine values at the
same point in time.
a. To compare or combine cash flows that occur at different points in time, you first
need to convert the cash flows into the same units of move them to the same point
in time.
2. Moving cash flows forward in time: to move the cash flow forward in time, you must
compound it
a. Compounding = the process of moving a value or cash flow forward in time.
b. Compound interest: earning “interest on interest”
c. Future value of a cash flow: 𝐹𝑉𝑛 = 𝐶 × (1 + 𝑟)𝑛
i. C = cashflow
, ii. n = periods
iii. r = interest rate (constant)
3. Moving cash flows back in time: to move the cash flow back in time, you must discount it.
a. Discounting = the process of moving a value or cash flow backward in time; finding
the equivalent value today of a future cash flow.
b. Present value of a cash flow: 𝑃𝑉 = 𝐶 ÷ (1 + 𝑟)𝑛
i. C = cashflow
ii. n = periods
iii. r = interest rate (constant)
𝐶
Present value of a Cash Flow Stream: 𝑃𝑉 = ∑𝑁 𝑁 𝑛
𝑛=0 𝑃𝑉(𝐶𝑛 ) = ∑𝑛=0 (1+𝑟)𝑛
Future value of a Cash Flow with a Present Value: 𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟)𝑛
Net present value (NPV): 𝑁𝑃𝑉 = 𝑃𝑉(𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠) − 𝑃𝑉(𝑐𝑜𝑠𝑡𝑠) = 𝑃𝑉(𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠 − 𝑐𝑜𝑠𝑡𝑠)
Perpetuity: a stream of equal cash flows that occur at regular intervals and last forever.
- The first cash flow does not occur immediately, it arrives at the end of the first period →
payment in arrears
𝐶
- 𝑃𝑉 = ∑∞
𝑛=1 (1+𝑟)𝑛
o Cn = C because the cash flow for a perpetuity is constant
𝐶
- 𝑃𝑉 (𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) = 𝑟
Annuity: a stream of N equal cash flows that is paid at regular intervals. The annuity ends after some
fixed number of payments.
𝐶
- 𝑃𝑉 = ∑𝑁
𝑛=1 (1+𝑟)𝑛
1 1
- 𝑃𝑉 (𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑜𝑓 𝐶 𝑓𝑜𝑟 𝑁 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑤𝑖𝑡ℎ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑟 = 𝐶 × 𝑟 (1 − (1+𝑟)𝑁)
o C = periodic payment = r x P
o P = initial investment
o N = periods
- 𝐹𝑉 (𝑎𝑛𝑛𝑢𝑖𝑡𝑦) = 𝑃𝑉 × (1 + 𝑟)𝑁
1
- 𝐹𝑉 = 𝐶 × 𝑟 ((1 + 𝑟)𝑁 − 1)
Growing perpetuity: a stream of cash flows that occur at regular intervals and grow at a constant rate
forever.
- The first payment occurs at date 1
- The first payment does not include growth
- g<r
𝐶(1+𝑔)𝑛−1
- 𝑃𝑉 = ∑∞
𝑛=1 (1+𝑟)𝑛
o g = growth
𝐶
- 𝑃𝑉 (𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) = 𝑟−𝑔
- The present value of the perpetuity is the first cash flow divided by the difference between
the interest rate and the growth rate
Growing annuity: a stream of N growing cash flows, paid at regular intervals. It is a growing
perpetuity that eventually comes to an end.
