5. Average daily balance,average monthly balance

Question

Suppose that you never carry cash. Your paycheck of $1,000 permonth is deposited directly into your checking account on the firstday of the month, and you spend your money at a constant rate sothat at the end of each month your checking account balance iszero.
What is your average daily money balance for the month?


How would each of the following changes in assumptions affect youraverage monthly balance? Be specific with your answers.
You are paid $500 twice monthly (1st and 15th) rather than $1,000each month.


You are uncertain about your total spending each month.


You spend a lot at the beginning of the month (e.g., for rent) andlittle at the end of the month.


Your monthly income increases.

Explanation / Answer

Your average daily value is $500. The logic behind it is this, onthe first day you have $1000, the last day you have $0. Average thetwo and you have $500. Then the 2nd day and the second to last dayyou have (1000-33.33) + (33.33)/2 = $500 (33.33 is the amount youwould spend in one day if you spend money at an equal, constantrate). This holds for every day, so that's why you get thatanswer. If you are paid twice a month, this number falls to $250. Use thesame reasoning as above. I'm not certain what to say about the uncertain about your spendingeach month. I would say that's rather ambiguous; you could beuncertain in either direction. If you spend a lot at the beginning of the month and less at theend, your average balance declines. If your monthly income increases, ceteris paribus, your averagebalance increases.

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