There are two "interest" rate here. You MUST be able to discern (differentiate) them. To avoid confusion, we call one as COUPON RATE, the other as YIELD TO MATURITY (YTM) Therefore we do not just use the word interest rate here. We use COUPON RATE and YTM.
Suppose a bond has 5 years of maturity, its COUPON RATE is 8%. If you have 10,000 unit of it, each you bought it at $100, the nominal value, or the size, (like how many shares) is $1,000,000. If the price of this bond is $100, the MARKET VALUE of the bond is $1,000,000. Each year you will receive $80,000 as income. You receive $80,000 for 5 years. The income is FIXED. Therefore we call bond as FIXED INCOME. What is the YTM of this bond now? Since the coupon is 8%, and the bond price is $100, the YTM must be 8%. (You discount the cashflow of $80,000 each year for the coming 5 years, and the terminal value of $1,000,000. You will get a present value of $1,000,000)
Now suppose the demand of BOND is smaller. The BOND is now trading at $99. What is the COUPON? It is still the same. $80,000 per year for the coming 5 years. Terminal value is also $1,000,000. Well, the present value is $99 (This is the market price, we do not analyse why it is lowered, just take it as an input, or an assumption). What is the YTM now? It MUST be higher than 8%. Why? If you use $100 to buy this investment, your YTM is 8%. But now you use $99 to buy this investment, the YTM must be above 8%. To get the exact value of the YTM, you equate 990,000 = Present Value of (5 years of $80,000 and 1,000,000 in 5 years time, under a certain rate r). This r is the YTM.
On the other hand, if the bond price is above $100, the YTM will be lower that 8%. Therefore, the YTM can also be said as the discount rate. YTM is the "actual" yield the investor received. (Given an investment product which gives a fixed cashflow, the price you bought it certainly affects the return, or the yield, or YTM, right?)
The interest rate in your question is actually YTM. Maybe you bought the bond at $100. If the YTM is now 10%, the bond price would certainly be lower than 8%. (Actually, in the real financial world, investors of every kind buy and sell every kind of product. It is that the demand is smaller, which drove down the bond price. Then people use the lowered bond price and calculate back the YTM at that moment.) Therefore YTM is also another way to see how expensive a bond is. The higher the YTM, the better the return, the lower the bond price must be. The smaller the YTM, the lower the return, the higher the bond price must be.
One last word as summary. There are 3 things here. BOND PRICE, COUPON RATE, YIELD TO MATURITY. Once you know any two of them, you can get the third one. Usually the COUPON rate is known, so it is either you have the YTM, and find the BOND PRICE; or you have the BOND PRICE, and work to get the YTM.
