Basic TVM Calculations

1. The Power of Compound Interest

Compound interest is like the secret power of a superhero because it gives you powerful growth of your savings over time. Compound interest is when you earn interest every year not only on the original amount you deposit into an account, but you also earn interest on interest! Let’s look at an example to see how compound interest and different interest rates, the percentage at which money grows over a specified period, impact the growth of your money over time. What if you have a choice to put $1,000 into three investment options that earn an annual compound interest rate of 6%, 12%, or 18% when you are age 25? What amount will you have in your account at the end of 40 years when you turn 65 years old with each of these options?

6% over 40 years = $10,286

12% over 40 years = $93,051

18% over 40 years = $750,378

As you can see, the power of compound interest is amazing! We call this the snowball effect, when you earn interest on both your original deposit and interest on the interest.

EXAMPLE

To get another glimpse into the power of compounding, let’s consider an example where you put $1,000 into a savings account that earns 10% annual compound interest. At the end of the year, you earn interest of $100 (i.e., $1,000 x .10). So, at the end of year one, you have $1,100 in your account. You then earn an additional $110 in interest by the end of the second year. Now your account is worth $1,210 and the interest earned in year two is $110. By the end of year three, your savings account is worth $1,331. The point is that you earn interest every year not only on the $1,000 you originally deposited but also earn interest on interest! And the earlier you start, the more time you have for your interest to snowball.

terms to know

Compound Interest

You earn interest on the amount deposited into your account and on the interest you already received.

Interest Rate (“r” in formulas)

The percentage at which money grows over a specified period. In time value of money calculations, the interest rate can also be referred to as a rate of return, opportunity cost, or discount rate. It represents the return earned when investing or the cost of borrowing money, reflecting the trade-off between using money now versus in the future.

Snowball Effect

When your money grows through compounding by earning interest on both your original deposit and interest on the interest.

2. Time Value of Money

As we can see in the above example, time can make a big difference as to how much money invested today can grow over time. The concept of valuing money at different points in time is called the time value of money, which says, an amount of money is worth more today than the same sum received in the future due to the ability to earn a return from this date forward.

We can use a formula to help us determine a future value when the calculations get more complex with more years. Let’s start with a formula that helps us determine the value of an amount of money at a future date, or the future value formula.

formula to know

Future Value (FV) Formula

Where:

• FV is the future value

• PV is the present value

• r is the annual rate of interest as a decimal (5% is .05)

• t is the number of years your money is invested

Note: the “t” or time value is an exponential value

Note: You will want to save this formula as a reference since you will see the formula again in a later lesson when we revisit the time value of money.

Using the Future Value (FV) Formula above, let’s calculate the future value (FV), or amount grown by an interest rate over time to a later period, of a savings amount using 10% interest as an example. Although interest rates vary given current economic conditions, we will use 10% since the calculations are simpler.

EXAMPLE

What is the value of an account at the end of three years that earns 10% compounded annually on $5,000 deposited today?

Using the Future Value Formula:
Where:
PV = $5,000
r = annual interest rate which is 10% or .10
t = number of years, 3

try it

After you figure this out, check if you are correct by selecting the “+” icon to reveal the answer.

Your turn, using the Future Value formula, calculate the future value that you will have if you deposit $1,500 today, or a present value amount today, that grows for 15 years compounded annually at 10%?

Using the Future Value Formula:
Where:
PV = $1,500
r = annual interest rate which is 10% or .10
t = number of years, so 15



Not bad for a $1,500 deposit!

Let’s try some situations that you may want to consider yourself.

EXAMPLE

Take Alex, who is age 25 and saved $3,000. He would like his savings to grow to $5,000 so he can return to community college part-time when his children attend elementary school in six years. What will he have in his savings account in 6 years if Alex figures he can earn 8% in a savings account at his local bank?

Using the Future Value formula:
Where:
PV = $3,000
r = annual interest rate which is 8% or .08
t = number of years, 6



Alex is just shy of his goal of $5,000! Earning a higher rate of return would likely allow him to surpass his goal within 6 years.

terms to know

Time Value of Money

The concept that an amount of money is worth more now than the same sum received at a future date due to the ability to earn a return from this date forward.

Future Value (FV)

An amount grown by an interest rate over time to a later period.

3. Solving for the Future Value (FV)

Let’s look at another example and consider the other options you may have when making a financial decision.

EXAMPLE

Let’s say that today is your twentieth birthday, and your parents plan to give you $1,000 next year on your twenty-first birthday. You’d like to buy a used car to drive to make it easier to work at your internship, so you plan on asking for the money now. If you received the $1,000 today, what would it be worth? Another way of saying this is, how much would $1,000 of cash received one year from now be worth today?

To understand the time value of your deposits or cash flows (cash payments made or received over time), we need to explore the relationships between their values in the future and their equivalent today, or their present value (PV, the value if money were received today). The present values today will be less than if we simply added the sum of each future cash deposit in the future to determine their total. Why? Because the cash flows are discounted or decreased by the amount of interest growth you could receive if you had all the cash today and invested it.

Understanding this discounted calculation is important to gaining insight into the relationships between time, risk, opportunity cost, and value.

To discount a cash flow to be received in the future, you need to know:

• what the future cash flow(s) (CF) will be
• when the future cash flow(s) will be received
• the interest rate at which you could invest the cash if received today (Interest)

Regarding the interest rate, you will need to determine what might be your options, opportunity costs, and risks of not having the money today.

EXAMPLE

You may be able to put the cash in an account that earns 5% interest or use the cash to pay down a credit card that you are paying 25% on the remaining balance. These two options are examples of your opportunity cost.

Obviously, if you receive your $1,000 birthday check today at age 20, you could deposit it into an interest-bearing account. By your twenty-first birthday one year later, you would have more than $1,000 because you would earn interest on the money over the year. If your bank pays 5% per year on your account balance, then you would earn $50 in interest over the year, or $1,000 × .05 = $50. On your twenty-first birthday, you would have $1,050 in your savings account.

Using this Future Value (FV) formula to calculate the future value of your $1,000 deposited birthday check:

Let’s look at the numbers used in the formula

PV	Interest Rate “r”	Time (years) “t”	Future Value
$1,000	0.05	1	$1,050 = $1,000 x (1 + 0.05)
PV	r	t

EXAMPLE

If you left the $1,050 in the bank until your twenty-second birthday, a year later you would have:



If we calculated the rate from your nineteenth birthday, the formula will now calculate interest for two years by using an exponential value of two to indicate two years in the future:

try it

Calculating a future value (FV)

If you deposit $200 into an account that receives an 8% annual interest rate, what is its future value in 5 years?

The future value (FV) is found using the following equation: $200 will grow to $293.86 in 5 years at an 8% annual interest rate.

big idea

The Relationship between Future Value (FV) & Time Value of Money

The Future Value (FV) formula is helpful to calculate what money will be worth at a later date. It is important to understand how the length of time (t), also known as number of periods, and the annual interest rate (r) affect the future value (FV) of money. Both of these variables have a positive relationship with FV (positive relationship means the variables reflect each other). In other words, and holding all else constant:

As t or r increases	the FV of money received in the future increases
As t or r decreases	the FV of money received in the future decreases

learn more

Practice using a FV calculator found here: www.calculator.net/future-value-calculator.html

Note: This website uses “Number of Periods (N)” for time (t) and Periodic Deposit (PMT) would be 0 if you are not planning on adding more deposits periodically.

terms to know

Cash Flows

Cash payments made or received over time that are the basis for calculating equivalent present or future values in time value of money calculations.

Present Value (PV)

The value of the money received in cash today.

Discounted

When a value is decreased by the amount of interest growth you could receive if you had all the cash today and invested it.

Discount Rate

The percentage earned over time on money because time decreases its value.

4. Solving for the Present Value (PV)

We can view your birthday gift another way. What if your parents offered to give you the value of what $1,000 to be received in the future is worth today instead of having to wait one year? The value of the $1,000 today is called the discounted value (or present value) and is simply the future value minus the interest you would receive over the next year.

So, let’s look at a formula you can use to determine the present value today of money to be received in the future. Using our same example above of earning 5% interest compounded annually, the rate at which time affects your value is 5% because that’s what having the choice to invest it would earn for you if only you had the $1,000 now. The 5% interest is your opportunity cost. We will need to take the Future Value (FV) formula and modify it to get the Present Value (PV) equation.

Using the FV formula:

Spin it around.

Then divide each side of the equation by , we now have our Present Value (PV) formula.

formula to know

Present Value (FV) Formula

Where:

• FV is the future value
• PV is the present value
• r is the annual rate of interest as a decimal (5% is .05)
• t is the number of years your money is invested

Note: the “t” or time value is an exponential value

Using the PV formula:

Our answer tells us that your birthday gift is worth $952.38 if you were to receive its equivalent value today. In other words, if your parents could give you $952.38 on your twentieth birthday, you could deposit the $952.38 into a savings account earning 5% compound annual interest today and it would grow to $1,000 one year from now.

try it

Calculating a present value (PV)

If you receive $10,000 in 8 years, what is it worth today using a 5% annual interest rate (also known as a discount rate)?

The present value (PV) is found using the following equation:

$10,000 given in 8 years is equivalent to $6,768.39 today at a 5% annual discount rate.

big idea

The Relationship between Present Value (PV) & Time Value of Money

The Present Value (PV) formula is helpful to calculate the value today of money that will be given to you in the future. It is important to understand how the length of time (t), also known as number of periods, and the annual interest rate (r), also known as the discount rate, affect the present value (PV) of money. Both of these variables have an inverse relationship with PV. In other words, and holding all else constant:

As t or r increases	the PV of money decreases
As t or r decreases	the PV of money increases

learn more

Practice using a PV calculator found here: www.calculator.net/present-value-calculator.html?c1futurevalue=1%2C000&c1yearsv=1&c1interestratev=5&x=Calculate#future-money

term to know

Discounted Value

The current cash amount determined by subtracting interest to be earned over time from the future cash amount.

5. Rule of 72

By now you may be wondering if there is an easy way to figure out how long it takes for your money to double? Yes! Here is another golden nugget of many that you’ll find throughout this course. 

It’s called the Rule of 72, which tells you the number of years it takes for your money to double at a given annual rate of return. How? You divide 72 by the return you would receive to find the number of years.

EXAMPLE

If you’ll earn 10%, your money will double in . Or, if you earn 6%, your money will double in .

 

learn more

If you would like a simple way to try various deposit amounts to see what they grow to over time, check out this link: www.calculator.net/interest-calculator.html

term to know

Rule of 72

A quick way to determine the number of years it takes for your money to double at a given annual rate of return