There was a question in a textbook to the effect:
Suppose you have inherited \$10,000 and are considering how to invest it. You can:
- Lend the \$10,000 to your friend at an interest rate of 7%, but you believe there is a 7% chance that your friend will default.
- Pay \$50 to find out extra information about your friend that would tell you with certainty whether they would default.
Which has the higher expected return?
Re. the question itself, I take it that, if $p$ is the probability of default, $q=1-p$ is the probability that they do not default, $V$ is your principal, $c$ are the information costs, $i$ is the interest rate, and $r$ is the fraction recovered upon default, then we are comparing expected pay-outs $E=qV\left(1+i\right)+\left(1-q\right)Vr$, versus $E'=q'V\left(1+i\right)+\left(1-q'\right)Vr-c$. So, taking $r=0$ and $q'=1$, the scenarios are:
- $E=0.93 \cdot \$10000 \cdot 1.07 = \$9951$
- $E' = \$10000 \cdot 1.07 - 50=\$10650$
... right? But this got me thinking: which of $c$, $q$, $i$, and $r$ are most important to determining $E$? Rearranging, I get that $E'=E$ when
$$f=\color{blue}{\left(q'-q\right)}V\color{green}{\left(1+i-r\right)}c^{-1}-1=0$$
When $f>0$, $E'>E$ (i.e. it is worth paying $c$ to reduce the probability of default to $1-q'$).
When $f<0$, $E'<E$ (i.e. it is not worth paying).
But I noticed something when plugging in some numbers. E.g. keeping the values in the question above $V=\$10000, r=0, i = 0.07, c = 50$ and varying $q'$, it becomes worth your while to pay \$50 to reduce the probability of default from 7% to about 6.5%. Let's call this 6.5% figure $p^{*}$.
However, doubling $i$ from 7% to 14% keeps $p^{*}$ at 6.5% (to 1 s.f.). Whereas doubling $c$ from \$50 to \$100, with $i$ at 7%, changes $p^{*}$ to 6.0%.
So it seems like $c$ is more important to $p^{*}$ (and to $E$) than $i$ is? I have come up with a few tentative explanations for this - my question is, which is correct (if any)?
- Letting $x=\color{blue}{q'-q}$ and $y=\color{green}{1+i-r}$ (since $i$ and $r$ seem to do broadly similar things) and taking the derivative:
$$df=\frac{\partial{f}}{\partial{V}}dV+\frac{\partial{f}}{\partial{c}}dc+\frac{\partial{f}}{\partial{x}}dx+\frac{\partial{f}}{\partial{y}}dy=\frac{xy}{c}dV-\frac{Vxy}{c^{2}}dc+\frac{Vy}{c}dx+\frac{Vx}{c}dy$$
Explanation 1a: $c$ is the most important because the $dc$ term has a $c^{2}$ in the denominator; and all the others have a $c$ in the denominator.
Explanation 1b: $c$ is the most important because the $dc$ term has three terms multiplied together in the numerator; and all the others have two terms.
- Explanation 2: considering again:
$$f+1=\color{blue}{\left(q'-q\right)}\color{green}{\left(1+i-r\right)}\color{orange}{\frac{V}{c}}$$
- Since probabilities range between 0 and 1, and we are assuming that $q' \geq q$, $x \in \left[0,1\right]$.
- $r$ is a proportion of something, so $r \in \left[0,1\right]$. Let us suppose that interest rates are capped at 100% (they were in the numbers I was playing around with), so $i \in \left[0,1\right]$. So $y \in \left[0,2\right]$.
- But $\frac{V}{c} \in \left(0,\infty\right)$; this is why $c$ dominates.
Explanation 3: the relevant proportional change to consider is the one to $1+i$, not the one to $i$. Hence I increased $1+i$ proportionally much less than I increased $c$.
I was wondering whether any of these explanations were right, or am I totally barking up the wrong tree?
The original textbook question is not very clear (at least as it is expressed here), but with the assumptions you have in your solution, your general formula and calculated value for $E'$ are not correct. You have effectively assumed that paying for the information makes the probability of default zero. This is not right: you pay for the information before you know whether your friend will default.
Before you pay the $50 for the extra information you believe there is
The expected value calculation gives
$$E′=0.93(10700)+0.07(10000)-50=10601$$
The general formula in your notation is
$$E'=q(1+i)V+(1-q)V-c$$
and
$$E'-E=(1-q)V(1-r)-c.$$
Thus it is better to pay for the information if
$$c<(1-q)V(1-r).$$
Notice that $i$ does not appear in this inequality. All else equal, decreases in $q$ (increases in your prior that your friend defaults), decreases in $r$ (the proportion recovered upon default) and increases in $V$ (the amount you have to lend) make paying for the information more attractive. However, note that it is likely that an increase in the amount lent ($V$) is likely to be accompanied by an increase in the probability of default ($q$ will fall). The proportion that can be recovered upon default could also be related to $V$.