Let’s say you want to maximise your expected utility. You know the probability density function of the closing price, p(x). Let’s also suppose you know your own utility function, U(y). Let’s say last night’s closing price was x’, and you currently hold Z in assets.
Then if yesterday you were all in cash, your expected utility is U(Z) if you stay in cash, and integral x=-infinity to x=infinity of U((Z-2) (x / x’) p(x)) dx if you switch to stocks.
If yesterday you were all in stocks, your expected utility is U(Z-2) if you switch to cash, and integral x=-infinity to x=infinity of U(Z (x / x’) p(x))dx if you stay in stocks.
So choose the larger utility.
Or, to make it much simpler, let’s say you’re trying to maximise your expected return the next day. If you’re in cash, and integral (Z −2) (x/x’) p(x)dx > Z, switch to shares, otherwise stay in cash. If you’re in shares, and integral Z * (x/x’) p(x)dx < Z −2, switch to cash, otherwise stay in stocks.
Reasonable but still missing a piece. Even after determining your wealth, your utility function has to take whether-you-are-currently-holding-stocks as an input, because it affects the probability that you incur a transaction cost in future time steps. I think this piece cannot be evaluated without supposing some pdf of future pdfs. I think this is why people are saying the problem is “underspecified”.
Let’s say you want to maximise your expected utility. You know the probability density function of the closing price, p(x). Let’s also suppose you know your own utility function, U(y). Let’s say last night’s closing price was x’, and you currently hold Z in assets.
Then if yesterday you were all in cash, your expected utility is U(Z) if you stay in cash, and integral x=-infinity to x=infinity of U((Z-2) (x / x’) p(x)) dx if you switch to stocks. If yesterday you were all in stocks, your expected utility is U(Z-2) if you switch to cash, and integral x=-infinity to x=infinity of U(Z (x / x’) p(x))dx if you stay in stocks.
So choose the larger utility.
Or, to make it much simpler, let’s say you’re trying to maximise your expected return the next day. If you’re in cash, and integral (Z −2) (x/x’) p(x)dx > Z, switch to shares, otherwise stay in cash. If you’re in shares, and integral Z * (x/x’) p(x)dx < Z −2, switch to cash, otherwise stay in stocks.
Reasonable but still missing a piece. Even after determining your wealth, your utility function has to take whether-you-are-currently-holding-stocks as an input, because it affects the probability that you incur a transaction cost in future time steps. I think this piece cannot be evaluated without supposing some pdf of future pdfs. I think this is why people are saying the problem is “underspecified”.