### Ex: Present Value of One Time Investment Given Future Value – WE’RE GOING TO DETERMINE
THE PRESENT VALUE OF \$30,000 DUE 4 YEARS FROM NOW IF INVESTED
AT 4% CONTINUOUS INTEREST. SO THIS QUESTION IS ASKING US WHAT AMOUNT DO WE NEED
TO INVEST TODAY AT THIS INTEREST RATE
TO GROW TO \$30,000 IN 4 YEARS. THIS IS CALLED
THE PRESENT VALUE. AND HERE’S THE FORMULA
FOR PRESENT VALUE. IT LOOKS VERY SIMILAR
TO THE FUTURE VALUE FORMULA AND I’LL SHOW YOU
WHY THAT IS IN JUST A MINUTE. THESE VARIABLES TO REPRESENT
THE EXACT SAME THING AS THEY DID IN THE FUTURE VALUE
FORMULA WHERE P SUB 0 IS THE PRESENT OR INITIAL VALUE
OF THE INVESTMENT ACCOUNT. K IS THE INTEREST RATE
EXPRESSED AT A DECIMAL. T IS THE TIME IN YEARS
AND P IS THE FUTURE AMOUNT. SO BEFORE WE SET THIS UP LET’S TALK ABOUT
WHERE THIS FORMULA COMES FROM. WHEN SOLVING FUTURE VALUE
PROBLEMS WE USE THIS EQUATION HERE. AND IT WAS A GRAPH
OF THIS EXPONENTIAL. AND WE’RE ACTUALLY GOING TO USE
THIS SAME EQUATION TO DETERMINE PRESENT VALUE. WE’RE JUST GOING TO WRITE THE
EQUATION IN A DIFFERENT FORM. BUT WHAT’S HAPPENING NOW IS WE’RE GIVEN THE AMOUNT
WE NEED IN THE FUTURE AND WE’RE DETERMINING
WHAT P SUB 0 WOULD NEED TO BE IN ORDER TO HAVE THE AMOUNT P
IN THE FUTURE. SO IF WE WERE TO START
WITH THE FUTURE VALUE EQUATION WE COULD SOLVE THIS FOR P SUB 0
FOR THE PRESENT VALUE EQUATION BY MULTIPLYING BOTH SIDES
BY E TO THE -KT POWER. NOTICE WHEN MULTIPLYING THESE
TWO THE BASES ARE THE SAME SO WE’D ADD THE EXPONENT. SO WE’D HAVE E TO THE 0 AND E
TO THE 0=1. SO WE JUST HAVE P SUB 0 OR THE PRESENT VALUE IS EQUAL
TO P x E TO THE -KT POWER. SO THIS IS THE REASON WHY IN THE PRESENT VALUE FORMULA
THE EXPONENT IS -KT. NOW LET’S GO BACK TO OUR EXAMPLE
AND SOLVE THE PROBLEM. SO THE FUTURE VALUE
OR P=\$30,000. THAT’S HOW MUCH WE WANT
IN THE FUTURE. T OR TIME IS 4 YEARS. AND OUR INTEREST RATE
AS A DECIMAL WOULD BE 0.04. AND NOW WE’LL JUST SUB
THESE VALUES INTO OUR EQUATION FOR PRESENT VALUE. WE’LL HAVE THE PRESENT VALUE
IS EQUAL TO \$30,000 x E TO THE -0.04 x 4. AND NOW WE’LL GO
TO THE CALCULATOR. SO WE HAVE 30,000, SECOND NATURAL LOG BRINGS UP
E RAISE TO THE POWER AND WE HAVE -0.04 x 4. SO THE AMOUNT WOULD BE
\$25,564.31 WHICH MEANS WE WOULD NEED
TO MAKE A ONE TIME INVESTMENT OF \$25,564.31 TO HAVE \$30,000 IN 4 YEARS IF THE ACCOUNT EARNS
4% CONTINUOUS INTEREST.

### Chester McEachern

Author Since: Mar 11, 2019