Chapter 1: The Time Value of Money, Part 1
At the end of this chapter you should be able to:
- Understand the Time Value of Money
- Know the difference between simple and compound interest
- Solve for present value, future value, interest rate and time.
1.1 The Time Value of Money: Future Values (FV)
In finance, money has a time value because you can invest it to earn interest and that interest accumulates over time. To begin our discussion about this very important topic, we will start with a simple example.
Imagine that you deposit $100 into an account which you hope will earn 10% interest per year. What would the value be at the end of the year?
It is likely that you were able to do this computation in your head. If your investment earned a 10% return, you would earn $10 in interest (10%*$100). So you would have the $10 plus your original investment of $100, so you would end up with $110. The calculation looks like this:
Now what would happen if you let your money grow for one more year? This would depend on whether this is a Simple Interest account or a Compound Interest account.
Simple vs. Compound Interest
There a two primary ways that interest can be earned (or charged). Simple Interest is applied only to the original value of the investment (the principle). Whereas Compound Interest is applied to the original value plus all previously earned interest.
Simple Interest – Interest is earned or charged only on the principle amount of an investment or loan. The amount of interest earned or charged is the same every period.
Compound Interest – Interest is earned or charged on both the principle amount and the interest earned on an investment or loan. The amount of interest earned or charged increases every period.
If you are investing, which do you like better, SIMPLE or COMPOUND interest account (all else equal)?
If you are borrowing, which do you like better, simple or compound interest account (all else equal)?
Now, to get back to Example 1, if this account earns Simple Interest, you would earn $10 in interest for every year that you remained invested. In this case, if you invest your money for 2 years, you would earn a total of $20 in interest, so you would end up with $120 in total. The calculation looks like this:
If this was a Compound Interest account, you would earn $10 the first period but MORE THAN $10 each year after. This is because Compound Interest investments earn interest on the original deposit, PLUS interest earned on previously earned interest. After one year, we would have $110, just like with the simple interest account, but after 2 years, we would have $121 because the first year’s interest of $10 would earn interest too. The $1 difference is the interest earned on the $10 interest from the first period (.10*$10=$1). Here is what the calculation looks like:
This simplifies to:
The table below shows how the difference between the simple interest and compound interest investment increases over time.
The difference between the future value of the simple interest investment and the compound interest investment is called the effect of compounding because the difference between the two accounts comes from interest earned on previously earned interest.
→ At the end of one year, both accounts have exactly the same value: $110.
→ The extra $1 in interest earned in the second year is from the $1 in interest earned on top of the $10 in interest earned in the first year.
→ The difference between the account increases as time increases and interest accumulates.
The Effect of Compounding
From the graph above, you can see that for this $100 investment at 10%, the value of both the simple and compound interest account increases over time. Notice also that the difference between the simple interest investment and the compound interest investment becomes greater over time also. In particular, remember that there is no difference between the simple interest and compound interest account after one period, but as you can see, in this case, after 50 years that difference increases to over $11,000.
What is the effect of compounding for a $1,000 investment that earns 5% each year for 30 years?
(Hint: First compute the future value given both compound interest and simple interest and then subtract the values.)
Notice that, the simple interest investment will earn $50 each period for 30 periods, so that is $1,500 in interest.
The compound interest investment will earn $50 in interest in the first year, but MORE THAN $50 in interest for each subsequent year. This is because, after the first year, interest grows on the original investment (principle) and the interest earned in previous periods.
The effect of compounding is the difference between the simple interest and compound interest future values:
$4,321.94-$2,500 = $1,821.94
Imagine that 50 years ago, your aunt made an investment of $10 into an account that earned 12% each year. How much would the investment be worth today? (Round to 2 decimal places, do not include dollar signs in your response.)
In Practice 1, we see that even small investments over long periods of time can accumulate considerably when interest compounds.
As discussed previously, we like compound interest accounts when we invest, but we prefer simple interest when we borrow. Here’s an example to see how that would work.
Suppose that you would like to borrow $500 to do some traveling this summer. Both your brother and your sister offered to lend you the money at a 10% monthly rate. Your brother will charge you compound interest, while your sister will charge you simple interest. If you plan to pay the money back in 12 months, how much more would you have to pay your brother? (Notice that you are given a monthly rate. This means that 't' must be number of months, not years.)
Future Values: Relationships Between Variables
In most cases, investments earn compound interest, so from now on, we will assume that, unless otherwise noted, we are working with compound interest. Let’s do a few more examples so that we can see how our variables (FV, PV, r and t) relate.
Suppose that you deposit the $120 you save on this textbook into an account that you hope will earn 8%. How much will you have in 2 years?
Now let’s look at what happens to the future values as the variables PV, r, and t change.
If the present value is higher, will the Future Value be higher or lower?
The future value will be higher for two reasons. First, we began with a higher amount and, second, the interest each period will be greater.
To test this, calculate the future value given a present value of $130.
Future Values are higher when Present Values are higher.
Now decrease the PV to $110.
Future Values are Lower when Present Values or lower.
If the interest rate is higher, will the Future Value be higher or lower?
The future value will be higher because the investment is growing at a higher rate.
To test this, calculate the future value at an interest rate of 9%.
Future Values are higher when Interest rates are higher.
Now calculate the future value with an interest rate of 7%.
Future Values are lower when Interest Rates are lower.
If the number of periods is greater, will the Future Value be higher or lower?
The future value will be higher because the investment will have more periods of growth.
To test this, calculate the future value after 3 years.
Future Values are higher when the number of periods are greater.
Now, calculate the future value after 1 year.
Future Values are Lower when the number of periods are fewer.
In summary, there is a positive relationship between Future Values and PV, r, and t. As present values, interest rates and number of periods increase, future values increase. As present values, interest rates and the number of periods decrease, future values decrease. The table below summarizes these relationships.
Which of the following will have the higher future value? (No calculations needed!)
An investment with a present value of $1000, an interest rate of 5% that will be invested for 10 years.
An investment with a present value of $1000, an interest rate of 6% that will be invested for 10 years.
1.2 The Time Value of Money: Present Values (PV)
We will now learn how to answer questions like, “How much do I have to invest today to have $2,000 in 2 years?” or “How much is $5,000 to be received in 2 years worth today?”
Remember, that we will assume all interest is compound unless otherwise noted.
To begin with, recall the future value formula:
If we know any of the three variables, we can solve for the fourth. In this case, we can rearrange the formula to solve for the present value (PV). It looks like this:
You would like to have $3,000 in 2 years so that you can move into a new apartment when you graduate. How much must you deposit today if you think you can earn an interest rate of 7% per year?
This means that if you invest $2,620.32 and it grows at 7% as expected, then you will have the $3,000 at the end of the 2 years.
Calculator Tip: Make use of the parenthesis on your calculator to help you with the order of operations. The keystrokes, for most calculators, would be like this: 3000/(1.07^2=
This also ensures that you are not rounding too much because the calculator will store all decimal places.
Rounding Guidelines: Make sure that you are using at least 4 decimal places for intermediate steps. For dollars and cents, you can round to two decimal places for your final answer.
A friend has asked for your help. She wants to save some money to buy a house in 10 years. She heard that there is an investment that could earn 8% per year. If she would like a down payment of $50,000 for her house, how much must she invest today?
Given an interest rate of 8%, your friend must invest $23,159.67. If the investment earns at least 8% per year, she will meet her goal in 10 years. But notice that “IF”. There are no guaranteed returns that will earn 8%. In order to expect to earn a return of 8%, you must be willing to take on some risk – this means that there is a chance you will earn a return that is lower than 8% - it could also mean that there is a chance that you could earn a return that is higher than 8%. “Guaranteed” returns earn very low interest rates – we call these “risk-free” investments. A common “risk-free” investment is a savings account. As of July 2018, savings accounts earn on average, about 1.75% How much would your friend have to invest if her expected return were just 1.75% per year?
You would like to have $5,000 in 5 years to do some traveling. You are looking at an investment that has had an average return of 9% over the past 3 years and you are hoping that return will continue. How much would you have to invest today if you can earn an interest rate of 9%? (Note: Enter only numbers in your response. Round to 2 decimal places.)
Assume that you invest $3,249.66 and your investment grows at 9% for each of the first 2 years. At that point, you learn that the investment is likely to only earn 6% for the remaining 3 years. This means that you will not be able to reach your goal unless you invest more money. How much more would you have to deposit at the end of 3 years so that you can meet your goal? (Note: Enter only numbers in your response. Round to 2 decimal places.)
Present Values: Relationships Between Variables
Time Value of Money Concepts are at least as important as the computations. Using an example, let’s think about the relationships between present values and the inputs- – FV, r, and t.
You want to save up to buy a motorcycle in 5 years. You expect that the bike will cost $15,000 at that time. If you think that you can earn 10% on your investment, then how much must you invest today?
If the future value is higher, will the present value be higher or lower?
The present value must be higher
To test this, calculate the present value given a future value of $16,000.
Present Values are higher when Future Values are higher.
Now calculate the present value given a future value of $14,000.
Present Values are Lower when Future Values or lower.
If the interest rate is higher, will the Present Value be higher or lower?
The present value will be lower because the investment is discounted at a higher rate. (The time value of money costs more!)
To test this, calculate the present value at an interest rate of 11%.
Present Values are lower when Interest rates are higher.
Now calculate the Present Value at an interest rate of 9%.
Present Values are higher when Interest Rates or lower.
If the number of periods is greater, will the Present Value be higher or lower?
The present value will be lower because the investment has more years of lost interest.
To test this, calculate the present value with 6 years.
Present Values are lower when the number of periods are greater.
Now calculate the Present Value after 4 years.
Present Values are higher when the number of periods are fewer.
In summary, there is a positive relationship between present values and future values, but an inverse relationship between interest rates and present values and time and present values. The table below summarizes these relationships.
Which of the following will have the higher present value? (No calculations needed!)
An investment that will pay $500 in 5 years at an interest rate of 6%.
An investment that will pay $500 in 6 years at an interest rate of 6%.
Here is a present value problem from a different perspective.
Suppose that your parents are looking at purchasing a lot of land in an up-and-coming neighborhood. Their financial advisor told them that she thinks it will be worth $150,000 in 10 years. If your parents require a return of 10% on this type of investment, how much should they be willing to pay today?
This means that if they pay $57,831.49 today for the land and it sells for $150,000 in 10 years, they will have earned 10% on their investment.
Think…What if the land costs $60,000, should your parents invest?
No. Your parents should not invest if the land costs $60,000. The highest price that your parents should be willing to pay is $57,831.49. This is the price that will give them exactly their required return. Any price higher than that will provide a return that is too low, but a price lower than that will give them a return that is higher than 10%. In that case, they should take it.
Notice that, instead of asking how much must I invest to reach a goal, we are asking how much is something worth today given a required interest rate.
Here’s some more practice.
An investment will pay $3,000 in 7 years. The fair return on investments of this risk is 6%. What is the fair price of the investment today?
If the investment costs $1,900, should you invest?
1.3 The Time Value of Money: Rates (r)
We move on now to calculating the interest rate “r”. Again we begin with the general future value formula:
And rearrange to solve for the rate (r):
As always, the order of operations is really important. If you have not made use of the parenthesis on your calculator, now is the time.
Refer again to Example 7. We already know that your parents should not buy the land because it is too expensive. Another way of making this decision is to compute the implied return and compare it to the required return. Let's do it! Solve for the rate given a price of $60,000 for an investment that we expect will sell for $150,000 in 10 years.
It looks like this:
Notice that the answer displayed will be in decimals, you must convert to a percent.
For most calculators, the keystrokes would be: 150000/60000=^(1/10=-1
Notice that by pressing the = key after each computation you do not need parenthesis in the first step, or to close parenthesis in the second step.
You really want to be able to travel when you graduate in 4 years. Right now you have an extra $300 that you want to invest. You are hoping to have at least $500 for your trip. What interest rate % must you earn to reach your goal? (Include on numbers in your answer. Round to 2 decimal places.)
When first introduced in 1985, Nike Air Jordan’s sold for $65. Today (July 2, 2018) on stockx.com, a pair sold for $3023. What is the annual rate of growth of the price of the Air Jordan’s? (Note – assume t=33 years.) Include only numbers in your answer. Convert to percent. Round to 2 decimal places.
Risk and Comparing Rates
As investors we like high rates, as borrowers, we like low rates. However, it is not so simple. The most important thing as investors is that we are compensated for the amount of risk that we are bearing. It should make sense that if an investment is high risk, it should have a higher return than one that is low risk. Putting your money in a savings account is not the same as investing money in any stock. Your savings deposit is insured, for one thing, so you can be sure that you will not lose any money; however, you will also not gain much either – the average rate on a savings account is about 1.75%. For example, by some measures Netflix stock should have a return of about 10%. The higher return expected from Netflix is compensation for the higher risk of Netflix stock. When investing in any stock, it is possible for you to lose all of your money and the upside, while it could be greater, is not guaranteed. This is why it is important to be sure to compare interest rates of investments with the same risk level.
You are looking to earn some interest on $200 you have saved. Your father has offered to pay you a rate of 4.00%. Alternatively, your mother told you that if you give her your $200, she will pay you back $250 in 5 years. Assuming that you trust your mother and your father equally, (in other words, the offers have the same risk) which option should you choose?
Choose your mother’s offer – she will pay you more interest.
1.4 The Time Value of Money and Number of Periods (t)
At last we look at the fourth variable in our Time Value of Money computations. We can rearrange our future value formula to solve for t. Like this:
Your mother wants to take you and your family on a trip. She estimates that it will cost $30,000. She already has $17,000 saved. If you think she could earn 7% on her investment, how long until she has enough money for the trip?
If your mother earns 7% on her investment every year, she will have $30,000 in 8 years and 4.7376 months (.3948/12=4.7376 months).
You just received $1,000. You are think about investing it into an account that you hope will pay 8.45% per year. How many years until you have $1,500? (Round to 2 decimal places.)
- The Time Value of Money – In finance, money has a time value because we can earn interest on it. The higher the interest rate, the higher the time value. The longer the time period, the greater the time value.
- Simple Interest – In a simple interest account, interest is calculated only on the original value. This means that the account earns (or is charged) the same amount of interest every period.
- Compound Interest – In a compound interest account, interest is calculated on the original value and also on all previously earned interest. This means that amount of interest an account earns (or is charged) increases over time.
- Investors prefer compound interest, all else equal.
- Borrowers prefer simple interest, all else equal.
- Rates of return should compensate investors for the amount of risk associated with the investment.
- When making investment choices, it is important to compare investments of the same risk level.
- Decimal to percentages – you must use decimals in all formulas. To change from percent to decimal, move the decimal place over 2 places to the left.
- Percentages to decimals – for final answers, you must always express rates as a percentage. To change from decimals to a percentage, move the decimal two places to the right and add a percentage.
- Use parenthesis if you have them.
- Round to 4 decimal places for intermediate steps.
- Round to 2 decimal places for dollars and cents.
FV = the future value. This is the ending value of the investment.
PV = the present value. This is the beginning value of the investment.
r = the interest rate. This is the interest rate per period. Convert to a decimal.
t = the number of periods.
An account that earns simple interest will always have a lower future value than one that earns compound interest.
An investment will pay you $5000 in 5 years if you pay $3000 today. What is the implied % rate of return? (Enter only numbers. Answer as a percent. Round to 2 decimal places.)
Savings account A and B pay 3% interest per year. Account A pays simple interest, account B compounds annually. In which account should you deposit your money if you plan to invest it for 5 years?
A and B make $1,000 deposits into an account that earns 7%. A takes his money out in 5 years. B withdraws her money in 8 years. Whose withdrawal is greater?
Suppose that 2 years ago you bought an old record player at a yard sale for $10. You saw today on E-bay that the same record player is selling for $79.99. If you were to sell the record play at that price today, what would be the implied return percentage? (Enter only numbers in your response. Round to 2 decimal places.)
If you have $500 today and you think you can earn an interest rate of 7% per year, how many years until you have $750?
All other graphs and tables created by author using author’s calculations.