Concept
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Another way to express interest rates are with so-called discount rates. Instead of using a rate to increase ๐๐0 to make it equal ๐๐๐๐ (as in Equation 5) we can decrease ๐๐๐๐ to make it equal ๐๐0 using the following equation:
๐๐0 = ๐๐๐๐(1 โ๐๐)๐๐
(9)
where ๐๐ is known as the annual discount rate. When we add the idea of a compound period to the equation, we get the following formula for the ๐๐-compounded annual discount rate:
๐๐(๐๐) = ๐๐โ๐๐(๐๐0
๐๐๐๐) 1
๐๐๐๐
(10)
Would you be able to describe the logic behind this formula in full, in other words, can you write new versions of Equation (5) and Equation (8) using Equation (9)?
The term annual discount rate (without the specification of a compounding frequency) refers to Equation (9) in which ๐๐= 1. Therefore, if the compounding is not mentioned, it is assumed to coincide with the annual period of the return.
Discount rates, which shouldnโt be confused with discount factors, are quantitatively similar to interest rates. Suppose a deposit of 90 grows to 100 after one year, then the (annual) interest rate is 11,111%, while the (annual) discount rate is 10%. Both numbers are of similar magnitudes, and both describe the quantitative relationship between the principal value and return.