At what constant, continuous annual rate should you deposit money into an account if you want to have $1,000,000 in 25 years? The account earns 5% interest, compounded continuously. Round to the nearest dollar.
FV = Pe^Yr
where FV = future value = 1,000,000
r here = .05
Y = 25
1,000,000 = P e^(1.25)
P = 1,000,000 / 3.49
P = 286,533
online calculator:
http://www.moneychimp.com/articles/finworks/continuous_compounding.htm
dP/dt = r P
dP/P = r dt
ln P = r t
e^ln P = e^(rt) + C
P = C e^(rt)
when t = 0, e^(rt) = 1
so C = value of P when t = 0
so
P = Po e^(rt)
How do I find the continuous rate though?
Oh, sorry
Try this, sinking fund "Continuous compounding at nominal rate r, uniform series"
http://ece.uprm.edu/~s016965/ININ%204015%20-%20Analisis%20Economico%20Para%20Ingenieros/Engineering%20Economic%20Analysis%208th%20ED.pdf
To find the constant, continuous annual rate at which you should deposit money, we need to use the formula for compound interest with continuous compounding:
A = P * e^(rt)
Where:
A = the future amount ($1,000,000)
P = the initial principal (unknown)
r = the annual interest rate (5% or 0.05)
t = the time in years (25)
e = Euler's number (approximately 2.71828)
We need to solve for P in this equation.
Dividing both sides of the equation by e^(rt), we get:
A / e^(rt) = P
Substituting the given values, we have:
$1,000,000 / e^(0.05*25) = P
Now, we can calculate this using a scientific calculator or an online tool.
Solving this equation gives us:
P ≈ $163,756.12
So, you would need to deposit approximately $163,756.12 in the account initially to have $1,000,000 in 25 years, with continuous compounding.
To find the constant, continuous annual rate at which you should deposit money, we divide P by the number of years:
$163,756.12 / 25 ≈ $6,550.25
Rounding to the nearest dollar, you should deposit approximately $6,550 per year at a continuous annual rate to reach $1,000,000 in 25 years, with continuous compounding.