Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 1
Content:
• The corporation and the financial market
• General evaluation principles
• Present value of some common investments
The corporation and the financial market
In the US there are four major types of firms:
• Sole proprietorships: owned and run by one person
who has unlimited personal liability for the debts.
• Partnerships: has more owners, who are fully liable,
except for limited partnership, where general
partners are fully liable while limited partners’
liability is limited to their investment.
• Limited liability companies (LLC): it is a limited
partnership without a general partner.
• Corporations
The corporation and the financial market
The corporation is a legal entity separate from its
owners (i.e., the latter are not liable for the obligations
of the former, and vice versa).
The ownership is divided into shares. The set of shares
owned by an investor is called “stock”, but these terms
are often used interchangeably to indicate shares.
The collection of all the outstanding shares of a
corporation is known as the equity of the corporation.
The corporation and the financial market
The owners can profit from:
• dividends payment;
• capital gain.
The direct control of the corporation is held by the
board of directors and the CEO.
Agency problem
These issues are dealt with by corporate finance.
The corporation and the financial market
Some or all the shares of a corporation can be privately
held, and they are traded on private markets.
Publicly traded stocks are traded on the stock market.
Primary market: where new shares are first issued.
Secondary market: where existing shares are traded.
The corporation and the financial market
Liquid stocks can be easily converted into cash without
affecting its market price. This allows for flexible
investments and efficient pricing of the investment.
Bid-ask spread: the difference between the price at
which it is possible to buy the stock (bid price) and the
price at which it is possible to sell the stock (ask price).
Bid price is always higher than ask price, but this spread
is small if the market is liquid.
The corporation and the financial market
Market makers are intermediaries that quote both a
buy and a sell price to profit on the bid-ask spread.
They increase the liquidity.
Limit order book: the collection of all limit orders
(orders to buy or sell a set amount at a fixed price).
They provide liquidity.
These and other issue are studied by a branch of
finance called market microstructure.
The corporation and the financial market
Time value of money: the difference in value between
money today and money in the future.
Interest rate: The rate at which we can exchange money
today for money in the future
Risk-free interest rate (Rf): the rate at which money can
be borrowed or lent without risk over a certain period.
The corporation and the financial market
Arbitrage: the practice of buying and selling equivalent
goods in different markets to profit from a price difference
Security: an investment opportunity that trades in a
financial market
Short selling: selling a security you do not own by
borrowing it and returning it to the owner at a later date.
The corporation and the financial market
Trading securities implies facing transaction costs:
• the commissions paid to the broker
• the bid-ask spread
Arbitrage keeps the prices of equivalent goods and
securities close to each other: prices cannot deviate
more than the transaction costs of the arbitrage.
General evaluation principles
Given the interest rate R and the number of periods n, the
value of an investment:
• in terms of dollars today is the present value (PV):
𝑃𝑉 =
𝐹𝑉
(1 + 𝑅) 𝑛
• in terms of dollars in the future is the future value (FV):
𝐹𝑉 = 𝑃𝑉 ∗ (1 + 𝑅) 𝑛
(Risky cash flows must be discounted at a rate equal to the
risk-free rate plus an appropriate risk premium.)
General evaluation principles
General process for pricing securities (in a market without
arbitrage opportunities):
1. Identify the cash flows that will be paid by the security
2. Determine the PV of the security’s cash flows
Unless the price of the security equals this present value,
there is an arbitrage opportunity:
No-Arbitrage Price = PV(All cash flows paid by the security)
General evaluation principles
It is only possible to compare or combine values at the
same point in time.
To move a cash flow C forward in time you must compound
it (compute the future value):
𝐹𝑉𝑛 = 𝐶 ∗ (1 + 𝑅) 𝑛
To find the equivalent value today of a future cash flow C
you must discount it (compute the present value):
𝑃𝑉 =
𝐶
(1 + 𝑅) 𝑛
General evaluation principles
Present Value of a Cash Flow Stream:
𝑃𝑉 = ෍
𝑛=0
𝑁
𝑃𝑉(𝐶 𝑛) = ෍
𝑛=0
𝑁
𝐶 𝑛
(1 + 𝑅) 𝑛
Future Value of a Cash Flow Stream with Present Value PV:
𝐹𝑉𝑛 = 𝑃𝑉 ∗ (1 + 𝑅) 𝑛
Present value of some common investments
A perpetuity is a stream of equal cash flows that occur
at regular intervals and last forever.
The present value of a perpetuity with payment C and
interest rate R is given by
𝑃𝑉 = ෍
𝑛=1
∞
𝐶
(1 + 𝑅) 𝑛
Because the first cash flow is in one period, C0 = 0.
It can be shown that:
𝑃𝑉(𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝐶
𝑅
Present value of some common investments
An annuity is a stream of N equal cash flows paid at
regular intervals (the first one at date 1). At the end of it,
you get your initial investment back.
The present value of an N-period annuity with payment
C and interest rate R is 𝑃𝑉 = σ 𝑛=1
𝑁 𝐶
(1+𝑅) 𝑛
It can be shown that the PV of an annuity is:
𝑃𝑉 = 𝐶
1
𝑅
1 −
1
(1 + 𝑅) 𝑁
The Future Value of an Annuity is:
𝐹𝑉 = 𝐶
1
𝑅
((1 + 𝑅) 𝑁
−1)
Present value of some common investments
A growing perpetuity is a stream of cash flows that occur at
regular intervals and grow at a constant rate forever. A
growing perpetuity with a first payment C and a growth rate
g has present value
𝑃𝑉 = ෍
𝑛=1
∞
𝐶(1 + 𝑔) 𝑛−1
(1 + 𝑅) 𝑛
It must be g < R, so that each successive term in the sum is
less than the previous term and the overall sum is finite.
It can be shown that:
𝑃𝑉(𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝐶
𝑅 − 𝑔
Present value of some common investments
A growing annuity is a stream of N growing cash flows, paid
at regular intervals. The first cash flow arrives at the end of
the first period, and the first cash flow does not grow. The
present value of an N-period growing annuity with initial
cash flow C, growth rate g, and interest rate r is given by
𝑃𝑉 = 𝐶
1
𝑅 − 𝑔
1 −
1 + 𝑔
1 + 𝑅
𝑁
Because the annuity has only a finite number of terms, we
can have g > R. The formula does not work for g = R, but in
that case growth and discounting cancel out: 𝑃𝑉 = 𝐶
𝑁
1+𝑅