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Discussion
Imagine you make an investment of $100 dollars that increases in value by 10% each year.
What will your investment be worth after a year?
What will your investment be worth in a decade?
Part (a)
After one year, an investment grows by a factor of 1+100r, where r is the annual interest rate. The key idea is that the “1” keeps the original amount while the 100r adds the extra r%.
So a 10% interest rate means the value is multiplied by:
So the value after one year is:
Part (b)
Every year you start with last year’s total and give it a ten‑percent boost, which you can do by multiplying by 1.1 just like we did in (a).
Year 0 (now): You have $100.
After year 1 (from prior):
After year 2:
After year 3:
Do you see the pattern? After n years the investment's value is equal to:
In ten years (n=10) this gives
So after a decade the investment is worth about $259.37.
We can generalize the ideas above to develop a formula to calculate the future value of some given investment after some given period of time.
What would we need to know?
The present value of the investment, which represents the initial value. We will call this PV for short.
The interest rate (or growth rate) expressed as a percentage each year. We will call this r.
The number of years the investment grows for. We will call this n.
How do these variables come together?
We start with the present value of the investment at n=0. The formula to find the investment's value after n years is just the present value, multiplied by the growth factor we found, raised to the power of n years.
This yields the compounding annual interest formula:
Checkpoint
You invest $75 into a company that produces annual returns of 20%.
What is the future value of your investment two years from now?
Select the correct option
Formula aside, let's analyze the behavior of compounding a bit further. Take a look at the graphic below.
Powered by Desmos
Discussion
What do you notice?
Each year, we can see that the 10% chunk we take becomes greater and greater, resulting in the original $500 doubling in under a decade. This animation reflects the power of compounding: each year we grow more because we are adding on 10% of a larger amount each year!
The "base" gets larger and larger: it's like rolling a snowball down a hill. As it accumulates more snow, it's able to grow more on the next roll. The key idea here is that a bigger and bigger chunk grows by 10% each period, so the growth snowballs upwards.
Discussion
While investments are typically expected to appreciate (grow) in value, some things we buy do the opposite: they depreciate in value.
Can you think of some purchases that lose value over time rather than gaining it?
Items such as cars, electronics, appliances, textbooks, and many other goods tend to depreciate at a fixed rate over time.
Suppose we have a airplane worth $20,000,000 that loses 4% of its value each year. How much will it be worth in a decade?
Every year the airplane loses 4% of its value, so you keep 100%− 4%= 96% of the previous year’s worth by multiplying by 0.96.
Year 0 (now): Value = $20,000,000
After year 1:
After year 2:
Hopefully the pattern becomes evident: each year we multiply by a factor of 0.96.
Repeating this process until year 10:
So after a decade the plane is worth about $13,296,652.72.
How might we be able to measure this depreciation mathematically?
The key idea here is very similar to that of an investment appreciating. The only difference is that rather than using a growth rate of 1+100r, we use a depreciation rate of 1−100r.
So the formula changes to:
Depreciation
where FV is the future value, PV is the present value, n is the number of years, and r% is the annual depreciation rate of the item.
While a fixed-percentage growth speeds up in growth over time, a fixed-percentage decay slows down in loss over time. Once some value has been shaved off from the original price, there is less to be lost: taking the "percent of a smaller pie" leads to a smaller chunk being lost each time.
The key idea is that decay eases off as time progresses.
Exercise
Hannah Lee, the CTO of CanopyCart, has invested her wealth into 1250 shares of CanopyCart stock, each valued at $200. The shares of CanopyCart depreciate by 20% of their current value each year.
Find the value of the shares, rounded to the nearest dollar, after 4 years.
Select the correct option
When banks quote you an interest rate, say, 12% per year, they're giving you a nominal annual rate, (explain this better). This is often called the APR, which stands for "Annual Percentage Rate", but you won't see this term on your exam.
But in real life, interest is sometimes added more frequently. Say your account yields 12% interest but has a catch -- interest is gained every month.
This is a key idea:
The interest rate is annual, but we get paid monthly. So instead of the account growing 12% all at once, we gain
On IB exams, the only compounding periods you will be asked about are
annually,
semi-annually (every 6 months),
quarterly (every 3 months), or
monthly.
You will always be given the annual interest rate, so if the payments per year (compounding periods) are shorter than a year, don't forget to compute the % per period, just like we did above!
Discussion
Alex and Jamie each go to the bank and take out a 1 year, $10,000 dollar loan with an interest rate of 12% per year, compounded monthly.
Each handles their monthly interest differently -- Alex pays off each month's interest right away, while Jamie lets each month's interest roll into the loan balance.
What is the total interest Alex pays over 12 months?
What is the total interest Jamie pays over 12 months?
Why are they different? Explain your reasoning.
Part (a)
The annual rate is 12% compounded monthly, so the monthly rate is
Alex pays each month’s interest as it arises. Each month he pays
so over 12 months the total interest is
Part (b)
Jamie lets interest roll into the loan, so we use our compounding formula, but have to apply it to each month, instead of multiple years like before. After 12 months,
His total interest is
Part (c)
They are different because one is compounded in multiple shorter periods which results in additional interest being added to the account.
Alex pays $1,200 because he never pays extra interest beyond what is owed on the principal amount.
Jamie pays about $1,268.25 because each month’s interest itself accrues additional interest in subsequent months.
Earlier, we learned the compounding annual interest formula:
But this version does not account for more than one compounding period per year! So we have to make some slight adjustments to the formula, to account for changes in compounding periods.
Recall that the 100r represents r% as a decimal. But now, instead of r% each year, we have kr% interest, compounded k times per year. So the 100r becomes
And, since we're compounding k times per year for n years, we have k×n compounding periods.
Compound Interest Formula
where FV is the future value, PV is the present value, n is the number of years, k is the number of compounding periods per year, and r% is the nominal annual rate of interest.
Notice that this is formula takes the shape of a geometric series with:
First term: PV (your initial investment)
Common ratio: (1+100kr) (the growth factor each period)
Term number: The specific period which you want to find FV
Interestingly enough, the compound interest formula: FV=PV×(1+100kr)kn represents exactly the kn'th term of this geometric series.
Discussion
Say you make two separate investments of $500. Each grows at 10% annually, but one is compounded quarterly and one is compounded annually.
Which investment will grow more after one year?
Conceptually, we can tell immediately that the investment compounded quarterly will grow more. After the first period, each quarter's interest itself accrues additional interest, which will result in further gains compared to the investment compounded annually.
Mathematically, the investment compounded annually grows to
But when interest is added every quarter, the nominal 10% rate is split into four 2.5% periods. Each quarter’s interest then itself earns interest in the remaining quarters. After four quarters the amount is
Because quarterly compounding pays interest earlier (and that interest earns more interest), the growth after one year ends up becoming about 10.38%. Since $551.90>$550, the quarterly‐compounded investment grows more in one year.
Your calculator includes a powerful tool called the TVM (Time Value of Money) Solver, which simplifies complex compound interest problems by handling all the calculations for you. Instead of plugging in manually, you just enter a few key values and the calculator takes care of the rest.
It uses our formula for compound interest:
...but instead of plugging in manually, you fill out fields like a form on your calculator, and the TVM solver can solve for any unknown in the equation (not just the future value)!
Say you get a question where you have to find the interest rate r given values for all the other variables. If you try solving directly using the formula, the algebra would get pretty complicated. The TVM Solver is your best friend in a situation like this -- all you have to do is enter in the parameters and it does all the heavy lifting!
Using TVM Solver (Calculator) - Compound Interest
You should understand the meaning of each variable and know how to use your calculator's Finance/TVM Solver:
To solve for an unknown, move your calculator's cursor to the unfilled slot and press alpha → enter.
Be very careful if P/Y is different from C/Y. The letter N will always be the number of payment periods, or in other words the number of years times P/Y.
Positive & Negative Cash Flows (TVM)
Whenever you use the Finance App (TVM Solver) on your calculator, it's critical that you enter and interpret the signs correctly:
When you receive money from a bank or savings account, that value is positive, because you're gaining money.
When you send money to a bank, that value is negative, because you're losing money.
In the context of savings or investments:
PV = Negative (you give money now)
FV = Positive (you receive money later)
An easy way to build this understanding is to think about the flow of cash during the transaction. Think about if you are losing or gaining money in the moment, and if you will lose or gain money when completing the transaction.
Exercise
A savings account with SuperBank offers an annual interest rate of 5% compounded monthly. Adam deposits $2500 into an account.
Find the balance of Adam's account after five years, giving your answer to the nearest dollar.
Select the correct option
Discussion
Both James and Arjun invest $1000 in an account with an annual interest rate of 10%, compounded quarterly. They both want to calculate how much they have after 5 years.
James uses the formula for compound interest:
while Arjun attempts to solve using his calculator. He enters in the following values and solves for FV:
N = 20
I% = 10
PV = -1000
PMT = 0
FV = ?
P/Y = 1
C/Y = 4
PMT: END
Do these calculations yield the same result? Is there anything incorrect you notice with Arjun's entries?
Arjun's calculations are incorrect. The issue lies in his inputted value for P/Y and it's relationship with N.
N describes the total number of payment periods. With one "payment period" per year, N = 20 implies a 20 year timeframe, instead of the 5 we are looking for.
There are two ways Arjun could input this into his calculator:
Set N = 20, and set P/Y = 4. So N counts quarters and we find the FV after 5 years, as intended.
Set N = 5, and set P/Y = 1. So a payment period is a year. Behind the scenes, the TVM Solver finds the effective annual rate that would arise as a result of compounding quarterly with a nominal annual rate of 10%.
Both approaches are correct; just remember that whatever you choose for P/Y defines what one period means (i.e. how many there are in a year), and N must count exactly that many periods.
Generally, the first option is simpler. The key idea to remember is whenever you use a TVM solver for pure compounding, set C/Y (and P/Y) equal to the compounding frequency and make sure N = (number of years) × C/Y.
We learned earlier that many assets (items with value) tend to depreciate and lose some of that value over time.
As you might know, money tends to behave similarly: the value of money depreciates over time through a process known as inflation, where the inflation rate i% represents the average percentage increase in prices over a year. In other words, money becomes i% less valuable over the course of the year.
The value of money after being adjusted for inflation is known as its value in real terms.
Checkpoint
The inflation rate for this year was 8%. You had $200 at the start of the year.
What will your money be worth, in real terms, at the end of the year?
Select the correct option
Inflation & Real Value
The real interest rate (needed when a question involves inflation) is given by r%=c%−i%, where c% represents the given interest rate (the nominal rate) and i% represents the inflation rate.
Note: You can calculate the real interest rate r% and enter it directly into the TVM solver (when required) as the nominal annual interest rate (I% on your calculator), since the TVM solver does not account for inflation effects in its standard calculations.
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