Exercise II: Continuous interest rate, again

Remember that HW $7$ introduced the notion of continuous interest. Here is another way

to think about it$.$ Assume that M$($t$)$ describes the amount of money in the bank at time

t days, starting from an amount A at time t$=0.$ Say that there is stated annual interest

rate r$>0.$ If interest is compounded n times per year, we saw that it means that we

receive $($r$)/($n$)$ times the amount present in the bank, n times per year. In other words, it

means that if interest is compounded every h$=(1)/($n$)$ years, then we receive $($r$)/($n$)=$rh times

the amount present in the bank during this period. Mathematically

M$($t$+$h$)=$M$($t$)+$rhM$($t$)$

For continuous interest, we have an interest compounded every very small amount

of time. Explain what this means mathematically, and deduce from the previous

equation that we have

M$^(\text{'})($t$)=$rM$($t$)$

for all t$.$ This is called a differential equation.

Let f$($t$)=$M$($t$)$e$^(-$rt$).$ What is f$(0)?$

Compute f$^(\text{'})($t$)$ and deduce f$($t$).$ Make sure to name the result that you use.

Deduce M$($t$),$ and compare to HW $7.$

Assume now that you also withdraw a continuous flow of money, that is you with$-$

draw a small amount bhb$>0$ h years. As above, we then have

M$^(\text{'})($t$)=$rM$($t$)-$b

Let f$($t$)=($M$($t$)-($b$)/($r$))$e$^(-$rt$).$ As above, compute f$^(\text{'})($t$),$ deduce f$($t$),$ then M$($t$).$

Compute

$\backslash$lim$\_($t$->+\backslash$infty $)$M$($t$)$

Your answer should depend on a relationship between A$,$b$,$ and r$.$ Explain what

this means in practical terms.