Now that we understand natural logarithms, let me give you another – let’s go back to finance and give you a problem in finance to work on.
Suppose a person who’s thinking about retirement has a certain target wealth that he or she must arrive at before they think they have enough money to retire.
And suppose – you know, we go back to this equation. Imagine that your money is invested at a continuously compounded interest rate of r.
And presently, the amount of money you have is A dollars, that’s what you have right now.
And you want to figure out how long it will take until your money gets to your target level of V before you can retire.
To make this specific, let’s suppose that currently your wealth is 1 million dollars.
But you think you need 2 million dollars before you can retire.
If you have 2 million dollars, you figure out that’s enough, I can put it in the bank and I can live off the interest.
And that’ll be enough money for me.
Now, suppose the interest rate is 3 percent.
The continuously compounded interest rate, let’s say, is 3 percent.
Now, the question is
How long will you have to wait before you can retire?
How long do you have to wait before this 1 million dollars becomes 2 million dollars?
And, actually, we’re making this problem a little bit simple because we’re assuming you’re not going to be saving in the interim.
If you were saving money in the interim, you could be building up your assets.
But you’re just going to depend on the power of compound interest.
So, how would you figure out how much time you have to wait in order for your 1 million dollars to become 2 million dollars
if your money is invested at a continuously compounded interest rate of 3 percent?
[Student comment]
You know the value of V.
You know the value of A and you know the value of r.
so, let’s just substitute in. 2 equals 1 times e to the 0.03t.
And, we’re trying to solve for t.
And your suggestion is a good one.
Let’s take the natural log of both sides.
So, taking the natural log of both sides, what do we get?
The natural log of 2 equals the natural log – this 1, you can just omit – the natural log of e to the 0.03t.
And then, what can we do with that right-hand side?
Is there anything we can do with that righthand side to simplify it?
[Student comment]
We can use this rule #3. Notice e is raised to the power 0.03t.
You could think of that as being the little a in rule 3.
So then, you can bring this down and make this right-hand side 0.03t times e to the 1 power.
Excuse me. I forgot the ln, didn’t I?
And, by the way, what is the natural log of e?
[Student comment]
1. Because what power to you raise e to in order to get e?
1.
Sometimes these simple questions can throw you off. But this is just equal to 1. So, this is 0.03t.
Now, if we divide both sides by 0.03, you get the natural log of 2 divided by 0.03 is equal to t star, where t star is the value of t that we wanted to solve for.
We’ll put a star there.
Now, what is the natural log of 2?
This is something you’re going to have to be able to do on the exam.
What’s the natural log of 2?
[Student comment]
0.6931
So, you divide that by 0.03 and when you do that, what do you get?
[Student comment]
23.1.
Sort of a discouraging answer, isn’t it?
if you’re just going to depend on the power of compound interest, it’s going to take 23.1 years for your 1 million to become worth 2 million.
Of course, had you had your money invested in the stock market, and assuming the stock market delivers a long-run historical average return, say 9 percent, then let’s see what it would be.
Suppose you redo this problem.
Instead of the continuously compounded interest rate being 3 percent, suppose it’s 9 percent.
Then, you would just redo this problem by making this 0.09.
So, again, we would take the logs of both sides.
the natural log of 2 equals the natural log of 0.09t.
And now you bring this 0.09 down. This becomes a 0.09 here.
And now, t is equal to the natural log of 2 again but it’s divided by 0.09.
And now, what does your answer give you?
[Student comment]
Ah, that’s much better. 7.7 years.
So, that shows you the power of compound interest at higher interest rates. Dramatically shorter
time.
