3.2 Timelines

We can make our lives much easier by visualising the concept of time value of money using timelines. A timeline allows you to visually present cash flows, most importantly, the timing of the cash flows as well as their nature as either inflows or outflows. This helps you better understand an investment project, an event, or any other forms of financial transactions. 

3.2.1 Timelines Introduction

3.2.2 Comparing Cash Flows

3.2.2.1 The Basic Concept

Let’s have a look at a simple example. Assuming you can earn interest of 5% per annum in a bank savings account, let’s compare:

(A) receiving payment of $100 today

(B) receiving a payment of $100 next year.

Which option is better?

3.2.2.2 Comparing Cash Flows

Whichever computation you use, Option A, receiving $100 today is always worth more! The key to this answer is that if you were to receive $100 today, you could earn 5% interest on that sum for one year. If the interest rate were higher, the difference between the value of Option A and Option B would be even greater!

Based on this example, we can make several generalizations:

• The value of a dollar invested at a positive interest rate grows over time. Thus, the further in the future you receive a dollar, the less it is worth today.
• The trade-off between money today and money at some future date depends in part on the rate of interest you can earn by investing. The higher the rate of interest, the more likely you will elect to invest your funds (assuming no additional risk of this higher interest rate).

Note that especially when we have multiple cash flows (Chapter 4) it will be important to draw a timeline. The video below explains this in more detail.

VIDEO: Tips for Time travel (YouTube, 2m15s)

3.3 Time Value of Money Basics

3.3.1 Definitions and Mechanics of Time Value Calculations

With the use of a timeline, you should be able to correctly identify cash flows, and their respective timing, and do some simple calculations by compounding or discounting cash flows. To do simplify our calculations, it would be handy to name the five variables involved in every time value of money problem.

Video: Time Value of Money – Definitions

3.3.1.1 Time Value of Money – Definitions

1. Present value (PV) – While present value can refer to the value today of a cash flow in the future, it can be helpful to think of present and future as relative terms. Whenever we are moving a cash flow backwards in time we will be using present value (discounting) computations. So, computing the value in 2025 of a cash flow promised in 2030 is a present value computation; it tells us what we will be willing to pay in 2025 for the promised cash flow in 2030.
2. Future value (FV) – Value of a cash flow at some specific future time – it is the money you have in your hand at a point in time in the future. Again, future is a relative term. Whenever we are moving a cash flow forwards in time we will be using future value (compounding) computations. So, computing the value in 1990 of a cash flow we expected to receive in 1985 is a future value computation; it tells us what that cash invested in 1985 was worth in 1990.
3. The number of periods (n) – This is the number of compounding periods (interest computation periods) during the investment. It is usually measured in years, but it could be any scale of time such as quarterly, monthly, or even daily.
4. Interest rate (r) – This is the growth rate of your money over the lifetime of the investment. This is also called the rate of return, discount rate, interest rate or growth per period.
5. Cash flows (CF) – These are a series of either single, equal, or mixed, evenly-spaced or irregular cash flows. Cash flows can be positive or negative. If we’re receiving money (cash inflows) we use positive numbers, if we’re paying money (cash outflows) we use negative numbers. As every investment has two sides (borrower/lender; company/shareholder; bank/depositor), each side will be looking at a mirror image of what the other side sees.

The variables are illustrated for the example we saw previously of $100 received today, earning 5% per annum, and solving for the value at year 1.

3.3.1.2 Future Value (FV)

Future Value and Compounding

Future Value (FV) of an investment is what the investment will be worth after earning interest for one or more time periods. The process of converting the initial amount into a future value is called compounding.

VIDEO: Future Value Formula (YouTube, 4m17s)

The video has derived the Future Value formula as follows:

Where:

1. PV is how much you have at the start of the time period you are evaluating
2. r equals the interest rate you are earning on the investment
3. n equals the number of periods you are putting the money away
4. FV equals how much you have at the end of the period you are evaluating, the future value
5. The term (1+r)ⁿ is the future value interest factor, often also known as the future value factor.

The Power of Compound Interest

The concept that interest compounds – that is, that you earn interest on the interest earned in previous periods – is very powerful. To see the effect of compounding, compare the compound interest to the amount of interest you would earn without compounding. We call interest computed without compounding “simple interest”, and it’s computed as the interest rate times the original principal. For one period it’s the same as compound interest:

But for subsequent periods it’s quite different:

and

3.3.2 Re-arranging the FV formula

In the previous sections, you learned everything that you need to know in order to solve time value of money problems. That is quite a bold statement, but it is true! The key is that we derived basic time value of money formula to solve for future value:

From this formula, we can easily derive other formulas for solving for any of the variables in a time value problem:

Note that the mathematical re-arranging is the “easy” part, the more difficult part is to figure out from a problem which variable you need to solve for. But don’t worry, your timelines and practice will help you!

3.3.3 Present Value (PV)

Present Value and Discounting

When calculating the Present Value (PV) of an investment, we calculate the value of a cash flow at an earlier point in time. This can be useful to understand for many reason, such as how much you need to save to achieve a desired outcome in the future. 

In the video below we demonstrate an example for the following present value problem: After finishing his apprenticeship, Joe wants his own food truck. He has 2 years to save $1800 for the deposit. He knows his bank pays 6% interest, but how much does he need to deposit now to have $1800 in 2 years’ time?

VIDEO: Present Value Formula (YouTube, 2m26s)

The process of converting the future cash flows into present value is called discounting. The present value formula is:

[latex]PV=\frac{{\rm FV}_n}{\left(1+r\right)^n}[/latex]

Remember that n is the number of periods in the future that the future cash flow is located (relative to the present value). Note that as we are dividing the future value, the present value will always be less than the future value (at a positive interest rate). Accordingly, the present value can be thought of as the current value of a future cash flow that has been discounted at the appropriate discount rate.

Just as we have a future value factor (1+r), we also have a present value factor, which is more commonly known as the discount factor. The discount factor, which is [latex]\frac{\mathrm{1}}{(1+r)}[/latex] is the reciprocal of the future value factor. This expression may not be obvious in the equation above, but note that we can write that equation in two ways:

3.3.4 Interest Rate (r)

There are times where we know the cash flows that occur at different points in time, and would like to know what the interest is that is earned on the investment. For this, we would need to use our re-arranged FV formula for r:

Take the previous example where you received a term deposit from your mum. Let’s say you didn’t know the interest rate of this deposit, but you know that if you would currently sell it you receive $100 (the present value), whilst the value in two years would be $121, what would be the interest rate you earn if you held this investment for two year? Intuitively you may say 10%, but as calculations become more tricky it would be handy toknow the formula that can give you an answer every time.

Using the formula above, we see that a term deposit with a future value of $110, a present value of $100, and 1-year time period, gives a discount rate of 10%:

 3.3.5 Number of periods (n)

Being able to calculate the number of periods can come in useful if you, for example, want to know how long it would take to save for a holiday if your money is sitting in the bank.

Again, for the term deposit example, you may say answering the following would be easy: if a term deposit pays 10% p.a., costs $100 now, and pays $110 at maturity, how many years until the term deposit matures? The answer is quite straightforward after going through the example a few times now: it would be one year. The formula we re-arranged for n however is useful when problems become more complex (which they almost always are!).

3.4 Compounding in Non-Financial Settings

The concept of compounding is not restricted to money. Any number that grows over time, such as the population of a city or the number of frogs in a pond, could grow at some compound growth rate. Compound growth occurs when the initial value of a number increases or decreases each period by a factor (1+ growth rate). As we go through the course, we will discuss many different types of interest rates, such as the discount rate on capital budgeting projects, the yield on a bond and the internal rate of return on an investment. All of these “interest rates” can be thought of as a growth rate (g) that relates future values to present values.

When we refer to the compounding effect, we are really talking about what happens when the value of a number increases or decreases by [latex]\left(1+growthrate\right)^n[/latex]. Does this sound familiar? If we want, we can rewrite the future value equation in a more general form as a compound growth rate formula, substituting g, the growth rate, for r, the interest rate.

Where:

Future value of the economic factor, such as sales or population, at the end of period n. 

PV  =   Original amount or present value of economic factor

g      =   Growth rate per period

n   =   Number of periods; a period may be a year but can also be a quarter, month, week, day, minute, or any other length of time.

The following graph shows how different world regions have grown since 1820.

Since the 1950’s the world has seen unprecedented and exponential growths in population and wealth, which have gone paired with numerous socio-economic changes such as energy use and fertiliser consumption for food production. These changes have made our lives a lot better and more comfortable. In turn, these anthropogenic (=human caused) changes are affecting our earth system in unprecedented ways, such as increasing the concentration of greenhouse gases in the atmosphere and the altering of land from forests to farms or domesticated land, causing a loss of biodiversity on a global scale.

For additional materials on the topic, you can view the following video on The Exponential Function where, decades ago, physics professor Albert Bartlett argued that “The greatest shortcoming of the human race is our inability to understand the exponential function”

VIDEO: Dr. Albert Bartlett: Greatest Shortcoming of Human Race and the Exponential Function (YouTube, 2m4s)

3.5 Interest Rates: Compounding Period and Interest Rate

Each of the time value formulas contains an interest rate, r, and a number of time periods or payments, n. For each problem or investment, interest is compounded at a given frequency – this could be annual, quarterly, monthly, etc. Compounding frequency depends on the facts given. If you borrow money from a bank, the contract will state how frequently interest will be compounded. We care about compounding frequencies for 2 main reasons:

• Interest rate, r, have to match with the frequency of the payments n.
• Interest rates with different compounding periods cannot be compared (next section)

The problems we’ve dealt with so far have all involved annual compounding. In that case, the compounding period is a year, and the appropriate interest rate is the annual rate itself. Things are a little more complicated with non-annual compounding periods.

3.5.1 Annual Percentage Rates (APR)

Banks and other credit providers disclose to consumers what is called an annual percentage rate (APR). The APR, otherwise known as the nominal rate, is the annualised interest rate using simple interest (thus without compounding!).

Let’s say you observe an interest rate of 10% on your bank’s website for a credit card. This observed rate is what we call the quoted rate, or APR (it is required to be disclosed by law). Does this mean you pay 10% per year? Not necessarily. It depends on the compounding frequency. You find out that the interest rate is compounded monthly. This means you have actually only been provided with a monthly rate, which you have to calculate. How does that work??

VIDEO: Interest rates (YouTube, 2m20s)

To convert the APR into the variables for time value formulas, APR is divided by the number of compounding periods per year to get the “periodic” interest rate, or the interest rate per compounding period to get r for the PV or FV formula. And the number of years is multiplied by the number of compounding periods per year to get the n for the time value formulas.

Let’s consider quarters as an example:

Suppose we have a time value problem that runs for five years and has an APR of 12%, compounded quarterly. If we have quarterly cash flows that run for 5 years, what is our r and n?

The r rate for that period is 12%/4=3%. The time dimension (=n) of the problem is 20 quarters rather than five years (5 years x 4 quarters/year = 20 quarters). Hence r and n for the problem should be 0.03 and 20, respectively.

3.5.1.1 More examples of compounding periods and interest rates

VIDEO: Compounding frequencies (YouTube, 2m20s)

3.6 Equivalent Interest Rates

3.6.1 Working with Different Compounding Frequencies

Often we need to compare different investments, and as we’ve seen, not all interest rates are quoted the same way. Compounding frequency matters. For a 12% APR, would you rather borrow with quarterly compounding or fortnightly compounding? How would you compare?

Let’s consider the credit card from the last section. Remember that the APR is 12% and compounding is monthly. You charge $100 on the card at the beginning of the year. At the end of the year, you go to pay off the credit card balance as planned. It seems reasonable to assume that with an APR of 12%, your credit card balance at the end of one year would be $112 (1.12 × $100 = $112). Is that right? Wrong! The bank is actually charging you 1% per month, meaning that the bank will compound your credit card balance monthly, 12 times over the year. The bank’s calculation for the balance due is $112.68 ($100 × 1.01¹² = $112.68). The bank is actually charging you 12.68% per year, and the total interest paid for the 1-year loan is $12.68 rather than $12. This simple example raises a question: What is the correct way to annualise an interest rate?

VIDEO: Introduction to EAR (YouTube, 3m3s)

3.6.2 Computing the Effective Annual Interest Rate

Given the credit card example above, we can conclude that when interest on your investment or loan is paid more frequently than once a year, you may in fact be earning (or paying) a higher rate of interest on your savings than the quoted rate given to you by your financial institution. Every interest rate quoted has a compounding frequency associated with it. If this is not stated explicitly confusion can (and often does) arise about exactly what that interest rate means.

VIDEO: Calculating the EAR (YouTube, 2m40s)

The Effective Annual Interest Rate (EAR) is an important tool that allows the evaluation of the true return on an investment or true interest rate on a loan. More specifically, EAR is defined as the annual growth rate that takes compounding into account. Mathematically, the EAR can be stated as follows:

where

m = the number of compounding periods during a year

APR = quoted or nominal interest rate

The EAR conversion formula accounts for the number of compounding periods, and thus effectively adjusts the annualised quoted interest rate for the time value of money. In the case of compounding more frequently than once per year, the EAR is always higher than the stated annual interest rate. Because EAR is the true cost of borrowing and lending, it is the rate that should be used for comparing different investment options. Note however that simply converting the APR to the EAR does not take fees into account which should be considered when deciding between investment options (more on this in Chapter 4!).

We will use our earlier bank credit card example to illustrate the EAR equation. Recall that the credit card has an APR of 12% (1% per month). The APR is the quoted interest rate, and the number of compounding periods per year (m) is 12. Applying the EAR equation, we find that the effective annual interest rate is:

The EAR of 12.68% is the true cost of borrowing the $100 on the bank credit card for one year. The calculation adjusts for the effects of compounding and, hence, the time value of money.

Finally, notice that interest rates are quoted in the marketplace in three ways:

1. The quoted interest rate (APR). This is an interest rate that has been annualised by multiplying the rate per period by the number of compounding periods per year.
2. The interest rate per period (periodic interest rate). The bank credit card rate of 1% per month is an example of this kind of rate. You can find the interest rate per period by dividing the quoted interest rate by the number of compounding periods per year.
3. The effective annual interest rate (EAR). This is the interest rate actually paid (or earned) over a year, which takes compounding into account. Sometimes, it is difficult to differentiate a quoted interest rate from an EAR. Generally, however, an annualised consumer rate is an APR rather than an EAR.

Notice the shift in rankings that takes place as a result of the EAR calculations. When we initially looked at the APR quotes, it appeared that Westpac offered the lowest rate and Commonwealth had the highest. After calculating the EAR, we find that when we account for the effect of compounding, Commonwealth actually offers the lowest effective interest rate.

Another important point is that if all the interest rates are quoted as APRs with the same compounding frequency, such as monthly, the interest rates are comparable, and you can select the correct rate by simply comparing the APRs. That is, the lowest APR corresponds with the lowest cost of funds. Thus, it is correct for borrowers or lenders to make economic decisions with APR data as long as interest rates have the same maturity and the same compounding period. To find the true cost of the loan, however, it is still necessary to calculate the EAR.

3.7 Summary

This chapter has introduced the basic principles of a very important concept in finance: the time value of money. The basic equations for present value and future value equations are two of the most fundamental relationships in finance and will be applied throughout the remainder of this course.

• Timelines are a critical first step in organizing cash flows in a financial problem.
• There are three rules of time travel:
• Only cash flows that occur at the same point in time can be compared or combined.
• To move a cash flow forward in time, you must compound it.
• To move a cash flow backward in time, you must discount it.

We also learnt that the concept of compounding does not only occur in financial settings. A variable growing at a constant rate (=compounding) occurs in many different settings, and thus we can use the same formula to find out, for example, at what rate a population has been growing. It was shown how our socio-economic systems have been growing exponentially over the past 50 years has caused rapid changes in our natural environment.

Finally, we discussed the importance of interpreting interest rates appropriately. Interest rates are often quoted as Annual Percentage Rates (APRs), compounded at a certain frequency. From this information we can find: 1) r and n for our time value of money calculations and 2) the EAR for comparing different interest rates with different compounding frequencies.

In this chapter we applied the time value of money concept to a single cash flow, next chapter we will apply the concept to multiple cash flows.

Note that the “discount rate” can also be used to compound.

3.7.1 Key Formulas