What Is the Relationship Between Interest Rates, Bonds, and Stock Prices?

Let me try to answer this with some examples.

Bond prices goes up when interest rates goes down, and bond prices goes down when interest rates goes up.

Here's an example why:
Let's say you have a bond that is worth $1000 and pays 5% coupon interest. You then decide to sell that bond when interest rates increased. You will now have to sell that bond for less than $1000. Since interest rates increased, investors can now get a bond that is worth $1000 and pays 6%. So in order to make your bond appealing to investors you will have to lower the price of your bond so that they can make up for the difference in interest that your bond is paying. (Remember your bond is paying 5% and they can easily get one that pays 6%)

Here is an example of how it works the other way: Let's say you are still holding the same 5% bond and interest rates dropped to 4%. If you decide to sell your bond now you can charge more for it because the investors can only find bonds paying 4% while you have one for sale offering 5%. The investors will be willing to pay more for that bond to have the higher coupon rate.

The relationship between stocks and interest rates are a little more complicated because it has more variables but the relationship is usually inverse as well. If interest rates rise stock prices tend to fall. Interest rates drop stock prices increase.

There are many variable as to why this can occur. When interest rates are low it is cheap for companies to borrow money. They can now use the funds to expand their business, refinance higher interest debt… and so on causing the stock price to increase. When rates are high it is more expensive for them to expand their business therefore making the stock prices drop.

Another reason why stock prices have an inverse relationship to interest rates is because most money managers use the interest rate as one of the variable in calculating the intrinsic value of the stock. So if interest rates change the intrinsic value of the stock changes with it.

The examples I gave above are very general and simplified. You also have to keep in mind that there are other variables at play. For example there is the credit rating of the bond, maturity, coupon rate… and the list can go on. I hope that made it easier to understand.