how do you answer this problem,Find the present value, using the present value formula and a calculator. (Round your answer to the nearest cent.)

Question

Achieve $225,500 at 8.15% compounded continuously for 8 years, 145 days.

Can someone please walk me through the process of solving this problem?

Answer 1

In case of continuously compounded interest, the relevant formula is

A = Pe^rt. 

In this case you want to achieve A=225,500, r=8.15% or 0.0815 and t = 8+145/365 =8.39726 years.

So, you can find the amount of money that you want to invest today, P, to achieve A in 8 years and 145 days. The answer is   P = A/e^rt or P = 225,500/e^{0.0815*8.39726}.  To compute the exponential in the denominator, it depends on what calculator you are using but it should be something like (0.0815*8.039726) [2nd] e^x =1.98254 so that P =225,500/1.98254 = $113,743.22.

On most calculator the PV formula is not meant to treat problems where interest is continuous compounded, so you would have to use the formula.  You can sort of trick your calculator (I am using Texas Instruments BAII Plus, a financial calculator). For example, using daily compounding, we find

N= 8*365+145=3065 (days) ; I/Y=8.15/365 (daily interest), PMT=0, FV=225,500, CPT -> PV

This gives 113,751.91

You can try in half-days... then

N=(8*365+145)*2=6130, I/Y=8.15/(365*2) (semi-daily interest), PMT=0, FV=225,500, CPT-> PV 

which gives 113,747.56

so we are approaching the true answer, but to get the correct answer you need to follow the formula we used first as that is the only one that will give you the accurate answer you are looking for.

Answer 2

Laura,

Continuous compounding of interest involves euler's number "e" which is irrational, approximately 2.7.  Generally, the amount in the account at time t, A(t), is equal to the initial amount in the account, A(0), times e raised to the power of (the interest rate times the time the money is in the account).  A(t)=Ae^(rt).  You should have this formula in whatever text you are using; sometimes instead of A for account, P is used for principal.

The trick for these problems usually involves remember that the formula only works for r as a decimal rate, not a percentage and the amount of time has to be expressed in years.  So you need to convert 8.15% into a decimal and convert 8 years, 145 days into 8 point some number of years.  (I'll leave those processes to you).

This problem is also asking the question in the reverse order, i.e. you need to know how much to initially put into the account (A), to produce 225,500 in 8+ years, so you need to solve for A, because you've been given A(t). 

I hope this helps.