How does interest rate affect the price of a share/stock?

I have this little box. Every year it spits out a dollar. I want to sell it. What's it worth?

The answer comes by considering what else you could do with your money to get a dollar a year. If you had a savings account with 2% interest you would have to deposit $50 to get a dollar a year, so my little box is worth $50. If there were another investment that paid a 10% dividend, you would only have to put in $10 to get a dollar a year. So the value of my little box depends on the available interest rate.

Another thing that would influence the value of my little box is the chance that it might spit out more or less than a dollar. But let's just consider the basic example: if the prevailing interest rate is 0%, the value of my little box is infinity dollars. It's only a question of whether it is reliable. If we assume my little box is reliable, but the value of the dollar is dropping 10% per year, then a fair price would be 10% more than what we would otherwise compute. That is to say if you expect the box to spit out 90 cents, and you expect each of your dollars to lose 10% of their value, then you break even if you pay 10% more than the computed value to buy the box.

This simple example explains why people are willing to pay more for stocks than we would think they are worth. Your assessment of the reliability of stocks is a separate issue. The role of gold in this example is hampered by a social problem: many people really, really don't like gold. The prices we see are generated by just a few people who are willing to buy gold in spite of its social stigma. When the herd sees reality, they will knock down doors to buy gold at any price. Because as long as the Fed is still trying to hold the interest rate to zero, gold is worth infinity dollars.