Real Investments - physical, tangible assets that are acquired in order to generate future cash flows.
Capital budgeting, a topic in Corporate finance, is focused on the decision to make real investments.
Example: A factory is a real asset and the decision to build a new one would be studied in corporate finance.
Financial Investments - Contracts that provide a claim to future cash flows and possibly ownership rights to the firm.
Are just electronic records (one exception is bearer bonds).
Corporations sell financial securities to investors in order to obtain the funds necessary to make real investments.
Investments boils down to buying and selling sets of cash flows.
FI 623: Chapter 1
2
The St.Petersburg Paradox
We could consider all financial investments to be “gambles” or “lotteries.”
The price of a financial investment is the price you pay to enter the gamble.
FI 623:
Chapter 1
A classic example of a lottery:
Round 1: Initial pot is $1. I flip a fair coin – if it comes up heads you win $1 and the game is over, if it comes up tails we double the pot and go to the next round.
Round 2: Pot=$2, heads you win the pot and game over, tails we double the pot and go to the next round.
Round 3: Pot=$4, heads you win the pot and game over, tails we double the pot and go to the next round.
Round N: Pot=$2N−1 dollars, heads you win $2N−1 , tails we double the pot and go to the next round...
How much would you personally pay to play this gamble?
FI 623: Chapter 1
3
The St.Petersburg Paradox
What is the expected amount to be won from this gamble?
FI 623:
Chapter 1
To answer this, let’s think about a simpler gamble: I flip a coin once and then if it turns up heads you pay me $5, and if it turns up tails I pay you $5.
It is clear that the expected winnings from this simple gamble are zero.
But how do we express this mathematically?
1
2
probability of heads
×
−$5 winnings if heads
+
1
2
probability of tails
×
+$5 winnings if tails
=0
We can use this approach to computing the expected value of any gamble:
Multiply the probability of each outcome by the winnings associated with the outcome Sum these terms representing the probability-weighted winnings
FI 623: Chapter 1
4
The St.Petersburg Paradox
Expected amount won if you play the St.Petersburg gamble:
FI 623:
Chapter 1
(probability of ending in round 1) × (pot in round 1)
+
(probability of ending in round 2) × (pot in round 2)
+
..
.
(probability of ending in round 3) × (pot in round 3)
+
(probability of ending in round N) × (pot in round N)
which can be calculated as:
1
1
=
($1) +
2
4
($2) +
1
8
=
1
2
4
2N−1
+ + + ... + N + ...
2
4
8
2
=
1
1
1
1
+ + + ... + + ...
2
2
2
2
=
∞
FI 623: Chapter 1
($4) + . . . +
1
2N
($2N−1 ) + . . .
5
The St.Petersburg Paradox
Would you pay $∞ to play this gamble? Of course not! But the important question is why not?
FI 623:
Chapter 1
Point #1: The price of a gamble (security) depends on more than just the expected winnings (expected return).
Point #2: Risk is a crucial input into the price or value of this gamble, but how do we measure risk?
Point #3: A realistic accounting of the value of this gamble must consider the decreasing marginal utility of wealth that nearly all people exhibit. This decreasing marginal utility of wealth is equivalent to risk aversion when it comes to gambling.
The amount a person is willing to pay to enter the St.Petersburg gamble is related to their level of risk aversion (#2) and their level of wealth (#3).
FI 623: Chapter 1
