Investment in CD and Stocks

Investing in CD, the interests $\alpha_i$ are defined by contracts. Only the reliabilities $p_i,\ i=1,...,n$ of banks are uncertain. Investing in stocks, in addition to reliabilities $p_i,\ i=n+j, \ j=1,...,m$ of companies, their future stock rates are uncertain, too. The predicted stock rates are defined by a coefficient $a_i$ that shows the relation between the present and the predicted stock rates. The prediction "horizon" is supposed to be the same as the maturity time of CD.

To simplify the model suppose that one predicts $L$ different values of relative stock rates $a_i^l,\ l=1,...,L$ with corresponding estimated probabilities
$p_i^l,\ \sum_{l=1}^L p_i^l=1,\ p_i^l \ge 0$.
In this case, one may define probabilities $p(y^i)$ of discrete values of wealth $y^i,\ i=1,...,n+m$ by exact expressions. The expressions for CD remains the same. Therefore, we shall consider only stocks assuming that $n=0$ and $L=2$. Then

$$p(y^0) = \prod_{i} q_i,$$

$$p(y^1) = p_1 p_1^1 \prod_{i \ne 1} q_i,$$

$$p(y^2) = p_1 p_1^2 \prod_{i \ne 1} q_i,$$

$$p(y^3) = p_2 p_2^1 \prod_{i \ne 2} q_i,$$

$$p(y^4) = p_2 p_2^2 \prod_{i \ne 2} q_i,$$

$$............. ...................................$$

$$p(y^{2n-1}) = p_n p_n^1 \prod_{i \ne n} q_i ,$$

$$p(y^{2n}) = p_n p_n^2 \prod_{i \ne n} q_i ,$$

$$p(y^{2n+1}) = p_1 p_1^1 p_2 p_2^1 \prod_{i \ne 1, i \ne 2} q_i ,$$

$$p(y^{2n+2}) = p_1 p_1^2 p_2^p2^2 \prod_{i \ne 1, i \ne 2} q_i,$$ (10)

$$............. ..................................................$$

Here
$y^0=0,\ y^1=a_1^1 x_1,\ y^2=a_1^2 x_1,\ y^3=a_2^1 x_2, y^4=a_2^2 x_2,\\ \ y^{... ...{2n}= a_n^2 x_n, \ y^{2n+1}=a_1^1 x_1+a_2^1 x_2, \ y^{2n+2}=a_1^2 x_1+a_2^2 x_2$. The reliabilities $p_i$, the stock rate predictions $a_i^l$ and their estimated probabilities $p_i^l$ are defined by experts, possibly, with the help of time series models such as ARMA For example, maximal values of multi-step prediction are considered as "optimistic" estimates. The minimal values- as "pessimistic" ones. The average values of multi-step prediction are regarded as "realistic" estimates.

Here is a simplest illustration were $n=m=1$ and $L=2$. In this case from (5) (10) the probabilities $p(y^k)$ of wealth returns $y^k,\ k=0,...,5$ are

$$p(y^0) = q_1 q_2,$$

$$p(y^1) = p_1 q_2,$$

$$p(y^2) = p_2 p_2^1 q_1,$$

$$p(y^3) = p_2 p_2^2 q_1,$$

$$p(y^4) = p_2 p_2^1 p_1,$$

$$p(y^5) = p_2 p_2^2 p_1.$$

Here
$y^0=0,\ y_1= a_1 x_1\ y^2=a_2^1 x_2,\ y^3=a_2^2 x_2, \\ y^4=a_1 x_1+ a_2^1 x_2,\ y^5=a_1 x_1+ a_2^2 x_2$.