DISCOUNTING TO PRESENT VALUE
Assume I owe you $X one year from today.
Assume you can invest moneyat an interest rate = i
How much should I payyou todayto clear the debt?
I owe you $11,000 oneyear from today
Assume i = .10 (or 10%)
PV = Present Value of the debt = Amount I should giveyou today
PROOF: Suppose I gaveyou $10,000 today.
You could invest it for 1 year at 10% interest.
You would get $1,000 in interest.
You would havea total of $11,000.
You could invest PV for one year at i = 10%
After one year, you would get 10% interest and would have
PV + .10 PV = PV x (1.10)
Thus, the amount PV should be the solution of
PV x (1.10) = $11,000
PV = $11,000 / (1.10) = $10,000
WHAT IF THE INTEREST RATE IS i = .05 (5%)? I should giveyou an amount PV todayso that in one year you would have
PV x (1.05) = $11,000
Therefore, the amount I should giveyou todayis
PV = $11,000 / (1.05) = $10,476.19
DISCOUNTING TO PRESENT VALUE:
GENERAL CASE FOR MONEY DUE IN ONE YEAR
F1= Amount owedin oneyear
i = Interest rate
PV = Present value
PV x (1 + i) = F1
PV = F 1 (1 + i)
What if moneyis owedin two years?
Suppose I oweyou $12,100 in two years
Let i = .10 (or 10%)
PV x (1.10) = Result after investing the amount PV for oneyear
PV x (1.10) x (1.10) = Result after investing for two years
PV x (1.10) x (1.10) = PV x (1.10) = $12,100
PV = $12,100 / (1.10) = $10,000