All asset markets, including shares and bank deposits typically compete for capital (money). So in theory, high interest rates should encourage money to flow from the stock market (hence lowering the XJO as share prices fall) into savings accounts take advantage of high rates on savings deposits. And vice versa, low interest rates would encourage funds to move out of bank deposits into the share market (hence increasing the XJO as share prices rise) as investors chase higher returns.
This is the theory, but I have no idea what the actual correlation figures look like. Would be good if someone could enlighted us.
The relationship between the CPI and XJO is far more complex. CPI, as it measures prices of a "basket of goods and services" can be thought of as almost another asset class which competes for money. Ie you have $1000, you can put it in a bank deposit, a stock, or you can spend it on goods and services. Whichever way you choose to use the money, it will add to prices in that asset class. So you can look at the CPI/XJO relationship as being two asset classes competing for funds. However, there is also the indirect relationship that occurs because CPI and interest rates are correlated, as are interest rates and share markets. As I said, far more complex.
Somebody here may have better ideas than me however.