# Compound Interest

What is compound interest and how does it help us to get ahead with investing? Professor Gilbert discusses compound interest in this section.
• When we put money into a savings account, the bank agrees to pay us interest at a particular point in time and at a specific rate.
• For instance, we might have a savings account that pays us interest monthly and is offering 6% interest (for simplicities sake) per annum (for the entire year). In essence we would receive 0.5% interest on our savings at the end of each month. Say we have $1000 in our savings account, that would mean that one month later we would receive$5 in interest. Now we have two options, we can take the money and spend it, or we can leave it in the bank.
• If we leave it in the bank and ‘reinvest’ it, we earn compound interest. Compound interest is earning interest on the interest that we earned earlier.
• So, we reinvest our $5, that means in the second month we have$1005 in savings that we earn 0.5% that month. So instead of earning $5 in interest we now earn$5.025. We have earned 2.5 cents more than we did in the first month.
• Doesn’t sound like much right, would not even get you a sweet. But what if we keep reinvesting our interest payments.
• Well at the end of the first year we would have $1,061.68, which is$1.68 more than we would have earned if we took the interest out and only kept our $1000 in the bank each month (what we call simple interest). If we took the interest out, we would have earned$5 per month and $60 over the entire year. • By the end of the second year, we would have$1127.16, compared with $1,120,$7.16 more and we would be earning $5.64 in interest in that month. Again, doesn’t sound like much, but it adds up over time. • After 5 years we would have$1,348.85 and be $48.85 better off than if we didn’t reinvest our interest. After 10 years we would have$1819.40 and be nearly $220 better off. More importantly, we would be earning$9.10 per month in interest, nearly double what we started with.
• After 15 years we would have $2,454 • After 20 years we would have$3,310
• After 25 years we would have $4,465 • After 30 years we would have$6,023
• After 40 years we would have $10,957 and we would be earning$55 per month in interest- 11 times what we started with
• After 50 years we would have $19,936 and we would be earning nearly$100 per month in interest
• In fact, the graph below shows the increase in the value of our savings

• What the graph demonstrates is that compound interest behaves exponentially, overtime the rate of increase gets faster and faster as we earn higher interest each period. Unfortunately, it takes a long time for it to really build up. Who wants to wait 40 years!
• So how can we supercharge it?
• Several factors influence the speed with which compound interest works.
1. Frequency with which we earn interest
3. Whether we make regular additions to our savings
• Let’s look at each in turn

### 1 Frequency

• In our previous example we had money in the bank earning 6% per annum but the interest was paid out monthly. This means that each month we earn 1/12th of the annual interest each month, but most importantly, we start earning interest on interest earlier then if we earn interest just once per year. But how much of a difference does it make?

• As you can see from the graph above, earning interest monthly as opposed to annually results in earning nearly an extra $7,000 70 years from now, this is nearly 12% higher so it’s not nothing! ### 2 Interest Rate • This obviously has a substantial impact, the higher the interest rate the more we earn each period and therefore the more there is to earn interest on next period. • In our initial example we used 6% per annum, or 0.5% per month. Unfortunately, this is not very realistic at the moment as interest rates are globally low for savings accounts. • But for the sake of simplicity, what if we assumed that we could get 3% per annum, or .25% per month, how big a difference would that make to our eventual savings? • The answer is a lot! Over the space of 20 years the difference in your savings between earning 6% and 3% is nearly double. Specifically, you would have earned$820 in interest at 3% and $2,310 at 6%, nearly 3 times as much interest. And over 50 years the difference is a 445% difference in the amount of interest earned,$3,473 vs $18,935! • This point is important to remember when considering investment options. • We will talk a little later about the concepts of risk and return, but what the above graph demonstrates is that a higher expected return (in this case a higher interest rate) can dramatically impact the value of your savings in the future. This is the potential payoff for taking on riskier investments. ### 3 Additional Savings • Another way to supercharge the power of compound interest is to add to your savings yourself. • In this case, small but frequent additions can quickly grow the size of your investment meaning you are earning more interest sooner. • What if, based on our original example, we added$5 per month to our savings. So, if we invest $1,000 for 50 years we get just under$20,000. If in addition we save an additional $5 per month, so$3,000 in total over the 50 years, we get to just under $39,000, nearly double. The video explains the concept of compounding. Also, how to calculate the future value of investing with compounding mathematically. Concept of Compounding The video explains the concept of compounding. Also, how to calculate the future value of investing with compounding mathematically. Compound interest introduction – Interest and debt – Finance & Capital Markets – Khan Academy This is an additional video, hosted on YouTube. ###### If you can’t see the video, please click on this link ## Time value of money • One of the most important ideas in finance is the present value of money. The way we value investments like stocks, money market instruments, bonds etc is based on the present value of their future cashflows. • So, what do we mean by the present value? • At its heart, time value of money is the idea that$10 today is not worth the same amount as $10 in one year from now. • Why is that? After all, I use the same$10 bill at both times. The difference is that $10 in the future is worth less to me then if I receive it now. • Why? If I receive the$10 today, I can invest it and earn more! Say I can get 5% in the bank, in that case my $10 would be worth$10.50 a year from now.
• The present value is a concept that allows us to convert future cashflows into the equivalent dollar value in today’s terms. So, for instance, say I was offered an investment that would pay out $1,000 in 5 years. • If the appropriate ‘interest rate’ for that investment is 5%, then the most I would be willing to pay for that is$783.53. If it costs me anything more than $783.53, I am better off putting the money in another investment that pays me 5% as I will have more than$1000 in 5 years. If it costs less, then it is a worthwhile investment.
• To calculate the present value of a cashflow we need to know the expected future cashflow, when it will occur and the appropriate discount rate for the cashflow.
• The discount rate is effectively the return we need to earn for this particular investment and is driven by how risky the investment is.
• We will discuss the relationship between risk and return later, but for now it is simply enough to recognise that the discount rate is a compensation for the risk related to a cashflow and that it can be thought of as the opportunity cost of your money i.e. what you could earn on an equivalent investment with similar risk.
• The discount rate is important as it has a significant impact on the present value of future cashflows. A smaller discount rate, which equates to a lower interest rate, means that future cashflows are worth more today than if you have a higher discount rate.
• Thinking about this from an investment perspective though, if you are wanting to build higher future wealth then a higher discount rate would indicate that you can turn less money today into the same amount in the future.

### Exercise

• Investing is about achieving financial goals, saving for the sake of saving rarely works, but if you have a target or goal in mind then you are more likely to stick with it.
• So, let us assume my goal is to take a luxury holiday in 5 years time. I estimate that the holiday will cost $25,000. • If I can invest my money in the bank at 4% per annum compounded annually, how much money would I need to stick in the bank today to get to my target? • Use the formula and skills shown in the video above. What about if I can save at 6% or 8% instead? ### Solution At 4% you would need to invest$20,548.18 today. Alternatively at 6% you would need to invest $18,681.45 and at 8% you would need to invest$17,014.58.

You use the formula PV = FV/(1+r)^n, for 4% this would look like 25000/(1.04)^5.

### End of Topic Videos

This video explains how to calculate the present value of cashflows and provide details on time value of money.

Introduction to present value – Interest and debt – Finance & Capital Markets – Khan Academy

Present Value 2 – Interest and debt – Finance & Capital Markets – Khan Academy