Explain the mechanism behind the "1" in the Present value calculation of money.
The formula is PV=A/(1+DISCOUNT RATE)^NO.OF YEARS.It would be helpful if someone could explain the 1,and if there's a simple math rule,law, convention,or algebraic expression behind it, please expatiate on that as well. Imagine yourself as explaining to a grade schooler when you do it(unfortunately, my math skills are that bad). Thank you for your time.
On
So many people answer this question by stating the formula and not the calculation behind it; they expiate on the contents of the formula,$P, N, R$, etc, but never refer to the "$1.$" An excellent explanation is contained in "The Handbook of Financial Mathematics" $(1929)$ pgs $65-67$, by J. Moore. Based on Moore's explanation, the $\#1$ represents the Principal at the beginning of the $1$st year, or thereafter, at the beginning of a compounded time period. For instance: What will $\$350$ amount to in three years at $4\%$ if interest is compounded annually? [Principal or compounded amount at beginning $2$nd year] = [Principal at beginning of $1$st year + interest for one year on original principal]. or $$364 = 350 + 350 \times .04$$ Any quantity is equal to itself multiplied by "$1$" Hence, instead of 350, we write $350 \times 1$, giving $364 = 350 \times 1 + 350 \times .04$ But in each multiplication operation on the right hand side $350$ is a common factor. Therefore, write: \begin{align}364 &= 350(1 + .04)\\ &= 350(1.04)\\ &=350 \times 1.04 \end{align}
Rule: to get the amount of any sum of interest for one year, multiply the principal by "$1$" plus the annual interest rate, expressed as a percent.
If I loan you $\$100$ at the rate $r\%$ for one year. At the end of the year you will pay me $\$r$ interest plus my original $\$100$
$100 + r = 100(1+r\%)$
If you can't pay me back and I agree to extend the loan, I expect you to pay me interest on $(100 + r)$ and the pay back the loan amount.
$(100+r)\times r\% + (100+r) = (100+r)(1+r\%) = 100(1+r\%)^2$