Is there a rule of thumb to calculating the inverse relation between a bond’s price (or any fixed income payout) and its interest rate? I’m actually more interested in bond ETFs, so something along the lines of “if an ETF yields 3% and the yield then rises to 4%, its price will fall by X%”
Here's an example for the fixed income payout of a preferred stock:
The current interest rate is 3% so if the current price (CP) is $25 then the interest/dividend is 75 cents.
INT/CP = .03 or INT = .03 * CP
If the interest rate becomes 4% and FP is the future price then:
INT/FP = .04 or INT = .04 * FP
Solve the equations and you have:
FP = 3/4 * CP
In this example, FP would then be $18.75
Worth 3/4 of original price is a 25% loss.
You could also come at this with:
Position Loss = (New Price - Original Price)/Original Price
= (18.75 - 25.00)/ 25.00 = - 25% (same answer)
Note that these calculations are for basic interest rate risk. Some bonds are more sensitive to interest rate changes than others (duration risk). This is the risk associated with the sensitivity of a bond’s price to a one percent change in interest rates. That's a different beast and you can Google for details if that calculation interests you.