# Johnny kept a secret from Sara

Sara is in love with Johnny. They have been going together for almost four years. Sara and Johnny even had one big joint high school graduation party for their close friends well ahead of the end of the school year. They tell each other everything about their life events – well almost everything. Life went on as usual but this is the one time that they had a quarrel when Sara had come to Johnny’s place after the school and they were sitting and chatting.

Johnny: I am happy that I did very well in school, and in large part it was thanks to you. I told you that my dad wanted me to attend XYU and also that I got the admission.

Sara: Yes, I know. I also wanted to get into a science program at ZYU and got an admission letter from them last week. You know that. I also told you long time ago that Nana gave me some money, and that my parents have been putting money into a scholarship fund. Together that should cover at least 70% of the cost but Nana would chip in if I need it. I might also get a merit scholarship from the university.

Johnny: Yes, you told me about the scholarship fund almost two years ago. At that time, I also told you that my parents have a similar scholarship fund for me. I also have a secret fund I haven’t told you about.

# Sara got mad at Johnny

Sara: Since when did we start keeping secrets from each other? I have always told you everything about me – where I want to go for university, who is going to fund it, how my parents met, about my Nana and everything else. Why are you keeping secrets from me? I am going home.

Sara was pretty mad and left. Johnny tried to call her on the phone but she would cut him off. She didn’t know why Johnny had been secretive. Usually she was quite stable and well balanced but she couldn’t take it that Johnny would keep secrets from her. She thought, “What else was he keeping secret? If he has secret girl friends, he can go to hell.” Days went by and she didn’t talk to Johnny. She even took a different path to and from school so she wouldn’t run into him.

Several days later, Sara was at home and the doorbell rang. It was Johnny. Nana had let him in. Johnny had a piece of paper in his hand.

Johnny: Sara, I am sorry. I should have told you about it a lot earlier. Here is what I did not tell you.

Sara: Why would you not tell me about it? It says that you have lot more money than in your scholarship fund. So Johnny, do you think I would have been jealous of you? I am very happy for you but why keep it secret from me?

Johnny: Because I am stupid. I don’t understand what it says.

# 5% annual interest compounded continuously

Sara: It says that at the time of your birth, someone gave you a gift of $10,000 which was money deposited in a bank that would give you 5% annual interest compounded continuously until you withdraw the money.

Johnny: That’s what I don’t understand “interest compounded continuously”. I never heard of that. That’s why I felt stupid talking to you about it.

Sara: Come here my love.

Sara gave Johnny a long kiss and then said: Once my dad talked to me about it. I don’t quite understand it but we will figure it out together.

Johnny felt relieved that he wasn’t that dumb after all. Here was something that even his idol math whiz Sara did not know, and she had a 98% grade point average. He wasn’t alone. They chatted and smooched for a while and then Johnny went home. The next morning Sara did not avoid Johnny and they went to school together. Sara told Johnny that she talked to her dad who had explained everything to her. It would take a while and they could talk about it after school. After school, Sara went with Johnny to his home where they snacked and sipped on some pop, and then started talking about Johnny’s “continuously compounding” mystery.

Sara: The way dad explained it to me involved some basic algebra and some of the things we talked about when we were planning our graduation party.

Johnny: What? Who to invite out of our 100 friends or who sits next to whom?

# Binomial expansion

Sara: Both, mostly about the combinations but let’s start with something you know. What is (a + b)^{2} ?

Johnny: That’s simple – you multiply a + b by a and by b and then add the two together like

(a + b) x a = a^{2} +ab, (a + b) x b = ab +b^{2}, and then add them to get

(a+b) x (a +b) = a^{2 }+ 2ab + b^{2}.

Sara: We can determine the coefficients of the a^{2}, ab and b^{2} terms by considering different combinations of b. Because combinations of b can occur with a or with b, there are a total of two types of choices but none of them give a^{2}.^{ }Then the coefficient of the a^{2} term can be written as ^{2}C_{0 }^{ }which is 2!/(2!0!) or 1. The term ab comes from one choice – combination of b with a. Therefore, the coefficient of ab could be written as ^{2}C_{1 }which is 2!/((2-1)!(2-1)!) or 2/(1 x 1) which is 2. For the b^{2 }term, you must use b two times out of the total two times and thus its coefficient becomes ^{2}C_{2 }or 2!/(0!2!) or 1.

Johnny: Why muck around with all this, when I can get the answer the way I did?

Sara: Johnny boy, I am doing this to develop a pattern. Now, what is (a+b)^{3}?

Johnny: Do the same thing – multiply (a+b) with a+b two times and get the answer

or I could do this knowing that (a+b)^{2 }= (a^{2}+ 2ab + b^{2})

(a^{2}+ 2ab + b^{2}) x a = a^{3} + 2a^{2}b +ab^{2}

(a^{2}+ 2ab + b^{2}) x b = a^{2}b + 2ab^{2} +b^{3}

By summing up the two equations, I get

(a+b)^{3 }= a^{3} + 3a^{2}b + 3 ab^{2 }+ b^{3}.

Sara: It is the same pattern, ^{3}C_{0} choices which is 1 for a^{3}, ^{3}C_{1 }which is 3 for 3a^{2}b, ^{3}C_{2}which is 3 for ab^{2 }and ^{3}C_{3} which is one for b^{3}.

Johnny: I get it but what’s the big deal?

Sara: Now that we know the pattern, let’s expand (a+b)^{6}. You do it your way, multiply 6 times, I will follow this pattern to get the answer.

Very quickly Sara wrote:

(a+b)^{6 }= ^{6}C_{0} a^{6} + ^{6}C_{1} a^{5}b +^{6}C_{2} a^{4}b^{2} +^{6}C_{3} a^{3}b^{3} + ^{6}C_{4} a^{2}b^{4} + ^{6}C_{5} ab^{5} +^{6}C_{6}b^{6}

= a^{6} + 6 a^{5}b +15 a^{4}b^{2} + 20 a^{3}b^{3} + 15 a^{2}b^{4} + 6 ab^{5} + b^{6}.

Johnny struggled for a while and then came up with the same answer but then said,”Sara, I know that you are smarter than me but what are you trying to prove?”

Sara: You know this trend can continue, and so that you can write

(a + b)^{n }= ^{n}C_{0}a^{n }+ ^{n}C_{1 }a^{(n-1) }b+ ^{n}C_{2 }a^{(n-2) }b^{2 }………^{..}+ ^{n}C_{n-2 }a^{2}b^{(n-2) }+^{n}C_{n-1 }ab^{n-1 }+ ^{n}C_{n}b^{n}.

You could also summarize it as the Binomial Theorem

Johnny: I get that but what does this have to do with my problem of continuous compounding?

# Compound interests

Sara: If we say a is the principal and b equals the interest rate and that the compounding occurs n times, we could calculate the final amount as (a + b)^{n}.

Johnny: Yes, that is the basic idea of the compound interest. I know that.

Sara: The interest rate to be given to you is 5% per year. So if someone was to deposit one dollar, at end of the first year they would have $(1+.05) and at the end of the two years, it will be $(1 + .05)^{2}. The same way when the money has been in the account for 20 years. you would get the amount $(1 + .05)^{20}.

Johnny: I know that. I already calculated that with my calculator 1.05^{20} = 2.6533. I would get $26533. I didn’t need your Binomial Theorem for it.

Sara: So, if I tell you that you will get more money, would that make you happy or do I get that extra money?

Johnny: No, you don’t get the extra money because the bank will figure it out and give it to me if they think I earned more interest.

Sara: You have heard how some banks advertise that they give you an annual rate of 5% but they compound it monthly?

Johnny: Yes, I have heard of it. So the monthly interest would be .05 divided by 12, which is 0.004166667 and that the length of time is 20 x 12, which is 240 months. In that case, I would need to expand (a + b)^{n}, with a = 1, b = .05/12 or 0.004166667 and n = 240. Wow that’s a lot of compounding. One would have to do lot of multiplications. I see where one could use the binomial expansion. By compounding it monthly they would give me $10000 x (1.00416667)^{240 }which comes out to be $27126 which is $593 more than I thought I would get before. Sara, monthly compounding gave me $593 more.

Sara: For daily compounding your daily interest will become 0.05 divided by 365 but the it will be compounded 7300 times. What is (1+.05/365)^{7300}.

Johnny was excited and he changed to his laptop for doing these calculations. The number (1+.05/365)^{7300 }came out to be 2.7181 which meant that with daily compounding he would get $27181 which was even more money $27126 which he would get from monthly compounding. He wondered if they would to hourly compounding and asked Sara.

Sara: Why stop at hourly compounding? Let’s say if they would compound every second. Who cares if they don’t? It’s fun to think how much more you could get.

Johnny figured out the increase per second would be miniscule (0.05/365/24/3600) but the compounding will occur 20 x 365 x 24 times 3600 which is a whopping 567,648,000 times. So he calculated the final amount to be $10000 x 2.4596 or $27183, which was only two dollars more than from the daily compounding.

# Euler’s constant and binomial expansion

Sara: You see even if you keep compounding every second you don’t get that much more. You could keep shortening the compounding interval to microseconds and the compounding will become million times more often. You will see that sky is not the limit for the amount of money you could get. We could write 1 instead of a and 1/n instead of b, then

Since 1^{n-k } = 1, (1+1/n)^{n }= sum of (^{n}C_{k} (1/n)^{k}) from k =0 to n or you could write

When n →∞ (1+1/n)^{n}= 1+ 1 + 1/2 +1/6 +1/24 +1/120………………………….

This sum is called the Euler’s number or e. It cannot be calculated exactly but approximates to 2.7183. So that’s $27183 is approximately the maximum you can make, may be a few cents over if we calculated the series using lots of terms.

Johnny: I have two questions. First, is this the same Euler’s number we talked about when we were doing logarithmic functions?

Sara: Yes, the natural logarithms are to the base e instead of 10. What is your second question?

Johnny: We calculated all this for 20 years. What if I want to withdraw my money now which is 18 years after the initial deposit?

# Continuous compounded interest formula

Sara: Continuous compounded interest is based on the formula:

A = P e^{rt}, where P is the principal, r is the rate say 5% or 0.05 per annum for you and t is the time – in this case the number of years, and is the final amount with interest.

Johnny: So with r = 0.05 and t =18 years, I would get only $24596 but with t = 20 years, I got $27183. I also figure that if I withdrew after 50 years, I would get $121,825.

Sara: You could but you are lucky that I also discussed this part with my dad.

Johnny: What’s that?

# Income tax conundrum

Sara: Income tax. Right now, you are 18 and going to school. You have very little other income to speak of. There is no income tax on small gifts like this but the interest earned on them is considered taxable income.

Johnny: So, if I with draw now, my income will be $24596 minus $10,000 which is $14596. I wonder how much income tax I will have to pay.

Sara: Very little because you do not have much other income but if you withdraw later, you will have income from the scholarship fund and then from your jobs. The income tax amount will keep increasing.

Johnny: So, are you saying that I should withdraw this money now?

Sara; I didn’t say that. One of the possibilities is that you could put this money in a tax free savings account. If you do that you will have to pay income tax on the amount earned so far but you could keep earning more interest tax free.

Johnny: But if I do it now, I have to pay the income tax now from some other money.

Sara: Talk to your dad. I am surprised that you have not had this chat with him yet. I am sure you he will be glad to help you out. That way, you can have your cake and eat it too.

Johnny: Thanks Sara. I am really sorry that I kept this as a secret from you until now. I will talk to my dad.

Sara: The way I figure is that you can keep the money in the tax free savings account so that you withdraw it in future when we buy a house together. Just kidding!

Johnny told her that she was a fox and gave her a big hug and a long kiss.

*Challenge*

Farah, a grade 11 student, was sitting in the cafeteria and chatting with her friends. She was bragging about her inheritance. Her grandfather had left her an inheritance of $120,000 which was deposited on her 10th birthday but she would have full access to it on her 18th birthday. The deposited amount has been earning an annually compounded interest rate of 5 percent. Determine the amount on her 18th birthday using binomial expansion.

*Solution: *The amount at the end of the nth year is given by 120,000 x (1+.05)^{n }because the interest is 5% per annum compounded annually.

Binomial expansion can be written as the formula

(a + b)^{n }= ^{n}C_{0} a^{n }+ ^{n}C_{1 }a^{(n-1) }b + ^{n}C_{2 }a^{(n-2) }b^{2 }………^{..}+ ^{n}C_{n-2 }a^{2}b^{(n-2) }+^{n}C_{n-1 }ab^{n-1 }+ ^{n}C_{n }b^{n}

Using a =1 and n = 8 which is difference between the years 18 and 10, this expansion becomes

^{8}C_{0} + ^{8}C_{1 }b + ^{8}C_{2 }b^{2 }+ ^{8}C_{3 }b^{3}+ ^{8}C_{4 }b^{4}+ ^{8}C_{5}b^{5}+ ^{8}C_{6 }b^{6} + ^{8}C_{7 }b^{7}+ ^{8}C_{8 }b^{8}

= 1 + 8 b + 28 b^{2 }+56 b^{3}+ 70 b^{4}+ 56b^{5}+ 28b^{6} + 8b^{7}+ b^{8 }

With b = 0.05, it becomes 1. 1.477455. So the amount will be 120000 x 1. 1.477455 ≈ $177295.