### Ex: Present Value of One Time Investment Given Future Value

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**Chester McEachern** - 2 weeks ago
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– 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.