### Video Transcript

Bank A offers depositors four percent annual interest compounded once per year. Bank B offers 3.93 percent per year compounded monthly. Write an explicit formula for the return π
after π years on a deposit of π
sub zero dollars with both offers. Which bankβs offer is better?

So in this question, weβre given information about the interest rates for bank A and bank B. Bank A offers a higher interest rate of four percent, and thatβs compounded only once per year, whereas bank B offers an interest rate of 3.93 percent, but thatβs compounded monthly, Weβre asked to do two things in this question. The first thing is to write a formula for bank A and then for bank B. And secondly, we need to work out which of the banksβ offers would be better.

So letβs start by looking at how we would write a formula. And weβll start with bank A. Both of these banks will apply compound interest. And in bank A, that compound interest is applied once per year, so that means that we can use this simpler compound interest formula: π equals π times one plus π to the power of π¦. In this formula, π represents the value of the investment, π is the principal or starting amount, π is the interest rate, and π¦ is the number of years the money is invested for.

So letβs use the information in the question to work out what the values of π, π, π, and π¦ will be for bank A. Given that weβre asked to write a formula for the return π
, then that means that our value of π can be given as π
. Notice that thatβs a capital π
to make it different from the interest rate, which is a lowercase π. Next, weβre told that this formula applies for a deposit of π
sub zero dollars. So that means that our principal amount will be π
sub zero.

Then we have the annual interest rate as four percent. So that means that our lowercase π is four percent. Finally, weβre given that this money will be invested for π years, so that means that our number of years π¦ is equal to π.

When we fill these values into the formula, we need to take care with the interest rate, remembering that four percent is four over 100 or 0.04. So here, we have a formula in terms of the return π
. Itβs π
equals π
sub zero times one plus 0.04 to the power of π. And of course, we could further simplify within the parentheses to give us a simple formula if we wished.

But now letβs have a look at a formula for bank B. Because weβre told that bank Bβs interest is compounded more than once a year, then we need to use a slightly different compound interest formula. This formula is often written as π equals π times one plus π over π to the power of ππ¦, where π is the value of the investment, π is the principal amount, π is the interest rate, π is the number of times per year that the interest is compounded, and π¦ is the number of years that the money is invested for.

A handy tip at this point is that we might notice that we could get confused with this given value of π in the question. We have an π in the formula, so letβs change this to a different letter. We can choose the letter π₯ for example. So now in the formula, we have this value of π₯ in the parentheses and in the exponent. And that π₯ just represents the number of times per year that the interest is compounded.

And so just like we did with bank A, letβs find the values for each of these letters using the information given in the question. The value of the investment is simply the return π
. Then we have the same starting amount of π
sub zero dollars. Next, the interest rate is 3.93 percent. For the value of π₯, thatβs the number of times per year that the interest is compounded. And while weβre not specifically told that, we are given that the interest is compounded monthly. And since there are 12 months in a year, then that means that our value of π₯ must be equal to 12. And finally then, just like for bank A, the number of years is given as π, so π¦ is equal to π.

Before we put these values into the formula, it can be helpful to remember that the interest rate is 3.93 percent, which is 3.93 over 100. Or as a decimal, itβs 0.0393. When we fill these values into the formula, we get π
equals π
sub zero times one plus 0.0393 over 12 to the power of 12π. And so thatβs the first part of the question answered. We have two formulas which give the return π
after π years on a deposit of π
sub zero dollars.

Before we look at the second part of the question, thereβs one other thing that we can point out. In this second formula, which we use for bank B, we could have also used this to create a formula for the annual interest in bank A. In bank A, the number of times per year that the interest was compounded is one. So that would make each value here of π₯ equal to one. When π₯ is equal to one, that would give us the same formula that we used for bank A.

But now letβs clear some space to work out which of these banks has the better offer. In order to identify the bank with the better offer, that is, the one which gives us the higher return on the investment, then letβs see what happens after one year.

The value of π represents the number of years, so letβs substitute π equals one into both formulas. For bank A, we have π
equals π
sub zero times 1.04 to the power of one. This simplifies to π
equals 1.04π
sub zero. For bank B, when we substitute π equals one, we get π
equals π
sub zero times 1.003275 to the power of 12. When we simplify it and round this coefficient of π
sub zero to five decimal places, we get π
equals 1.04002π
sub zero. And so when we compare what we get for bank A and bank B, we can see that the coefficient of π
sub zero in bank B is a little bit higher.

It may be helpful here to illustrate this with an example for π
sub zero. Letβs say that the starting amount, π
sub zero, is 10,000 dollars. Thatβs whatβs invested in both of the banks. That would mean that the return of the investment π
for bank A would be 10,400 dollars. Then if we invested the same amount of money in bank B, the return on the investment π
would be 10,400 dollars and 20 cents. So even though itβs just by a small amount, bank B would be better.

Remember that we substituted the number of years to be equal to one in order to compare both banks after one year. Each year that money is invested in either bank, that will mean that the bank B return on investment will get proportionally larger each year. And so we can give the answer for the second part of the question that bank B has the better offer.