What is the difference between multiplying and dividing percentages with 1 added to the percentages and not

Question

r_{local currency} =

(1 + r_{foreign currency}) x (1 + Expected inflation_{local currency} ⁄ 1 + Expected inflation_{foreign currency}) - 1

vs.

Country risk premium = Country default spread x (σ_equity ⁄ σ_{country bond})

I'm probably missing some very fundamental understanding, but I want to know why an equation adds 1+ percentage, vs just using the percentage. Is there some intuition why some equations do this? Thank you!

Answer 1

Yes, there is a mathematical reason for this. When you are using an equation that adds 1 to a percentage, generally its is because you are increasing a base number by a percentage. In these equations, it’s important to retain the base amount, which is why you add 1.

Let’s use the example of calculating the balance in your savings account after 1 year. If you have $1,000 (P) in your savings account, and you earn 1% interest (r) , the formula for calculating your balance after 1 year (t) is P*(1+r)^t, which gives you $1,010. If you did not add 1 to the 1% interest, you would come up with $10, which is just the amount of interest you earn. Adding 1 retains the base P. Retaining the base becomes more important as you add years. For example, after 10 years, you should have $1000 * (1+.01)^10 = $1,105. Without adding 1 to the base, the formula would imply you have less than a penny in your savings account, which is obviously incorrect.

In the inflation/currency example you used, the base, P, is the base CPI (for inflation) or the base monetary value (for interest rates). It is important to retain the base levels of these values in order to compare increases/decreases across currencies/countries.