<p>It is this probability that is consistent with shares trading for their expected value. The expected value is given by:
<span class="math-container">$$E(S_1+S_2)=E(S_1)+E(S_2)=p_1 \times1+p_2 \times1=p_1+p_2$$</span></p>
<p>where <span class="math-container">$S_i$</span> is the payment of share <span class="math-container">$i$</span>, <span class="math-container">$p_i$</span> is the probability of payment of share <span class="math-container">$i$</span> (due to the sales figure falling in some range). So if the expected value is to equal the price of the security, then:</p>
<p><span class="math-container">$$E(S_1+S_2)=P_1+P_2 \Rightarrow p_1+p_2=P_1+P_2 $$</span></p>
<p>Where <span class="math-container">$P_i$</span> denotes the price of share <span class="math-container">$i$</span>. So the probability of one of them paying off equals the sum of their prices (note that the events of share 1 and 2 paying off are mutually exclusive).</p>
<p>In your case</p>
<p><span class="math-container">$P_1=$$\\\$$$0.20$</span></p>
<p><span class="math-container">$P_2=$$\\\$$$0.10$</span></p>
<p>Of course these calculations totally disregard the many complications that go into asset pricing (like discounting cash flows, accounting for having to wait for payment), but I assume the authors want to keep it simple to prove a point. The bigger picture you should take away from this answer is that share prices can be interpreted as reflecting information about (what investors believe to be) probabilities of future events.</p>

CC BY-SA 4.0
2021-04-11T10:45:21.503Z
2021-04-11T10:51:04.047Z
2021-04-11T10:51:04.047Z
user18214
<p>The discussion on prediction markets in the chapter on the price system in Tyler Cowen and Alex Tabarrok's <em>Modern Principles of Economics, Second Edition</em> (pp. 123-124) contains the following paragraphs:</p>
<blockquote>
<p>Members of HP’s sales team bought and sold shares that paidoff when sales fell within a certain range. A typical security would pay out <span class="math-container">$\\\$1$</span>,
if and only if future sales were, say, between <span class="math-container">$10,000$</span> and <span class="math-container">$15,000$</span> units. Another
might pay off if sales were between <span class="math-container">$15,000$</span> and <span class="math-container">$20,000$</span> units. The market con-
tained <span class="math-container">$10$</span> types of securities—a range broad enough to include all the relevant
possible sales outcomes.<br />
By examining the prices of all <span class="math-container">$10$</span> shares, HP could assign a probability to any
combination of outcomes. For example, <strong>if the price of the <span class="math-container">$5,000–10,000$</span> unit
sales security was <span class="math-container">$10$</span> cents and the price of the <span class="math-container">$10,000–15,000$</span> unit sales security was <span class="math-container">$20$</span> cents, this suggests that the probability of selling <span class="math-container">$5,000 –15,000$</span> units
was <span class="math-container">$30\%$</span>.</strong></p>
</blockquote>
<p>How did the authors arrive at this figure of <span class="math-container">$30\%$</span>?</p>

CC BY-SA 4.0
2021-04-12T09:24:40.840Z
If people are willing to pay only 10 cents for a chance of getting a dollar, that means they think they're about 10% likely to get a dollar.
Criticize SE actions means ban
<microeconomics><markets>
How did Cowen and Tabarrok calculate the probability of Hewlett-Packard sales?
Ray Bradbury
# How did Cowen and Tabarrok calculate the probability of Hewlett-Packard sales?

(Asked by: username_0 on 2021-04-10)

The discussion on prediction markets in the chapter on the price system in Tyler Cowen and Alex Tabarrok's *Modern Principles of Economics, Second Edition* (pp. 123-124) contains the following paragraphs:

> Members of HP’s sales team bought and sold shares that paidoff when sales fell within a certain range. A typical security would pay out $\\\$1$,
> if and only if future sales were, say, between $10,000$ and $15,000$ units. Another
> might pay off if sales were between $15,000$ and $20,000$ units. The market con-
> tained $10$ types of securities—a range broad enough to include all the relevant
> possible sales outcomes.  
> By examining the prices of all $10$ shares, HP could assign a probability to any
> combination of outcomes. For example, **if the price of the $5,000–10,000$ unit
> sales security was $10$ cents and the price of the $10,000–15,000$ unit sales security was $20$ cents, this suggests that the probability of selling $5,000 –15,000$ units
> was $30\%$.**

How did the authors arrive at this figure of $30\%$?

**Tags:** microeconomics, markets

> **Comments:**
> **username_1**: If people are willing to pay only 10 cents for a chance of getting a dollar, that means they think they're about 10% likely to get a dollar.  

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## 1 answers

### Answer by username_2 on 2021-04-11. Score: 3 | Accepted answer

It is this probability that is consistent with shares trading for their expected value. The expected value is given by:
$$E(S\_1+S\_2)=E(S\_1)+E(S\_2)=p\_1 \times1+p\_2 \times1=p\_1+p\_2$$

where $S\_i$ is the payment of share $i$, $p\_i$ is the probability of payment of share $i$ (due to the sales figure falling in some range). So if the expected value is to equal the price of the security, then:

$$E(S\_1+S\_2)=P\_1+P\_2 \Rightarrow p\_1+p\_2=P\_1+P\_2 $$

Where $P\_i$ denotes the price of share $i$. So the probability of one of them paying off equals the sum of their prices (note that the events of share 1 and 2 paying off are mutually exclusive).

In your case

$P\_1=$$\\\$$$0.20$

$P\_2=$$\\\$$$0.10$

Of course these calculations totally disregard the many complications that go into asset pricing (like discounting cash flows, accounting for having to wait for payment), but I assume the authors want to keep it simple to prove a point. The bigger picture you should take away from this answer is that share prices can be interpreted as reflecting information about (what investors believe to be) probabilities of future events.


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