Future and Present Value

We will go over investments, interests and compound interest, but for a previous explanation of these topics, please click here.

Investments and Interest

Let's say I want to start my own restaurant, or buy a new house. In either case, this would cost a lot of money, probably more than I have in my bank account. To overcome this obstacle, I can go to the bank and ask for a loan, and if the bank is convinced that I'll pay it back over time, they will lend me the money. As compensation for borrowing the money, I will also eventually have to pay the bank extra, called interest. The amount of interest to be charged is calculated as a percentage of the amount borrowed. The larger the loan, the larger the interest incurred.

However, did you know that every time you put money into a savings account, you are actually loaning the bank money? They use this money to help give out loans to others. It's all one big cycle! The bank will pay you interest on a regular basis for letting them use the money in your account, and you will make a small profit.

When you loan someone money in the hope of making a profit, you are making an investment. There are many other ways to invest money, such as putting money into a retirement fund, or purchasing government bonds. The present value of your investment will grow into some future value, and it is our task now to calculate the required present value of an investment if we hope to make some future value, or on the other hand, the expected future value of an investment if we invest some present value.

Compound Interest

Suppose I invest $\$1000$$1000 into a retirement fund, an amount which I expect to grow over the years in preparation for my retirement. Let's look at what happens to my investment as I earn interest over time.

Present Value	$\$1000$$1000
After $1$1 Year	$\$1100$$1100
After $2$2 Years	$\$1210$$1210
After $3$3 Years	$\$1331$$1331

At first glance, my investment seems to be growing randomly without any particular pattern. Is $\$100$$100 being added each year? No, that can't be right. How about if we check how big it has grown each year compared to the start? If we do the mathematics, we find that each year, it has grown by $10%$10%, then $21%$21%, then $33.1%$33.1% since the beginning. So no, there is no clear pattern there either.

Wait a minute. What if we check how big the investment has grown each year compared to the year before? In the first $12$12-month period, $\$1000$$1000 grew by $10%$10% from the start of the year. In the second $12$12-month period, $\$1100$$1100 also grew by $10%$10% from the start of the year, and likewise for the third $12$12-month period! Now we're getting somewhere. 

This is called compound interest and it involves interest being added to a growing investment. As your investment grows larger, it earns more interest!

In the case of my $\$1000$$1000 above, the interest rate was $10%$10% p.a. ('p.a.' stands for 'per annum', which means 'per year') and the interest was compounded annually.

Investments don't have to be compounded annually, though. They can be compounded semiannually (twice a year), quarterly (four times a year), monthly, weekly, or even daily. 

However, be careful! By convention, we only ever express interest rates per annum. If interest was being compounded monthly then, it would be your job to remember to divide this annual rate by $12$12.

The Compound Interest Formula

Let's go back to the example above involving the $\$1000$$1000 investment into a retirement fund at a $10%$10% p.a. interest rate. Remember that growing by $10%$10% is exactly the same as multiplying by $110%$110%. Let's call the initial amount $P$P (the principal amount) and set up a table for the amount $A$A after $n$n years.

$n$n (number of years)	$A$A (amount)	$A$A	$A$A
$0$0	$\$1000$$1000	$P$P	$P$P
$1$1	$\$1000\times110%$$1000×110%	$P\times110%$P×110%	$P\times110%$P×110%
$2$2	$\$1100\times110%$$1100×110%	$P\times110%\times110%$P×110%×110%	$P\times\left(110%\right)^2$P×(110%)2
$3$3	$\$1210\times110%$$1210×110%	$P\times110%\times110%\times110%$P×110%×110%×110%	$P\times\left(110%\right)^3$P×(110%)3

In the table, the column for $A$A has been written in three equivalent ways. It turns out that each year, we are just multiplying $A$A by another $110%$110%. However, when we multiply by percentages, what we are actually doing is multiplying by decimals. Multiplying by $110%$110% is the same as multiplying by $1.1$1.1. Our rate $r$r of $10%$10% can be expressed as $0.1$0.1.

Now, let's rewrite the above table in a different way.

 

$n$n (number of years)	$A$A (amount)	$A$A	$A$A
$0$0	$P$P	$P$P	$P$P
$1$1	$P\times1.1$P×1.1	$P\times\left(1+0.1\right)$P×(1+0.1)	$P\times\left(1+r\right)$P×(1+r)
$2$2	$P\times\left(1.1\right)^2$P×(1.1)2	$P\times\left(1+0.1\right)^2$P×(1+0.1)2	$P\times\left(1+r\right)^2$P×(1+r)2
$3$3	$P\times\left(1.1\right)^3$P×(1.1)3	$P\times\left(1+0.1\right)^3$P×(1+0.1)3	$P\times\left(1+r\right)^3$P×(1+r)3

And so we can come up with our general formula for compound interest.

Examples

Question 1

 

Different Interest Periods

Remember, be careful with $r$r and $n$n! They will change if the interest isn't being compounded annually!

Let's return again to the retirement fund example involving $\$1000$$1000. If the investment is compounded semiannually, suddenly I have a semiannual rate $r=5%$r=5%$=$=$0.05$0.05 and interest periods twice a year. If I want to calculate the future value of my investment after 2 years, then $n=4$n=4!

Hence, we get $FV=\$1000\times\left(1+0.05\right)^4$FV=$1000×(1+0.05)4$=$=$\$1215.51$$1215.51

You might ask: Is it better to compound more often, or less often? Or does it give the same value?

Let's return to the $\$1000$$1000 retirement fund investment. After $1$1 year it grew to $\$1100$$1100. What would happen if we compounded semiannually at the same interest rate of $10%$10% p.a.? Our interest rate would need to be divided into two, for each half of the year, giving $r=0.05$r=0.05. On top of that, we need to double the amount of interest periods that there would be with yearly interest only. 

Now, what would happen if we compounded quarterly? Or monthly? Look at the table below comparing the value of the investment under different interest periods.

 

	Annually	Semiannually	Quarterly	Monthly
	$r=0.1$r=0.1	$r=0.05$r=0.05	$r=0.025$r=0.025	$r=0.00833$r=0.00833
Present Value	$\$1000$$1000	$\$1000$$1000	$\$1000$$1000	$\$1000$$1000
After $1$1 Years	$\$1100$$1100	$\$1102.50$$1102.50	$\$1103.81$$1103.81	$\$1104.67$$1104.67
After $2$2 Years	$\$1210$$1210	$\$1215.51$$1215.51	$\$1218.40$$1218.40	$\$1220.29$$1220.29
After $3$3 Years	$\$1331$$1331	$\$1340.10$$1340.10	$\$1344.89$$1344.89	$\$1348.02$$1348.02

The above table shows us that for the same investment of $\$1000$$1000 at an interest rate of $10%$10% p.a., it is slightly better to compound more often. Banks are aware of this and have to adjust the kind of investment deals they offer so as not to give some investors an unfair advantage.

Experiment with changing the values below to observe what happens when you change investments, interest rates, and periods.

Calculating Present Value from Future Value

In some questions, we might be asked to work backwards from a given future value to find the present value that created it or would be expected to create it.

Let's say I want to save up to buy a car worth $\$20000$$20000 in $6$6 years. If I invest my money now into a fund that compounds interest annually at $5%$5%, what principal value do I need to invest at present so that I can afford the car in the future?

By rearranging the compound interest formula to make $PV$PV the subject, we get $PV=\frac{FV}{\left(1+r\right)^n}$PV=FV(1+r)n​. We input our values to get $PV=\frac{20000}{\left(1+0.05\right)^6}$PV=20000(1+0.05)6​$=$=$14924.31$14924.31 rounded to the nearest cent. So I need to invest at least that much to afford the car in $6$6 years time!

Examples

Question 2

Question 3