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金融代写|投资组合代写Investment Portfolio代考|INFINITE PAYOFFS, LIMITED COSTS-THE ST. PETERSBURG PARADOX

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金融代写|投资组合代写Portfolio Theory代考|INFINITE PAYOFFS, LIMITED COSTS-THE ST. PETERSBURG PARADOX

In a previous chapter, we briefly discussed Daniel Bernoulli’s critique of expected value theory. As a counterargument, Bernoulli proposed a gamble: An individual begins with an initial stake of $\$ 2$and we flip a coin. The initial$\$2$ bankroll is doubled every time heads comes up, and when tails comes up the game is over and the winnings are the value of the bankroll. What is interesting about this game is that it has an infinite expected value:
$$\sum_{n=1}^{\infty} \operatorname{Pr}i[\text { Heads }] \times V_i$$ where $\sum{n=1}^{\infty} \operatorname{Pr}i[$ Heads $]$ is the probability of getting heads on the $i^{\text {th }}$ coin flip, and $V_i$ is the payoff of that flip. The probability of winning the first flip is $1 / 2^1$ and the payoff is $2^1$. The probability of winning the second flip is $1 / 2^2$ and the payoff is $2^2$. We can generalize the expected value of the gamble as $$\sum{n=1}^{\infty} \frac{1}{2^i} \times 2^i$$
And it becomes clear that the sum is infinity:
$$\frac{1}{2^1} \times 2^1+\frac{1}{2^2} \times 2^2+\frac{1}{2^3} \times 2^3+\cdots=1+1+1+\cdots=\infty \text {. }$$

金融代写|投资组合代写Portfolio Theory代考|QUESTIONS OF PROBABILITY, QUESTIONS OF DOUBT

In 1953, as a critique of Von Neumann and Morgenstern’s axiom of independence, economist Maurice Allais ${ }^8$ conducted an experiment. He offered his subjects two sets of choices:
1A: $\$ 1,000,000$with$100 \%$probability 1B:$\$1,000,000$ with $89 \%$ probability $\$ 5,000,000$with$10 \%$probability$\$0$ with $1 \%$ probability
The second choice set:
2A: $\$ 1,000,000$with$11 \%$probability$\$0$ with $89 \%$ probability
$2 \mathrm{~B}: \quad \$ 5,000,000$with$10 \%$probability$\$0$ with $90 \%$ probability
Allais found that people tended to choose choice 1A over 1B and also chose $2 \mathrm{~B}$ over $2 \mathrm{~A}$. Though it is not immediately obvious, this is a contradiction!
From the first choice set we learn that $u(1 A)>u(1 B)$, or
$$v(\ 1,000,000)>0.89 v(\ 1,000,000)+0.10 v(\ 5,000,000),$$
which can be simplified to
$$0.11 v(\ 1,000,000)>0.10 v(\ 5,000,000) .$$
In other words, we learn that an $11 \%$ chance of gaining $\$ 1,000,000$carries more utility than a$10 \%$chance of gaining$\$5,000,000$. We learn from the second choice set that $u(2 A)<u(2 B)$, or
$$0.11 v(\ 1,000,000)<0.10 v(\ 5,000,000)$$
which directly contradicts the first choice set!
Allais concluded that the axiom of independence cannot be a valid one because it fails to predict “reasonable people choosing between reasonable alternatives.” Markowitz rebutted that people choosing the wrong alternative acted irrationally, but that people are irrational does not negate the axiom. So the behavioral-normative split was formed.

金融代写|投资组合代写Portfolio Theory代考|INFINITE PAYOFFS, LIMITED COSTS-THE ST. PETERSBURG PARADOX

$$\sum_{n=1}^{\infty} \operatorname{Pr}i[\text { Heads }] \times V_i$$其中$\sum{n=1}^{\infty} \operatorname{Pr}i[$头像$]$是$i^{\text {th }}$抛硬币得到头像的概率，$V_i$是抛硬币的收益。第一次抛掷获胜的概率是$1 / 2^1$，收益是$2^1$。第二次抛掷获胜的概率是$1 / 2^2$，收益是$2^2$。我们可以将赌博的期望值概括为$$\sum{n=1}^{\infty} \frac{1}{2^i} \times 2^i$$

$$\frac{1}{2^1} \times 2^1+\frac{1}{2^2} \times 2^2+\frac{1}{2^3} \times 2^3+\cdots=1+1+1+\cdots=\infty \text {. }$$

金融代写|投资组合代写Portfolio Theory代考|QUESTIONS OF PROBABILITY, QUESTIONS OF DOUBT

1953年，作为对冯·诺伊曼和摩根斯特恩的独立公理的批判，经济学家莫里斯·阿莱${ }^8$进行了一项实验。他给实验对象提供了两组选择:
1A: $\$ 1,000,000$与$100 \%$的概率 1B:$\$1,000,000$带$89 \%$概率$\$ 5,000,000$带$10 \%$概率$\$0$带$1 \%$概率

2A: $\$ 1,000,000$带$11 \%$概率$\$0$带$89 \%$概率
$2 \mathrm{~B}: \quad \$ 5,000,000$与$10 \%$概率$\$0$与$90 \%$概率

$$v(\ 1,000,000)>0.89 v(\ 1,000,000)+0.10 v(\ 5,000,000),$$

$$0.11 v(\ 1,000,000)>0.10 v(\ 5,000,000) .$$

$$0.11 v(\ 1,000,000)<0.10 v(\ 5,000,000)$$

MATLAB代写

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