Category: Financial Maths

Last time we saw that our $1000 invested at 3% compounded annually will result in $1343.92 in 10 years. To get even better results, the interest rate can be compounded more frequently than annually. Let’s say the interest is applied mid-year. Then the interest earned during the first half of the year will be added to the principal amount and that total will be used to apply the second half interest to.

This will change the formula A_n = A₀(1 + r)ⁿ a bit though. First of all, we can’t apply the full per annum interest rate to the initial investment as only half a year has passed. So only half the interest rate will be used. Also, the period length is now half a year so the number of periods refers to how many half-years have passed.

So in our example, the annual interest rate is 3%, so the 6 month interest rate, r, is 3/2 = 1.5% since there are 2 six-month periods in a year. If we want to know how much we will have after 10 years, the number of periods, n, is now 10×2 = 20 since there are 20 half -year periods in 10 years. So we can use the same formula with r = 1.5% = 0.015 and n = 20:

A₂₀ = A₀(1 + r)²⁰ = 1000(1 + 0.015)²⁰ = $1346.86

So the extra compounding has made us a bit more money.

You might ask (OK – I’ll ask for you), would we make more money by compounding more frequently? Yes we would!

Let’s compound every quarter-year. this means the interest rate we apply each quarter is 3/4 = 0.75% and the number of periods after 10 years is 10×4 = 40:

A₄₀ = A₀(1 + r)⁴⁰ = 1000(1 + 0.0075)⁴⁰ = $1348.35

Looks like we want more yet. What about monthly? Here r = 3/12 = 0.25% = 0.0025 and n = 10×12 = 120:

A₁₂₀ = A₀(1 + r)¹²⁰ = 1000(1 + 0.0025)¹²⁰ = $1349.35

Better, but notice that this is not much better that quarterly. It appears that we will reach a limit as to how much we can make. Let’s try compounding daily. Here r = 3/365 = 0.0082% = 0.000082 and n = 10×365 = 3650:

A₃₆₅₀ = A₀(1 + r)³⁶⁵⁰ = 1000(1 + 0.000082)³⁶⁵⁰ = $1349.84

Well that’s disappointing. There is only a 0.49 difference between compound monthly and daily after 10 years.

There is a maths formula that computes the amount of interest if the investment is compounded continuously. This is the limit of what you can make by compounding. For our example, after 10 years, the most compounding will get us after 10 years is $1349.86, just 2 cents more than compounding daily.

In my last post, I developed a recursion formula for compound interest where the interest is compounded annually:

A_n+1 = A_n + A_n × r/100 where you use the percentage value of r or
A_n+1 = A_n + A_n × r where the decimal equivalent of r is used.

These are good formulas if you want to know how much money you will have next year, but will take a while if you want to know how much you will have in 10 years. However, we can use the recursion formula to help develop the direct formula.

I will use the decimal equivalent equation so I don’t have to keep writing “/100”. I will add that at the end of this development.

Using the recursion formula, at the end of year 1 you will have

A₁ = A₀ + A₀ × r = A₀(1+r)

Please review my post on factoring, specifically the distributive property, if you do not understand why A₀ + A₀ × r = A₀(1+r).

At the end of year 2, you will have A₂ = A₁ + A₁ × r = A₁(1+r). But we have A₁ in terms of our initial investment A₀ above. So substituting A₁ = A₀(1+r) into A₂ = A₁(1+r) results in

A₂ = A₁(1+r) = A₀(1+r)(1+r) = A₀(1+r)²

Wait! The exponent 2 of (1+r) is the same as the current period (the subscript of A₂. Wouldn’t it be nice if this pattern continued? Let’s do one more iteration, substituting A₂ with the above equation:

A₃ = A₂ + A₂ × r = A₂(1+r) = A₀(1+r)²(1+r) = A₀(1+r)³

Wow! It looks like this is true. In fact, it is. The exponent of the (1 + r) part is the same as the period of interest. If you want to find how much you will have after 10 years, you can use the direct formula:

A_n = A₀(1 + r)ⁿ

So in our example of investing $1000 at 3% interest per annum (that is each year), compounded annually, you will have after 10 years

A₁₀ = A₀(1 + r)¹⁰ = 1000(1 + 0.03)¹⁰ = $1343.92

Of course, you need a calculator that can do exponents, or you can use the all-knowing internet. The percentage form of the direct equation is

A_n = A₀(1 + r/100)ⁿ

where you would substitute the “3” in for r instead of “0.03”.

I said last time that interest can be applied (compounded) more frequently than annually. I will continue this example next time where we do just that.

It is now time to introduce compound interest. The main difference between compound interest and simple interest is that in compound interest, the amount that is used to apply the interest, changes each period. With simple interest, this amount is the initial investment and never changes.

In compound interest, the interest earned from previous periods is added to the initial investment (the principal) before applying the interest for the current period.

Let’s continue with the example of investing $1000 at 3% interest per annum (that is each year). In compound interest, the interest can be applied (compounded) even within a period, but let’s start out simple and apply the interest at the end of each year.

At the end of the first year, the 3% interest is applied to the initial $1000, so we now have $1030 in the bank just like with simple interest. However, things are different at the end of the second year. In simple interest, the interest rate is applied to the initial $1000, but with compound interest the $30 earned after the first year is added to our initial investment so that the principal has changed to be $1030. So after the second year, instead of applying the 3% to $1000, it is now applied to $1030. The following table shows how the investment is growing for the first 5 years:

n	P	r	I	An
0	1000	0	0	1000
1	1000	3%	30	1030
2	1030	3%	30.90	1060.90
3	1060.90	3%	31.83	1092.73
4	1092.73	3%	32.78	1125.51
5	1125.51	3%	33.77	1159.28

Notice how the P for a current period is the A_n of the previous period. Compare this table with the one from my last post which showed the investment growth using simple interest:

n	P	r	I	An
0	1000	0	0	1000
1	1000	3%	30	1030
2	1000	3%	30	1060
3	1000	3%	30	1090
4	1000	3%	30	1120
5	1000	3%	30	1150

Looks like it’s better to invest using compound interest. In our example, we have $9.28 more after 5 years.

So what is the recursive formula for this, that is, how do we calculate the current year’s amount if we know last year’s amount?

Starting with A₀:

A₁ = A₀ + A₀ × 3/100= $1000 + $30 = $1030
A₂ = A₁ + A₁ × 3/100= $1030 + $30.90 = $1060.90
and so on until
A₅ = A₄ + A₄ × 3/100 = $1125.51 + $33.77 = $1159.28

Again, let’s make this more general. The “3” in the above is the interest rate r. So replacing the “3” with r, we get the general recursion formula:

A_n+1 = A_n + A_n × r/100

or if you use the decimal equivalent of r:

A_n+1 = A_n + A_n × r

This is OK for finding the amount next year, but what if you want to find the amount after 10 years? You do not want to apply this recursion 10 times. In my next post, I will develop a more direct formula.

So I am talking about simple interest. In my last post, I explained how to calculate the interest after the money has been invested for one period – one year in our example. It turns out that by investing $1000 at a simple interest rate of 3%, you earn $30 after one year. This means you have a total of $1000 + $30 = $1030 after one year. What if you want to know what you have after 5 years?

There are two ways of doing this: sequentially or directly. The sequential method has the advantage of showing how your money is growing each period, and the formula is very useful for entering in spreadsheet applications like MS Excel. Let’s first discuss this sequential method.

So after the first year, you have $1030. Each year, the interest rate of 3% is applied to the initial $1000 investment, and you get an additional $30. Below is a table of how the investment grows each year. I will explain the headings and the calculations afterwards:

n	P	r	I	An
0	1000	0	0	1000
1	1000	3%	30	1030
2	1000	3%	30	1060
3	1000	3%	30	1090
4	1000	3%	30	1120
5	1000	3%	30	1150

So according to this table, you will have $1150 after 5 years. So what are the column headings?

The first 4 were defined in my last post, but I’ll repeat them here. n is the period number. It starts at 0 since this indicates when time starts. You only get interest after the money has been invested for 1 period (a year in this case). P is the principal which in this scenario, is the amount originally invested. r is the interest rate. I is the amount of interest earned. From my last post, this is calculated as I = Pr or I = Pr/100, depending if you use the decimal equivalent of r or not (see my last post). A_n is the total amount you have after n periods. Using a subscript like this is very common in maths. A₀ is the initial amount after 0 periods. A₁ is the amount after 1 period. A₅ is the amount after 5 periods.

I know the table is a bit repeating with the $1000 and the $30 repeated throughout the table, but I did this so you can see the difference between simple interest and the eventual compound interest that I will talk about later.

Notice that the difference between A_n for each adjacent period is $30, that is, $30 is added to the previous A_n to get the next period total amount, A_n+1. So the sequential formula to calculate the next period’s amount is:

A_n+1 = A_n + 30

This is called a recursion formula as you recursively calculate the next period’s total amount by knowing the previous period’s amount. So starting with A₀:

A₁ = A₀ + 30 = $1000 + $30 = $1030
A₂ = A₁ + 30 = $1030 + $30 = $1060
and so on until
A₅ = A₄ + 30 = $1120 + $30 = $1150

Now let’s generalise this formula for any interest rate. The $30 in the above example is the interest I from the formula I = Pr or I = Pr/100. So the general recurring formula for the total amount of interest in a simple interest investment is:

A_n+1 = A_n + Pr (decimal equivalent r) or A_n+1 = A_n + Pr/100

That’s all well and good for a spreadsheet formula, but what if you only want to know how much money will you have after 10 years? Do you need to apply this formula 10 times to get the answer? The answer is “no” because we can get a formula that directly calculates an answer.

If you are adding the same amount each year, after 10 years, the total amount added is 10 × the amount after 10 years. So in our example, after 10 years, the total amount is $1000 + 10×30 = 1000 + 300 = $1300. Notice that I will get the same result as in the table above after 5 years: 1000 + 30×5 = 1000 + 150 = $1150. So you just need to multiply the same amount of interest each year by the number of periods desired. In general,

A_n = P + Prn (decimal equivalent r) or A_n = P + Prn/100

Next time I will introduce compound interest. But to prepare you for this a bit, notice that I can factor out a P from the above formula to get an equivalent one ( please see my posts on the Distributed Property if you need a review):

A_n = P(1 + rn) (decimal equivalent r) or A_n = P(1 +rn/100)

Knowing how to use this form of the equation will help you understand the compound interest formulas.

For some of my students, interest calculations are troublesome: you can say that they quickly lose interest in interest.

If I still have your interest after that bad joke, I will continue. The two main types of interest are simple and compound interest. In simple interest, the principal (the amount initially invested) stays the same and interest is calculated on that amount at all times. In compound interest, the principal grows and the value upon which interest is calculated changes.

I have previously talked about percentages and how to take a percentage of a number. Please review that if you do not know how to take a percentage of a number.

As always, in any new topic, there are some definitions to know so that we understand each other. The following are the main definitions with the abbreviations for them that will be used in equations:

Principal (P): the amount invested or borrowed
Interest rate (r): a percentage to be applied to the principal. This can be a percentage (eg. 15%) or its decimal equivalent (0.15).
Interest (I): the dollar amount which results when the interest rate is applied to the principal
Time (t): the amount of time to be used in a problem
Period: the basic amount of time used by the interest rate. For example, 15% per annum (abbreviated p.a.) means that the period is 1 year.
Number of periods (n): The number of periods to be used in a given problem. Note that equations can be in terms of time (t) or number of periods (n).

Let’s start out with a simple interest situation. Suppose I invest $1000 at a simple interest rate of 3% p.a., that is 3% each year. Though I haven’t asked a question yet, let me identify the key items of this set up:

P = $1000
r = 3% or 0.03
period = 1 year

So my first question is: how much interest do I earn after 1 year? At the end of each year, if I keep that initial amount 0f $1000 in the investment, I will earn 3% of $1000 in interest. If you remember, to take a percentage “of” something, the “of” means to multiply. So after 1 year:

I = 3% × $1000 = (3/100) × 1000 or 0.03 × 1000 = $30

Note that in equations where you can put the interest rate in directly (the “3”), there will be a “/100” part in the equation. In equations where the decimal equivalent of the interest rate (0.03) is to be used, there will be no “/100” part. So the formulas to find the amount of interest (I) earned in 1 period are:

I = Pr/100, if you like to use the interest rate number directly (the “3”)

I = Pr, if you like to use the decimal equivalent of the interest rate (0.03)

This is why you may see different formulas in different books.

In my next post, I will ask some more questions about this investment and provide more formulas.