Question

If the cost of going to the ATM is $2 and the nominal interest rate is 1 percent, someone who has a 9 percent probability of having his cash lost or stolen and spends $15 each day. How often will he go to the ATM (in days)?

Answer 1

Total annual cost = cost of visiting the ATM + opportunity cost + expected cost of theft or losing money

= (365 × cost of per ATM visit)/T + ($15 × T × i)/2 + ($15× T × probability of losing money or theft)/2

= (365*2)/T + (15*T*0.01)/2 + (15*T*0.09)/2

= 730/T + 0.15*T/2 + 1.35*T/2

The person will try to minimise this total cost, thus differentiating by T,

We get,

730*-1*T^-2 + 0.15/2 + 1.35/2 = 0

-730/T² + 0.075 + 0.675 = 0

-730/T² = -0.75

730/ 0.75 = T²

973.33 = T²

T = 31.2

So, the person would go 31.2 days in total to ATM to withdraw cash.