Problem $2$: Conditional probabilities, total probability theorem, efficient markets?

The dudes at NFQ Ltd$.($No Finance Quacks$)$ know that investors care whether the market

return is positive. It's probably all they know. They want to sell a weekly financial letter that

predicts the sign of the weekly US Market return. They consider $3$ strategies:

Always predict $>0$ return,

Always predict $<mn\; id="EditorElement\_419238">0</mn>$ return,

Randomly predicts $>0$ return $60\%$ of the time. Dudes read somewhere that this is called the

"randomization strategy" and relates to "adaptive expectations", big thing in psychology $!$

They read your published research showing that the weekly market excess return is positive

with probability $p(U)=59\%.$ They found this interesting, so they hired you as a consultant for

this project.

a$)$ Give them the weekly probability of correct prediction and the expected number of correct

predictions after $52$ weeks for each rule. Denote $"$C$"$ the event "correct prediction", $"$U$"$:

market's up$,"$D$"$: market's down. Round up all probabilities to $2$ decimals maximum.

Write the formula you will use: $p(C)=$

Rule $1$: $p(C)=$

Rule $2$: $p(C)=$

Rule $3$: $p(C)=$

b$)$ Explain in words, what is wrong with the "randomizing strategy" $(3).$ c$)$ NFQ is getting serious; they want to do conditional prediction. The rules in question a$)$

were unconditional rules. They ask you to investigate conditional rules, maybe markets are not

efficient! Even though you already know the answer, you will be able to charge them for some

data analysis. You hop on the $793$ Data web site and download the US stock market excess

return over the risk free rate from the weekly $3-$factor file, the one named RmRf$.{?}^1$ You use the

data from Jan $2015$ to July $2024$ included.

You first give them the three classic versionts of the the usual two$-$way table:

simple counts

joint probabilities

conditional probabilities.

T$1$: Counts Table

Probability Table in $\%$

T$2$: Joint

T$3$: Conditional Probability Table $(R_t|R_{t-1})$

Make sure you can compute these probabilities cold. Write the formulas and results below

$P(R_t>0|R_{t-1}>0)=$

d$)$ Use your results to compute the unconditional probability of success of the conditional

rule predicting a positive $($negative$)$ return follows a positive $($negative$)$ return

e$)$ Given your data, can a conditional rule improve on unconditional rules?

f$)$ Given these results, how efficient do you think the US market is$?$

g$)$ Wait wait! explain to the dudes why you used excess return on Rf$,$ not just the total return.