1 Time Value of Money

Time value of money (TVM) is arguably the most fundamental concept in modern finance. It is the foundation of asset valuation, funding, and wealth creation. It is used extensively to make personal and business finance decisions.

The importance of time value of money is illustrated with the concept map (Figure 1-1). All of the topics are connected through the time value of money. For now, don’t worry about the complexity of the map. You will build up the map throughout the course as your knowledge increases.

The two inputs (pieces of information you need) for any time value of money problem are the cash-flows involved and what we call the “discount rate.” First, we’ll examine the cash-flows and later in the course we’ll learn about the discount rate.

An important step in becoming an expert in any field is to be able to define and organize prob- lems according to similar structures or patterns. The first step toward this skill is to learn the vocabulary of the field—in this case, how to talk the language of finance. We’ll proceed with this in the next section.

Concept Map

Figure 1.1: Concept Map

1.1 Naming Cash Flows

Every discipline has its own unique way of talking and finance is no exception. Here we will focus on identifying and naming various types, or categories, of cash-flows. As we discussed in the introduction to this course, the Time Value of Money (TVM) is the most fundamental concept and set of techniques used in finance.

There are two inputs to any TVM calculation:

  1. The cash-flows associated with the situation

  2. The discount rate appropriate for the riskiness of the cash-flows (we will cover discount rates in future presentations)

Continuing with the “problem → solution” method used throughout the course, it is vital that we are able to identify patterns in the cash-flows that occur in actual personal finance and business situations. Concept map 1 shows how we will begin to build the material/concepts for the course.

This may be your first encounter with the notion of a concept map. It’s important to use the concept maps as “cheat sheets.” Concept maps will give you a quick visual reminder of how the pieces of the topic fit together.

Concept map 1

Figure 1.2: Concept map 1

The concept map is NOT a flowchart. Rather it is a way to compact a large amount of information into a diagram. The way to use the map is to pick a box and then follow the narrative to another box. For example, from concept map 1 we get two relations:

  1. Cash-flows are inputs to time value of money

  2. Discount rates are inputs to time value of money

We will continue with these summaries throughout the text.

Why Classifying Cash-Flows by Type Is Important

One of the key differences between experts and beginners is that, over time, experts come to organize different problems based on their underlying structures, while beginners tend to see each problem as unique. This leads to an overload on the memory of beginners. Experts reduce the memory overload by organizing concepts/problems based on patterns.

With time value of money, we will organize cash-flows based on their number, size, and tim- ing, which we will illustrate on a timeline.

Classification of Cash-Flows

There are two broad categories of cash-flow types:

  1. Lump sums

  2. Annuities

These are illustrated by expanding concept map 1 accordingly as shown in concept map 2.

Concept map 2

Figure 1.3: Concept map 2

Lump Sums

A lump sum is a single cash-flow arriving at some point in time. We will describe a lump sum based on when the cash-flow arrives. For example, the cash-flow shown on the timeline in Figure 1-4 would be classified as a lump sum today. Why? First, we know that it is a lump sum since there is just a single cash-flow. We add the qualifier “today” to indicate the lump sum arrives today.

Single Lump Sum Today

Figure 1.4: Single Lump Sum Today

A second possibility for a lump sum is a lump sum in the future. An example of this type of cash-flow is shown in Figure 1-5.

Single Lump Sum in the Future

Figure 1.5: Single Lump Sum in the Future

Again, we have the qualifier “lump sum” to indicate only one cash-flow, but now we have the additional qualifier of “future” since the cash-flow arrives in period 5.

Annuities

The second category of cash-flows is annuities. An annuity is a series of consecutive cashf-lows. Within the category of annuities, we can further divide the types of annuities by the pattern of the cash-flow. We have level annuities and growing annuities. With a level annuity, the cash-flows are all the same size. With a growing annuity, each subsequent cash-flow is a percentage greater than the previous cash-flow.

Annuities are further classified by when the annuity payment is made—at the beginning of the period, at the end of the period, or at some future date. An annuity where the first cash- flow is today is called an annuity due, while an annuity, where the first cash-flow is in period 1, is called an ordinary annuity. Finally, the payments for the annuity can start after period 1 which we call a delayed annuity. The figures below illustrate these types of cash-flows.

Figure 1-6 illustrates an ordinary level annuity. Since the first cash-flow payment starts in period 1, this is an ordinary annuity. Since the cash-flows are all the same size, this is a level annuity.

Ordinary Level Annuity

Figure 1.6: Ordinary Level Annuity

For the remaining cases, the logic for each categorization is shown to the right of each figure.

Ordinary Growing Annuity

Figure 1.7: Ordinary Growing Annuity

Level Annuity Due

Figure 1.8: Level Annuity Due

Growing Annuity Due

Figure 1.9: Growing Annuity Due

Delayed Level Annuity

Figure 1.10: Delayed Level Annuity

Growing Delayed Annuity

Figure 1.11: Growing Delayed Annuity

Concept Map Summary of Cash-Flows

Concept map 3 summarizes the different types of cash-flow categories that occur in most types of actual business and personal finance problems. Refer back to concept map 3 frequently and remind yourself of what each category means and how they are related.

Concept map 3

Figure 1.12: Concept map 3

Now that you have reviewed the various categories of cash-flows, watch the video “Cash-Flows: Identifying and Naming.”



Click here for: In-Class Exercise 1: Classification of Cash-Flows

1.1.1 Equivalence of Cash Flows

In this section of the Time Value of Money unit, we expand on the notion of the time value of money as a set of techniques that equates the dollar value of cash-flows that arrive at different points in time. This is illustrated in concept map 4.

Concept map 4

Figure 1.13: Concept map 4

Referring to the timeline below and from earlier material, you know that cash-flow A is a lump sum of $1,000 today and cash-flow B is a lump sum of $1,210 in period 2.

We say that two cash-flows are equivalent if an investor would be indifferent between receiving one cash-flow over the other.

An investor would be indifferent between receiving one cash-flow over another, if the investor can use financial markets to convert one cash-flow into the other.

Two cash-flows are equivalent if an investor would be indifferent between receiving one cash-flow over the other.
An investor would be indifferent between receiving one cash-flow over another, if the investor can use financial markets to convert one cash-flow into the other.

The two factors that we need to consider when we talk about equivalent cash-flows are
1. when the cash-flows arrive (the timing of the cash-flow), and
2. the market interest rate available to the investor

Continuing with the cash-flows in time line 1, let’s say the current interest rate you earn on your bank account is 10%, compounded annually. You have the option of getting cash-flow A, $1,000 today, or cash-flow B, $1,210 in two years. Which would you take?

To check whether cash-flow A and B are equivalent in value you check whether you can used financial markets to convert cash-flow A into cash-flow B. To make this comparison, you see how much you would have in your bank account in two years if you deposited the $1,000 from cash-flow A into your bank account.

After one year your $1,000 would earn 10% interest ($100) and you would have $1,100 in your account. After the second year you would earn $110 in interest (10% of $1,100) and you would have $1,210 in your bank account.

Notice that the $1,210 is the same amount you would have if you chose to get asset B. Since both asset B and asset A will get you $1,210 in year 2, asset A and B are equivalent in value. Make special note here that A and B are only equivalent when the interest rate is 10% and the timing of the cash-flows is what is shown on time line 1. In technical (TVM) terms, the $1,210 we calculated from the $1,000 is called the “future value” of cash-flow A. In addition, we illustrated that the future value of cash-flow A is equal to cash-flow B.

Now that you have some notion of what we mean by equivalent cash-flows, the video, “Equivalent Cash-Flows Part 1”, will walk you through additional examples.



1.1.2 Future Value, Discounted Value, and Present Value

At this point you have learned to name cash-flows based on their size, timing and pattern. In addition, you know that two cash-flows are equivalent if you can use financial markets to convert one cash-flow into another of the same size and timing. We now want to expand your vocabulary to be more precise with respect to the relationship between two cash-flows.

Here we introduce three new terms;

  1. future value (FV),
  2. present value (PV) and
  3. discounted value.

The first two terms can best be illustrated using one of our previous examples.

We know that two cash-flows are equivalent in value if you can convert one into the other using financial markets. In the previous example (repeated here) we know that cash-flow A is equivalent to cash-flow B, if the interest rate is 10%. When we have two cash-flows that are equivalent, we say that the cash-flow that arrives later in time is the “Future Value” of the cash-flow that arrives earlier in time. Therefore, in this example, cash-flow B is the future value of cash-flow A.

When we have two cash-flows that are equivalent, we say that the cash-flow that arrives earlier in time is the “Discounted Value” of the cash-flow that arrives later in time. Therefore, in this example, cash-flow A is the discounted value of cash-flow B.

Finally, if the earlier cash-flow happens to be TODAY, we say that the earlier cash-flow is the Present Value" of the later cash-flow. So, cash-flow A is the present value of cash-flow B.

Watch and study the video “Equivalent Cash-flows Part 2” to see additional examples of these terms as well as how to calculate future, discounted and present values.



Click here for: In-Class Exercise 2: Timeline → Present Value or Future Value (Lump Sum(s))

1.2 Basic Time Value of Money Operations Techniques

You are now beginning to use the vocabulary of finance and are building the tools necessary to solve personal and business finance problems. You know that two areas of critical concern for a finance analyst are cash-flows and discount rates. You know that time value of money gives us techniques that allow us to find the equivalence of cash-flows that arrive at different points in time. Next, we turn to the specifics of those time value of money techniques.

In this section on time value of money, we begin to explore in detail the specific calculations that compose the TVM techniques. We can illustrate how these fit into our framework by expanding our concept map as shown in Concept Map 5. The new material is shown inside the dashed line.

The two main operations (calculations) associated with TVM are present value and future value. These two operations are applied to lump sums and also to annuities. As a result we have four basic time value on money calculations.

We have four basic TVM calculations
1. future value of a lump sum
2. present value of a lump sum
3. present value of an annuity
4. future value of an annuity

1.2.1 Future Value of a Lump Sum

Type of Problems (where you use future value of a lump sum) Problems that are solved with the “Future Value” operation are often related to how much an investment you make today will grow to over time. As a clue to when to use future value techniques, look for phases such as

  • “how much will you have in five years if you invest $10,000 dollars today in an account that earns 5% per year, compounded annually?”

or

  • “If you save $10,000 today in an account that earns 6% annually, how much will you have in your retirement fund in 30 years?”

Recall the relationship between two equivalent lump sums: The earlier cash-flow that occurs today is the present value of the equivalent cash-flow that occurs later in time (the future value)

Present value and future value are determined by the

  1. timing of the cash-flows and

  2. the interest rate in the market.

If we are given present value and want to know future value we use the formula shown.

\[Future\;Value=Present\;Value\times(1+Interest\;Rate)^{Number\;of\;Compounding\;Periods}\]

Let’s apply this formula to the example problems above.

  • “How much will you have in five years if you invest $10,000 dollars today in an account that earns 5% per year, compounded annually?”

After 1 year you would earn $500 in interest, therefore you would begin year 1 with $10,500 in your account. After year 2 you would earn $525 in interest ($10,500 +5%) and begin year 3 with $11,025. Continuing in this fashion, you would end up with the $12,763 in year 5. You would have $12,763.

“If you save $50,000 today in an account that earns 6% annually, how much will you have in your retirement fund in 30 years?”

You would have $287,175.

Now that you’ve seen the basic idea, watch and study the video Future value of a lump sum to see how to apply future value techniques to finance problems you will encounter.


Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the future value of a lump sum using the practice questions widget.

1.2.2 Present Value of a Lump Sum

Type of Problems (where you use present value of a lump sum)

Present value technique can be used for problems that involve determining how to “fund” (a fancy way of saying pay for) some future bill or desired level of spending. That is to say, how much would you have to invest today to make sure that you have enough to cover some future liability. Example problems that suggest using present value include:

  • “How much would you have to invest today in a bank account that earns 5% compounded annually to have $25,000 in the account in 10 years?”

or

  • “You plan to go to graduate school two years from now. You will need $30,000 in two years. How much must you invest today to fund your graduate school if your investments earn 6% per year?” Once again, we see the relationship between present and future value.

If we are given information about the future value (the amount of money we want to have in the future), we can find the present value (the amount we would have to invest today) using the relationship

\[Present\;Value=\frac{Future\;Value}{(1+Interest\;Rate)^{Number\;of\;Compounding\;Periods}}\]

Let’s apply these formulas to the example problems.

  • How much would you have to invest today in a bank account that earns 5% compounded annually to have $25,000 in the account in 10 years?

You would have to invest $15,348 today in a bank account that earns 5% compounded annually to have $25,000 in the account in 10 years.

You plan to go to graduate school two years from now. You will need $30,000 in two years. How much must you invest today to fund your graduate school if your investments earn 6% per year?

You would need to invest $26,700 today to fund your graduate school if your investments earn 6% per year.

Notice in both timelines we emphasize with an arrow that we are “moving” the lump sum to the present time, hence the term “present value.”

Watch and study the video “Present Value of a Lump Sum” to see additional worked out examples and applications.



Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the present value of a lump sum using the practice questions widget.

1.2.3 Present Value of Unknown Future Cash-flows

You now know how to find the present value of a known lump sum dollar value. For example, given the question, “What is the present value of $2,500 that occurs five years from now when the interest rate is 5% compounded annually?” You know that the answer is

\[PV=\frac{\$2,500}{(1+0.05)^5}=\$1,959\]

In many cases however, we would like to know the present value of an unknown future cash-flow. We approach the present value calculations in the same way we would calculate a known future cash-flow, except we substitute “X” for the unknown cash-flow. We used this notation when we learned how to describe different types of cash-flows. As a reminder, we would describe the “X” in the timeline below as an “unknown lump sum” in year 10.

If the interest rate were 3% compounded annually, the present value of “X” would be

\[\begin{equation*} PV=\frac{X}{(1+0.03)^{10}}=\left[\frac{1}{(1+0.03)^{10}}\right]X=0.7441X \end{equation*}\]

If we had more than one cash-flow, we would just repeat the process more than one time. For example, let’s say we had the two cash-flows shown in the time-line and the interest rate was 4% compounded annually. The present value of the two cash-flows would equal 1.6122X.

So far we have just looked at the mechanics of finding the present value of an unknown cash-flow. We now want to see the types of problems that could occur where we would use what we’ve learned. For example, how would we solve the following problem?

“If you deposited $50,000 in a bank account that earned 4% compounded annually, how much could you withdraw (the same amount in each year) in years 4 and 5?”

First we would draw our time line and indicate the known and unknown cash-flows. Then calculate the present value of the two unknown cash-flows.

Next, we set the present value of the unknown cash-flows equal to the known cash-flow and solve for “X.”

\[\$50,000=1.6767X\]

\[X=\frac{\$50,000}{1.6767}=\$29,820\]

If you deposited $50,000 today in a bank account that earns 4% compounded annually, you would be able to withdraw $29,820 in years 4 and 5 and end up with zero dollars in the account in year five.

Today you deposit $50,000. Between today and year 1 you earn 4% interest which is shown as ($2,000 in year 1). This brings your account up to $52,000. Between years 1 and 2 you earn 4% on $52,000 which is shown as $2,080 in year 2. This brings your account balance to $54,080. Continuing in this fashion, you earn 4% on $54,080 between years 2 and 3. This is shown as $2,163 of interest in year 3. This brings your account to $56,243 in year 4. Next you earn another 4% of interest on $56,243 which is $2,250. From this you make your first withdraw of $29,820 which brings your balance in the start of year 5 to $28,673. Finally, you earn 4% on $28,673 of $1,147. This is just enough for you to make your second withdraw of $29,820 to leave $0 in your account. (there may be a small amount different from $0 because we have rounded to the nearest dollar)

Watch the video “Present Value Unknown Cash-flows” to see additional examples worked out for you. Pause the video at the end of each problem to see the full time-line, cash-flows and calculations.



1.2.4 Compounding Period

Up to this point we have used annual compounding for the interest rate. In many real world situations you will encounter, the compounding period will be more frequent than annually. For example, as a consumer you will almost certainly encounter these four types of loans at some point in your life:

  1. a car loan,
  2. a student loan,
  3. a mortgage
  4. credit card debt.

These loans typically use monthly compounding to correspond to how frequently you are required to make a payment. With receiving interest, your bank account or savings account may also use monthly compounding for the amount of interest you earn on your account.

When the compounding period is NOT equal to a year (not annual), we need to make a distinction between how the interest rate is stated and the actual amount of interest you earn or pay. The interest is stated as an Annual Percentage Rate (APR). The amount of interest you actually earn or pay depends on the APR AND the frequency of the compounding. We call the amount of interest you actually pay over a year the Effective Interest Rate.

To calculate the effective interest rate (which is what you really care about) when given the APR we use the following equation.

\[Effective\;Interest\;Rate=\]

\[\left[\left(1+\frac{APR}{\#\;compounding\;periods\;per\;year}\right)^{\#\;compounding\;periods\;per\;year}-1\right]\times 100\%\]

For example, let’s say you have a credit card with and APR of 14% with monthly compounding. The effective interest rate would be given as:

\[Effective\;Interest\;Rate=\left[\left(1+\frac{0.14}{12}\right)^{12}-1\right]\times 100\%\]

We enter the 14% APR as 0.14. We divide this by 12 since there are 12 months in a year. Next we take the interest factor to the 12th power (again because there are 12 months in a year). This result in and effective annual rate of 14.934%.

What this implies is that a $1,000 outstanding balance on your credit card would accrue $149.34 worth of interest over a year. Note that the monthly compounding results in you owing an extra $9.34 compared to what you would owe, if there were annual compounding (with annual compounding you would only owe $140 worth of interest.

Watch the video “Adjusting for Compounding” for additional examples



Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the Effective Rate using the practice questions widget.

Click here for: In-Class Exercise 3: Monthly Compounding — Present Value or Future Value (Lump Sum(s))

1.2.5 Present Value of an Annuity

Annuities are one of the two broad types of cash-flows (see concept map 4) we encounter in business. The present value of an annuity (or PVA) is one of the most frequently used TVM operations. Example situations that use the PVA operation include:

• "You would like to have $50,000 per year for your retirement income. Your retirement will last 25 years. You plan to retire 10 years from now. How much would you need to invest today to fund (pay for) your retirement if your investments earn 6% compounded annually?

or

• You plan to go to graduate school for 2 years, starting one year from now. Your tuition will be $30,000 per year for each of the years you attend graduate school. How much would you need to invest today in an account that earns 2% compounded annually to fund your graduate school?

The present value of an annuity (PVA) can be calculated using several methods:

  1. “Brute Force,”
  2. the PVA Formula or
  3. Financial Calculator Functions.

Let’s first look at the “Brute Force” method illustrated below.

We have an “ordinary level annuity” of three payments of $100. We would like to find the present value of this annuity when the interest rate is 5%. As shown, we can treat each individual cash-flow in the annuity as a lump sum and discount them one at a time back to today. We then just add-up the individual PVs to get the present value of this annuity to be $272.32. Since this is using just PV of a lump sum many times, we call this the “Brute Force” Method.

The second method we can use to find the present value of a level annuity is the present value of an annuity (PVA) formula. The PVA formula is given as:

\[PVA=PMT\times\left(\frac{1}{R}\right)\left(1-\frac{1}{(1+R)^N}\right)\]

Where • PVA is the present value of the annuity • PMT is the size of each payment in dollars, • R is the discount rate stated as a decimal and • N is the number of payments in the annuity.

If we apply the PVA formula to the previous annuity we get:

\[PV=\$100\left(\frac{1}{0.05}\right)\left(1-\frac{1}{(1+0.05)^3}\right)=\$272.32\]

Which is the same present value we got when we used the brute force method. Of course, if our annuity had 100 payments, the PVA formula would be a much faster way to get the answer!

One VERY IMPORTANT thing to know about the PVA formula is that it “converts” the level annuity into an equivalent lump sum in the period immediately before the annuity starts. To see exactly what we mean, look at the figure below.

The PVA formula converts the “delayed level annuity” of three payments of $100 starting in period 3, to a lump sum in period 2 of $277.51. If we wanted to know the true present value (the value today) of the delayed annuity, we would have to discount the $277.51 back two periods to today. We will do this operation in later material and an example is given in the video.

Watch the video “Present Value of an Annuity” to assist your understanding.



Click here for: In-Class Exercise 4: Present Value of Annuity Formula (PVA)

1.2.6 Calculator Functions

You have seen that there are a variety of ways to solve time value of money problems. The first method you used was the “brute force” method where every cash-flow was treated as a lump sum. The second method you used for annuities was the present value of an annuity (PVA) formula. We now introduce a third method, which is to use the functions of a financial calculator.

We know from concept map 5 that there are four basic time value of money operations. Using these operations in combination, we can solve ANY time value of money problem. If we examine the formulas for the four operations we will observe that there are only five inputs (an often only 4) to any time value of money operation. The five inputs are

NOTE
N= number of discounting periods OR number of payments
I/Y = per period interest rate (discount rate)
PV = present value cash-flow
PMT = the sice of the level annuity payment
FV = the future cash-flow

Most calculators with financial functions will use notation similar to this. However, you will have to consult the manual for the exact corresponding keys for your calculator model.

Watch the video “Calculator Functions” to see the correspondents between the formulas and the calculator key-strokes.

Setting UP BAII Plus


Calculator Functions



Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the present value of an annuity and a delayed annuity using the practice questions widget.

1.2.7 Future Value of an Annuity

The last formula you will use is the future value of annuity (FVA) formula shown below.

\[PVA=PMT\times\left(\frac{1}{R}\right)\left(1-\frac{1}{(1+R)^N}\right)\times(1+R)^N\]

To see example of using the FVA formula, watch and study the video below.



Click here for: Worked Problems 1: Annuities

Click here for: In-Class Exercise 5: Time Value of Money Story Problems

Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the future value of an annuity using the practice questions widget.

1.3 Funding Problems

You have expanded your understanding of TVM to include the fundamental TVM operations. We now turn our attention to applying TVM techniques to different types of finance applications. Once again we illustrate the relations among your expanding knowledge base using a concept map. Concept map 6 shows that

TVM solves four categories of problems:
1. funding,
2. asset valuation,
3. wealth accumulation and
4. choosing among alternatives

We will start with how TVM solves problems of funding.

The categories of funding problems are given with the blue-line box in Concept Map 6. Each type of problem is based on the cash-flow names you learned earlier.

At their most fundamental level, funding techniques solve problems of the form “how much do you have to invest at some earlier date to have a desired amount of money at some later date.” For example:

• “How much would you have to invest today, in an account that earns 4% APR, compounded monthly, to pay your tuition bill of $1,000 that is due four months from now?”

or

• "How much do you have to invest each month, starting next month, for 60 months, to buy a vacation home 61 months from now, if the home costs $500,000 and you earn 5% APR, compounded monthly, on your investments?

There are six basic structures of funding problems. Each type of problem is easily illustrated with a time line using the vocabulary you learned for cash-flows. The first funding problem structure is a lump sum funds a lump sum. The cash-flow pattern for this is shown below.

The pattern of the cash-flows given by the “X” and $1,000 indicates that we are trying to find out how big “X” would have to be today so that, when invested, it would grow to $1,000 by period 4.

A lump sum could also fund an ordinary annuity. In this type of problem, the cash-flows follow the pattern shown next.

The pattern of the cash-flows given by the “X” and the three $100 lump sums indicates that we are trying to find out how big “X” would have to be today so that, when invested, it would grow so that we could withdraw $100 in periods 1, 2 and 3. The full set of six funding cash-flow patterns is given in the figure 1, below.

When you encounter a personal or business finance problem that involves setting aside money to pay for some future desire or need, you are facing a “funding” problem. Refer back to figure 1 to see the type of funding problem you have. The solution method to each funding problem type will be illustrated below.

1.3.1 Lump Sum Funds Lump Sum

The general solution technique for a lump sum funds a lump sum is to just find the present value of the future cash-flow

Example 1

You earn 8% per year on your investments, how much do you have to invest today to “Fund” a payment of $1,000 due in 4 years?

You would have to invest $735.03 today to pay for a payment of $1,000 in year 4 if you earn 8% on your investments.

Example 2

If you earn 6% per year on your investments, how much do you have to invest today to “Fund” a payment of $500 due in 3 years?

You would have to invest $419.81.

Watch and study the video “Lump Sum Funds a Lump Sum” to see three worked examples of how to fund a lump sum with a lump sum.



Click here for: Worked Problems 2: Lump Sum Funds Lump Sum

1.3.2 Lump Sum Funds Ordinary Level Annuity

The second type of funding problem is a lump sum funds an ordinary level annuity. The general solution technique for this cash-flow pattern is to find the present value of the annuity using the PVA formula or calculator functions.

Example 1

How much would you have to invest today to fund three equal payments in periods 1, 2 and 3 of $100 each if the interest rate is 3%?

You would have to invest $282.86 today.

Example 2 How much would you have to invest today to fund two equal payments in periods 1 and 2 of $500 each if the interest rate is 5%?

You would have to invest $929.71 today.

Watch and study the video “Lump Sum Funds an Ordinary Level Annuity” to see three worked examples of how to fund an ordinary level annuity with a lump sum.



Click here for: Worked Problems 3: Lump Sum Funds Ordinary Level Annuity

1.3.3 Lump Sum Funds Delayed Level Annuity

The third type of funding structure is a lump sum funds a delayed annuity. You can recognize this from the time line below and your knowledge of how we refer to cash-flows. As the figure shows, the solution to this type of funding problem involves two TVM operations. First convert the annuity into a lump sum using the PVA formula. Pay careful attention that the equivalent lump sum occurs in the period just before the annuity starts. The second step is then to find the present value of the lump sum you calculated in step one.

Example 1

How much would you have to invest today to fund three payments of $200 each in periods 3, 4 and 5 if the interest rate is 6%?

Example 2

Your child is planning attend summer camp for three months, starting 7 months from now. The cost for camp is $1,000 per month, each month, for the three months she will attend.

If your investments earn 5% APR (compounded monthly), how much must you invest today such that your investment will grow to just cover the cost of the camp?

Watch the video “Lump Sum Funds Delayed Level Annuity” to see worked out examples. Then when you are finished, work the self-check to assess your understanding.



Click here for: Worked Problems 4: Lump Sum Funds Delayed Level Annuity

1.3.4 Ordinary Level Annuity “funds” lump sum

You’ve just completed the three cases where a lump sum invested today is used to pay for some future cash-flow need. We now turn to the cases where you fund future cash-needs with annuities.

Our first example is when an “ordinary level annuity funds a lump sum.” The cash-flow pattern for this case is shown below.

The solution technique for this cash-flow pattern is to discount the lump back to the period of the last cash-flow of the annuity and then use the future value of an annuity formula to find the annuity payments.

Example 1

You need $15,000 in 9 months to settle an overdue tax liability. If your investments earn 6% APR (compounded monthly), how much do you have to invest each month, starting next month, for 4 months, such that your investment will grow to just cover your property tax bill?

You would need to invest $3,630 in months 1, 2, 3 and 4 to fund your tax liability.

Example 2

You have purchased solar panels for your house and they will be installed in 10 months. You will pay $25,000 for the panels in 10 months. How much must you save (invest) each month, starting next month, for six months, to pay for the panels if your investments earn 3% APR, compounded monthly?

You would need to invest $4,100 starting next month for six months to fund (pay for) the solar panels.

For additional worked out examples, watch the video “Ordinary Level Annuity Funds a Lump Sum.”



Click here for: Worked Problems 5: Ordinary Level Annuity “funds” lump sum

Click here for: In-Class Exercise 6: Time Value of Money — Funding Problems

Click here for: Worked Problems 6: Loan Payments

Click here for: In-Class Exercise 7: Time Value of Money — Valuation, Funding, Choosing among Alternatives

1.4 Incorporating Inflation

Adjusting for Inflation

As we have already mentioned, one of the most important aspects of finance you will encounter in your personal finance plan is “funding.” Here we use the term funding to simply mean investing (saving) money at some earlier time periods to pay for some future liability or activity. You saw and worked many funding examples earlier in this unit.

One critical aspect of funding future activity that we have ignored up to this point is taking into account inflation. When you want to invest today to purchase some item in the future, you must somehow take into account that the price of the item you wish to purchase may increase from its price today.

Example 1

Let’s say you want to buy a house that currently costs $250,000, but you want to buy it in one year. Your investment account earns 5% APR, compounded annually. How much would you have to invest today to exactly pay for the house in one year?

From what you’ve done many times you know that the time line would look like the one shown below and you would have to invest $238,095 today.

In that example, there was no inflation over the one year you waited to buy the house.

How would the situation change if the prices on the type of house you wanted to buy were increasing at 2% per year? If you only invested $238,095 you would not have enough to buy the house in one year. In situations like this we need to adjust the future price of what we want to fund to take into account the price change (inflation).

We know that the price of the house is increasing at 2% per year, so if the house costs $250,000 now it will cost (1+0.02)x($250,000) = $255,000 in one year. Therefore, the adjusted time line is as follows

and we solve the “adjusted” problem as

You would have to invest $242,857 today to buy the house in one year. You’ve just learned one strategy to incorporate inflation into a time value of money problem.

Strategy 1 to Incorporate Inflation
1. Adjust the cash-flow that exists today to take into account inflation
future price of good = (current price of good)×(1+inflation rate)#of years into the future
2. Solve the problem as usual with the inflation adjusted cash-flows

Example 2

You want to buy a plot of land in 5 years. The land currently costs $100,000, but its price is expected to rise at 3% per year. If your investments earn 4% APR compounded annually, how much do you have to invest in years 1, 2 and 3 to be able to exactly pay for the plot of land in 5 years?

Apply the strategy.

  1. Adjust the cash-flows that exist today to take into account inflation.
  1. Solve the problem as usual with the inflation adjusted cash-flows.

You would have to deposit $34,335 in each of years 1, 2 and 3.

Example 3

You plan to go to graduate school for 2 years, starting 4 years from now. The current tuition for one year of graduate school is $15,000 but is expected to increase at 4% per year over the next 5 years. How much would you have to invest in years 1 and 2 to exactly pay for your graduate school tuition if your investments earn 5% APR, compounded annually?

Apply the strategy.

  1. Adjust the cash-flows that exist today to take into account inflation.
  1. Solve the problem as usual with the inflation adjusted cash-flows.

You would have to invest $15,454 in years 1 and 2 to exactly fund your graduate school education.

1.4.1 Nominal vs. Real

No study of inflation would be complete without discussing the concepts of nominal and real cash-flows and nominal and real rates of return.

Nominal and Real Cash-flows

When we refer to “nominal” cash-flows we simply mean the “number of dollars,” with no indication of what those dollars can buy. When we refer to “real” cash-flows you can think of substituting the number of dollars with “the amount of goods.”

We relate the “nominal amount” to the “real amount” using what is called the price level or price index. The relation among the three concepts is given as:

\[Nominal\;Value=Real\;Value\times Price\;Index\]

What do we mean when we talk about the “price level” or “price index?”

The price level is just an indication of the average price of a typical set of goods (called a basket of goods) that consumers buy.

The government chooses a particular year (called the base year) and measures the prices of the goods in the basket during that year. It then defines the price level in the base year to equal 100. One year later, it checks the prices of those same goods to see what the new prices are. Let’s say that one year later it costs 5% more to buy that same basket of goods as it did in the base year. In that year, the price level would be 105 (100+5%). This process continues in every year after the base year. We can use the price level to calculate the inflation rate, which we will illustrate next.

Consider the table below.

The inflation rate between any two years is just the percentage change in the price level between those two years as shown.

Over time, the base year may not correspond to the years you are interested in. For example, you may be interested in the following four years. The base year is not shown here, because none of the price levels are 100.

It is often more convenient to work with the price levels in decimal format (the price index or price adjustment factor) which we can calculate by dividing each of the price levels by the first price level in our data. To illustrate, look at the table below.

Now it’s easier to interpret what prices have done over time. Here we can see that if something costs $1 in year 1 (price index = 1), that same item would cost $1.05 in year 3 and $1.0688 in year 4.

To check your understanding, if some product costs $157 today and inflation is 6% per year, how much will that item cost in 10 years?

Answer: $157(1.06)10=$281.16

With our understanding of the price index in hand, we can examine how nominal values and real values are related.

Example 1 Nominal vs Real

Your annual income today is $50,000. After working in your career for 5 years your annual income is $61,000. Over those same 5 years, inflation has average 2%.

How much has your annual income increase in nominal terms and in real terms over the five years?

From the top timeline, you see that your nominal income has increased 22% over the five years. This simply means that you get 22% more dollars (think of $1 bills) for your income than you did five years earlier. Unless we know what happened to the price of goods over that period, we can’t say whether you are better or worse off.

The real values indicate how much more you can buy with your income. From the price index line, we can tell that over the five years, prices have gone up 10.41% and that is going to reduce the value of our 22% nominal raise.

To go from the nominal timeline to the real timeline we divide each nominal value by the price index in the corresponding time period.

Nominal value = (real value)*(price index)

This results in today’s real income to be $50,000 (note the nominal and real values are always the same today). The real income in year 5 is $55,249. Therefore, we have that our real income has increased by 10.498% over the time period. This means that we can buy 10.498% more in year 5 than we could today.

Example 2 Nominal vs Real

Twenty years ago the price of 1 gallon of gasoline was $1.12 per gallon. Today the price of 1 gallon of gasoline is $2.31. Assume over those same twenty years, general inflation has been 2.1% per year.

How much has the price of 1 gallon of gasoline increased in nominal and real terms over the twenty years?

Draw the time lines.

In nominal terms the price of gasoline has increase by 106.25% over the last 20 years. In real terms, its price has increased by 36.11%.

Nominal Rates vs. Real Rates

We saw from above the relationship between nominal and real rates is given by

(Nominal Value) = (real value)*(price index)

We also have a relationship between nominal returns and real returns. The formula is very similar to the one above.

\[(1+Nominal\;Rate\;of\;Return)=\]

\[(1+Real\;Rate\;of\;Return)\times (1+Expected\;Inflation\;Rate)\]

The concept is best illustrated with examples.

Example 1 Nominal vs Real Rate of Return

You deposit $1,000 in a bank account that pays 4% APR (compounded annually) and leave the money there for 5 years. Over that same 5 years inflation was 3% per year.

What was the per year nominal and real return you earned on your investment.

The per-period nominal return is 4%, while the per-period real return was 0.9708%. The real return indicates that our investment is earning a little less than 1% per year in terms of buying power.

Notice that a quicker way to get the real rate is to use the formula (1+nominal rate of return) = (1+real rate of return)x(1+expected inflation rate) Or using the numbers in this problem:

(1.04) = (1+real rate of return)x(1+0.03) –>

\[\left[\frac{(1.04)}{(1.03)}-1\right]\times100\%=0.97\%\]

Click here for: Worked Problems 7: Nominal vs Real rates

1.4.2 Incorporating Inflation into Funding Problems

Now that you’ve got the basic terms down and some problems, we want to look at some more examples of incorporating inflation into funding problems.

Finding a level nominal annuity to fund a real future amount.

  1. Draw the real cash-flows on a time line (label “Real”).

  2. Draw the price index line and corresponding price index values.

  3. Use the numbers in #1 ad #2 to find the nominal cash-flow.

  4. Use TVM to solve for the desired nominal cash-flows.

Example 1

You want to buy a vacation home in southern France in 10 years. The current cost of the types of homes you are considering is $600,000. The price of these types of homes is increasing at 2% per year.

How much do you have to save each year (the same amount each year), starting next year, for 5 years in nominal terms to exactly pay for the vacation home if your investments earn 5% APR, compounded annually and in nominal terms)?

Example 2

You plan on retiring in 30 years. Based on your desired standard of living, you calculate that you will need the current equivalent of $2,000,000 in 30 years. However, you estimate that inflation will average 2% over the next 30 years.

How much do you have to save each year (the same amount each year), starting next year, for 30 years in nominal terms to exactly pay for your retirement if your investments earn 5% APR, compounded annually and in nominal terms)?

Click here for: In-Class Exercise 8: Time Value of Money — Incorporating Inflation

Click here for: In-Class Exercise 9: Time Value of Money — Incorporating Inflation into Funding Problems

Click here for: Worked Problems 8: Funding problems with inflation

Click here for: In-Class Exercise 10: Time Value of Money — Review Questions

1.5 Interactive Practice Questions: Time Value of Money