Question 38411
{{{A = P(1+ r/n)^(nt)}}}
Suppose you deposit $20,000 for 3 years at a rate of 8%.
a) Calculate the return (A) if the bank compounds annually (n = 1).
{{{A = 20000(1+ .08/1)^(1*3)}}}
{{{A = 20000(1.08)^(3)}}}
{{{A = 25194.24}}}
b) Calculate the return (A) if the bank compounds quarterly (n = 4)
{{{A = 20000(1+ .08/4)^(4*3)}}}
{{{A = 20000(1.02)^(12)}}}
{{{A = 25364.84}}}
c) Calculate the return (A) if the bank compounds monthly (n = 12)
{{{A = 20000(1+ .08/12)^(12*3)}}}
{{{A = 20000(1+ .08/12)^(36)}}}
{{{A = 25404.74}}}
d) Calculate the return (A) if the bank compounds daily (n = 365)
{{{A = 20000(1+ .08/365)^(365*3)}}}
{{{A = 20000(1+ .08/365)^(1095)}}}
{{{A = 25424.31}}}
e) As the time or interest or frequency of compounded times increases, the amount of total money will increase.
f) If a bank compounds continuous, then the formula becomes simpler, that is {{{A=Pe^(rt)}}}
 where e is a constant and equals approximately 2.7183. Calculate A with continuous compounding. 
{{{A=(20000)e^(.08*3)}}}
{{{A=25424.98}}}
g) Now suppose, instead of knowing t, we know that the bank returned to us $25,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t). 
{{{25000=20000e^(.08t)}}}
{{{(5/4)=e^(.08t)}}}
{{{ln(5/4)=.08t}}}
{{{ln(5/4)/.08=t}}}
In about 2.789294
h) A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer? 
{{{A=Pe^(rt)}}}
{{{40000=20000e^(.08t)}}}
{{{2=e^(.08t)}}}
{{{ln(2)=.08t}}}
{{{ln(2)/.08=t}}}
In about 8.664340