# After 80 years of 5.8% interest compounded bi-weekly, an account has \$102,393.44. What was the original deposit amount today?

The original deposit is approximately \$991.45.

Suppose that each year contains 52 weeks: then, compounding occurs `n = 52 * 80 ` times and the rate for each compounding is `i = ( 5.8 % ) / 52 = 0.058 / 52 .`

If the original deposit amount is PV, then the future value FV is equal to

`FV = PV * ( 1 + i )^n = PV * ( 1 + 0.058 / 52 )^( 52 * 80 ) .`

We know the future value (it is given to be \$102,393.44), so it is not hard to find the present value:

`PV = FV / ( ( 1 + i )^n ) = (\$102,393.44) / ( ( 1 + 0.058 / 52 )^( 52 * 80 ) ) .`

The simplest way to compute this is to use a calculator:

`PV approx \$991.45 .`

Just for fun, we can simplify this expression with great precision. It is known that `( 1 + 1 / n )^n -> e , ` which easily implies that `( 1 + a / n )^n -> e^a . ` Here,

`( 1 + 0.058 / 52 )^( 52 * 80 ) = ( ( 1 + 0.058 / 52 )^52 )^80 approx ( e^0.058 )^80 = e^( 0.058 * 80 ) = e^4.64 ,`

That is, `PV approx FV e^( -4.64 ) .`

Also, we may suppose that compounding occurs exactly each week, not 52 times a year, and get slightly different answer.

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