Three years ago, Kuley invested $32,200. In 2 years from today, he expects to have $50,300. If Kuley expects to earn the same annual return after 2 years from today as the annual rate implied from the past and expected values given in the problem, then in how many years from today does he expect to have exactly $87,200

8.17 years(closest to 8 years )

Explanation:

The future value of $50,300, would be accumulated after 5 years of having made the investment(3 years+2 years=5 years)

As a result, we can determine the annual rate of return based on the future value in year 5 using the future value formula below:

FV=PV*(1+r)^n

FV=future value=$50,300

PV=amount invested initially=$32,200

r=unknown=annual rate of return

n=5 years

$50,300=$32,200*(1+r)^5

$50,300/$32,200=(1+r)^5

$50,300/$32,200 can be rewritten as ($50,300/$32,200)^1

($50,300/$32,200)^1=(1+r)^5

divide index on both sides by 5

($50,300/$32,200)^(1/5)=1+r

r=($50,300/$32,200)^(1/5)-1

r=9.33%

Our next task is to determine how long( in years) it takes to accumulate a future value of $87,200 from today's point, which means we need to determine the value of the investment today( 3 years after making the investment)

FV=$32,200*(1+9.33%)^3

FV=value of investment today=$42,079.82

Lastly, we can ascertain when $42,079.82 today would become $87,200

$87,200=$42,079.82*(1+9.33%)^n

n=number of years=unknown

$87,200/$42,079.82=(1+9.33%)^n

$87,200/$42,079.82=1.0933^n

take log of both sides

ln ($87,200/$42,079.82)=n ln(1.0933)

n=ln ($87,200/$42,079.82)/ln(1.0933)

n=0.72863604/0.08920065

n=8.17 years( from today, approx 8 years)