### Introduction to time value of money

Would you rather have $1,000 now or $1,000 in ten yearsβ time? When faced with such an offer, most people would undoubtedly go for $1,000 now as the better choice, but what about the choice between $1,000 now and $1,750 in ten years? Or, consider how much you would be willing to pay for the promise of $3,000 in five yearsβ time.

These scenarios involve the notion of **time value of money**. In Module 1, we learned that time value of money is an important financial principle. To recap, the difference in value between money today and what it will be worth in the future is referred to as the time value of money. As an extension of this, if you have money sooner, you can invest it and have more money in the future as a result.

In this module, we will examine what the time value of money principle tells us and explore how it impacts on financial decision making. Why do we need to understand the concept of time value of money? The answer is that, to make the best financial decision, like in the scenarios above, we need to be able to determine the value of the cash flows to our business at the point in time when we are making our decision.

Imagine your CEO asks for your recommendation on whether the business should invest in new technology costing $5m today that will generate $1.5m of revenue for the next 5 years. This $5m of investment has an **opportunity cost**. Should you decide not to invest in this new technology, this $5m can be redeployed somewhere else. The $1.5m in revenue you receive in year 1 can also be re-invested. In other words, whether it is the investment today or the future income streams, money has an opportunity cost that needs to be considered.

** Read**

To better understand the concept of time value of money, the creation of a timeline to visualise the cash flow series can be a very effective strategy. Refer to Checkpoint 5.1 on pages 140-141 of the textbook to learn more about timeline construction to visualise cash flows.

** Watch**

The LinkedIn Learning video Time value of money (04:34 minutes) further explains this key finance principle.

Patel, Y 2018 *Time value of money*, LinkedIn Learning video, viewed 12 June 2021, https://www.linkedin.com/learning/investment-evaluation/time-value-of-money?auth=true.

### Compounding and future value

#### Compound vs simple interest

**Compound interest** is where interest paid on an investment during the first period is added to the principal (the initial amount invested); and the interest during the second period is earned on the original deposit plus the interest earned during the first period. Therefore, money invested at compound interest accumulates at an increasing rate each period, exhibiting βexponential behaviourβ. In contrast, in transactions involving **simple interest**, the interest earned is based solely on the original deposit.

The following example illustrates the difference between calculating simple and compound interest.

An investment of $1,000 that earns **simple interest** of 10% per annum, will pay 10% x $1,000 = $100 interest per year, every year. After 20 years, the total future value of the investment will be $1,000 (the original investment) + $2,000 (20 years of $100 simple interest) = $3,000 total.

A $1,000 investment that earns **compound interest** of 10% per annum will pay $100 interest in the first year, which is then added to the principal. That means in the second year the interest will be calculated on the new principal of $1,100. So, 10% x $1,100 = $110. Again, this is added to the principal making it $1,210, which in the third year will generate $121 of interest.

Now, you can see that the value of the interest changes every year, and it would be onerous to calculate the final value after 20 years by calculating every single interest payment and updating the principal after each year (i.e. 1,000 + 100 + 110 + 121, and so onβ¦). To make the calculation faster, we can use the following formula for calculating the **future value** (formula **5-1a **on page 142 of the textbook):

πΉππ=ππΓ(1+π)πFVn=PVΓ(1+i)n

In general, in the above formula, you are taking a present value cash flow and projecting its value into the future (after adjusting for the interest rate it will compound at) in order to derive its future value πΉπFV. In this formula, πi represents the nominal annual interest rate in decimal form (0.1 = 10%, 0.08 = 8%, 0.03=3%, etc.); πn represents the number of years that the interest rate is compounded; and ππPV is the present value of the initial deposit or principal. We are solving for πΉππFVn , which is the future value after πn years have passed.

To solve our above example, we would put our variables into the equation like so:

πΉπ20=1,000Γ(1+0.1)20FV20=1,000Γ(1+0.1)20 — Note: that 10% interest is equal to 0.1 when expressed as a decimal number.

πΉπ20=1,000Γ(1.1)20FV20=1,000Γ(1.1)20 — Solve the equation in the brackets first.

πΉπ20=1,000Γ6.72750FV20=1,000Γ6.72750 — Then solve the power (usually the π₯π¦xy key or ‘^’ symbol on a calculator).

πΉπ20=$6,727.50FV20=$6,727.50

When you compare simple vs compounding interest, you can see the power of compounding. The future value of the initial deposit that earned compound interest on is more than double that of the deposit earning simple interest only.

If you look at the step-by-step solution above, you will notice that in the last step, we multiplied ππPV (in this case $1,000) by 6.7275. Now, if we wanted to discover the future value of a different initial deposit invested for 20 years at 10%, we would still multiply it by 6.7275 as that side of the equation will not change. In other words, any value invested for 20 years at 10% will grow in value around 6.7 times. This number is called the **future value interest factor **and can be found for any value of interest πi for πn years using the second half of the future value formula: (1+π)π(1+i)n.

** Read**

Pages 142-143 of the textbook look at the future value interest factor (πΉππΌπΉ)(FVIF) in more detail.

#### Different compounding periods

In the examples discussed so far, we have assumed the interest rate is compounding on an annual basis. A nominal rate of 12% **compounded annually** means the interest is calculated once and uses a value of 12%. But what about shorter compounding periods, such as monthly or quarterly? With a more frequent compounding, the πΉπFV will be higher.

For example, a nominal rate of 12% **compounded quarterly **means that the interest is calculated four times in a year at 3%. The effective interest rate here is more than 12%. It is in fact (1+0.03)4β1(1+0.03)4β1, which is = 0.125509 or 12.6%. The 3% is essentially 12% divided by 4 (quarterly) and πn is now 4 (4 quarters, rather than 1 being one year).

This is further illustrated in the table below. As can be seen, the more frequent the compounding period is (for an interest rate of 12%), the more the effective rate increases.

Compounding Period | Period | Effective Interest Rate |
---|---|---|

Annual |
1 |
12.0% |

Semi Annual |
2 |
12.4% |

Quarterly |
4 |
12.6% |

Monthly |
12 |
12.7% |

** Watch**

This short video (02:37 minutes) takes you through how to calculate πΉπFV in Excel for the scenario below, firstly by inputting values and formula, and then by using the easier **Financial** function under the **Formulas **tab. (*Watch in **full screen** mode).

Scenario: Imagine an elderly relative has bequeathed you $5,000 in their will. If you invest it at 10% interest for 5 years, what would be the future value?

** Read**

Pages 143β149 of the textbook cover an introduction to the concepts of compound interest and time, and also provide an explanation of the various approaches to calculating future values.

### Discounting and present value

We have seen that when we calculate the future value, we are in effect determining how the value of a sum of money in the present will grow over time, based on compounding interest. However, there are situations in finance when we want to determine the inverse situation and calculate the present value of a future sum of money, i.e. one that will be paid at a determined point in the future. We have already established that money today is worth more than the same value in the future, so we are now working backwards from a larger future value to a smaller present value.

To describe the rate of the change, we use the term **discount rate** instead of interest rate, and we are βdiscountingβ (reducing) the sum of money in the future to a present value, and not compounding. Just as with calculating the future value, we can also use a formula for calculating the present value of a future sum, at πn years in the future and at a discount rate of πi.

Formula **5β2** on page 149 of the textbook describes this mathematical relationship between the present value ππPV and the future value πΉππFVn.

ππ=πΉππΓ1(1+π)πPV=FVnΓ1(1+i)n

**Note**: Those of you with a maths background may notice that this is actually the same formula used for future value, just rearranged so it is now solving for ππPV and not πΉππFVn.

Letβs work through an example together: you want to know how much to invest now, at an interest rate of 6% compounding annually, so that in 15 yearsβ time you will have $200,000.

ππ=πΉππΓ1(1+π)πPV=FVnΓ1(1+i)n

ππ=200,000Γ1(1+0.06)15PV=200,000Γ1(1+0.06)15

ππ=200,000Γ12.397PV=200,000Γ12.397

ππ=200,000Γ0.4172=$83,440PV=200,000Γ0.4172=$83,440

$83,440 invested now at 6% compounding annually, will be about $200,000 in 15 yearsβ time.

We can check this by plugging the values into our established πΉπFV formula:

πΉπ15=83,440Γ(1+0.06)15FV15=83,440Γ(1+0.06)15

πΉπ15=83,440Γ2.397FV15=83,440Γ2.397

πΉπ15=$200,005.69FV15=$200,005.69

(Note that the $5.69 discrepancy is from rounding errors when calculating by hand).

** Watch**

Watch a demonstration (03:27 minutes) of how to calculate ππPV in Excel for the following scenario:

Imagine that in 5 yearsβ time, you wish to have $40,000 to pay for your MBA (upfront cost in 5 yearsβ time). If you can invest your money today earning 10% return, how much should you be putting aside to invest so that in 5 yearsβ time you will have $40,000? It can be worked out that you will need to put aside $24,837 today.

The video starts by showing how to input values and formula. Then it shows how to apply the same using the easier **Financial**function under the **Formulas** tab. (*Watch in **full screen** mode).

** Read**

Section 5.3 ‘Discounting and present value’ on pages 149β156 provides more in-depth explanation and illustration of how to calculate present value, πn and πi, given the relevant information.

**Optional reading**

If you are interested to learn more about how effective annual interest rates are calculated, read Section 5.4 ‘Making interest rates comparable’ on pages 156β162 of the textbook.

### Annuities

So far in the discussion of the time value of money you have been dealing with single lump sums of money, i.e. a principal accumulating to a future value or a future value discounted to a present value. However, in business, we most likely have to deal with multiple equal payments or receipts, e.g. rent, loan, superannuation payments or coupons from a bond. Such cash flow streams are called **annuities**. The textbook defines an annuity as “β¦a series of *equal *dollar payments that are made at the end of equidistant points in time, such as monthly, quarterly or annually, over a finite period of time, such as three years” (Titman et al. 2019, p. 170). As with a single lump sum, an annuity can have a future value and a present value. Unless otherwise advised, we assume that we are dealing with **ordinary annuities**, which simply means that the payments occur at the *end* of each period. If the payments occur at the *beginning* of the period as in, say, rental payments, then we are dealing with an **annuity due**. An annuity that continues forever is called a **perpetuity**. Perpetuities will be discussed further in the next section.

#### Future value of an annuity

The future value of an annuity is its value at the *end* of the period with the final payment. Figure 6.1 of the textbook below neatly demonstrates this with the example scenario of saving for a Masterβs degree and shows the timeline of the future value of a 5-year annuity:

Source: Figure 6.1, Titman et al. 2019, p.170.

The diagram above breaks down an annuity into separate lump sum payments, each of which compounds a decreasing number of times. From the diagram, we see the first $5,000 payment is invested for four years at an interest rate of 6% and compounds four times for a future value of $6,312.38. Then each subsequent annual payment compounds one less time, until the final payment of $5,000, which does not compound at all.

Pages 171-172 of the textbook illustrate how this method of breaking down the annuity into smaller fragments and solving for the future value can be condensed mathematically to a simpler annuity formula:

πΉππ=πππΓ[(1+π)πβ1π]FVn=PMTΓ[(1+i)nβ1i]

πππPMT is the regular payment deposited or received at the end of each period; πn is the number of payment periods; and πi is the interest rate applicable to each payment period.

**Example**

Letβs work though an example scenario step-by-step together: your grandmother has been investing $20 a month on your behalf in an account that pays 6% annual interest compounding monthly, and she has been doing it for 25 years.

**Step 1: Identify the variables**

The payment πππPMT is the regular sum of $20. The number of periods is (25 years x 12 months per year) = 300. You also want the interest rate per compounding period. In this case, the payments are made monthly and the account compounds monthly, so the interest per period is 6 divided by 12 = 0.5%.

**Step 2: Plug the numbers into the future value formula**

πΉπ25=20Γ[(1+0.005)300β10.005]FV25=20Γ[(1+0.005)300β10.005]

**Step 3: Solve all the equations in the brackets first**

πΉπ25=20Γ[(1.005)300β1]0.005]FV25=20Γ[(1.005)300β1]0.005]

πΉπ25=20Γ[4.64β1]0.005]FV25=20Γ[4.64β1]0.005]

πΉπ25=20Γ[3.460.005]FV25=20Γ[3.460.005]

**We get**

πΉπ25=20Γ692.99=$13,859.88FV25=20Γ692.99=$13,859.88

Without the impact of compounding (say, your grandmother just had the money hidden away under her mattress), the savings would have been $20 x 12 months x 25 years, which is $6,000. However, because your financially-savvy grandmother put the money in a bank account earning interest, the value of the investment is the much larger $13,859.88 instead of $6,000.

#### Present value of an annuity

In some circumstances, we might wish to know what the present value of a stream of regular future payments is. Think of a lottery winner who might be offered the choice between the full prize amount paid out over several years or a lesser lump sum payment upfront. This is similar to the example of radio contest winnings used on page 175 of the textbook and illustrated by Figure 6.2 below showing this timeline of a 5-year annuity discounted to the present value:

Source: Figure 6.2, Titman et al. 2019, p.175.

The diagram above breaks down the annuity into separate present value calculations with an incrementing number of periods. As illustrated, the first payment of $500 in a yearsβ time has a greater present value than the final payment of $500 in five yearsβ time. Rather than needing to calculate the ππPV for each separate payment individually, the textbook describes how the formula can be rearranged and (relatively) simplified to solve for ππPV with this single equation (Formula **6β2b** on page 176 of the textbook):

ππ=πππΓ[1β1(1+π)ππ]PV=PMTΓ[1β1(1+i)ni]

**Example**

Letβs work through a basic example together: letβs say your daughter has been accepted into an overseas university to study medicine. Rather than needing to wire money every month, you want to set up a financial instrument that pays her $1,500 a month over seven years, discounted monthly at the annual rate of 12%. What would be the present value (in this example, the cost to you) of such a stream of payments?

**Step 1: Identify the variables**

The payment πππPMT is $1,500, the number of periods is 12 months times by 7 years = 84, and the interest rate per period is the annual rate of 12% divided by the number of periods 12 = 1%.

**Step 2: Plug the numbers into the present value formula**

ππ=1500Γβ‘β£β’β’1β1(1+0.01)840.01β€β¦β₯β₯PV=1500Γ[1β1(1+0.01)840.01]

**Step 3: Solve all the equations in the brackets first**

ππ=1500Γ[1β12.3070.01]PV=1500Γ[1β12.3070.01]

ππ=1500Γ[1β0.4340.01]PV=1500Γ[1β0.4340.01]

**We get**

ππ=1500Γ56.65=$84,973PV=1500Γ56.65=$84,973

To help understand how discounting impacts the present value, consider the scenario whereby you decide to ignore compound interest. Instead, you decide to give your daughter $1,500 for each of the 84 months today. The cost to you *now *would be 84 x 1500 = $126,000, which is clearly a considerably larger amount. With the power of compounding, you are able to make your money work harder and therefore you would need to put aside an amount that is significantly lower than $126,000. In this case, putting aside $84,973 would allow you to meet the monthly payments of $1,500 for 7 years.

### Perpetuities

What if a cash flow goes on forever without an end date, that is to say, perpetually? For example, the owner of a share will receive dividends for as long as they hold the share; there is no ‘end date’ as in an annuity (assuming the company continues operating). This type of cash flow is known as a perpetuity and has its own formula for present value. There are two basic types: level perpetuities and growing perpetuities, and we will cover both here.

#### Level perpetuities

As the name suggests, level perpetuities have fixed value payments over time. To calculate the present value of a level perpetuity, we use formula** 6β5** on page 183 of the textbook:

ππ=ππππPV=PMTi

Imagine you just won the βSet up for Lifeβ lotto, which promises to pay the winner (and their descendants) $80,000 per year forever. If the annual interest rate is 10%, what is the present value of the perpetuity?

ππ=80,0000.1=$800,000PV=80,0000.1=$800,000

While it might be difficult to conceptualise putting a βfixedβ value on a constant stream of money at first, consider it from the perspective of the lottery company setting up your winnings’ payments. If it can invest a lump sum of $800,000 at a fixed annual rate of 10% indefinitely, then it can send you the $80,000 of interest earned each year forever without dipping into its pockets again.

#### Growing perpetuities

In a growing perpetuity, the payments increase at a fixed rate over time. The formula for this is:

ππ=πππ(in the first period)πβπPV=PMT(in the first period)iβg

Note that in this case, the growth rate of the payments πg must be less than the discount rate, or it results in an infinitely valuable revenue stream (as payments would be growing faster than the interest rate β so the payment growth cannot be indefinitely supported by passive interest).

To illustrate the real-world application of the perpetuity formula, letβs imagine you are preparing for your retirement and you find a reliable index fund that has a steady return of 5.5% annually. You calculate that you need around $58,000 per year to live comfortably, and inflation has been reasonably steady, so you only need the payment to grow by 1.8% to offset it.

How much will you need to invest to retire without any other source of income?

ππ=58,0005.5%β1.8%PV=58,0005.5%β1.8%

ππ=$1,567,568PV=$1,567,568

#### Estimating the theoretical price of shares

The concept of perpetuity valuation can be used to estimate the theoretical price of shares using the **Discounted Dividend Model**.

**Mini case study: Bia Ltd**

Letβs revisit our case study company, Bia Ltd, to illustrate the Discounted Dividend Model. Bia plans to issue a dividend of $2 per annum in perpetuity. The discount rate (similar to the interest rate πi) is 10%. Using the ππPV formula of a perpetuity, we can deduce a theoretical price for Bia to be

ππ=20.1=$20PV=20.1=$20

This means a rational investor may be willing to pay $20 for a share of Bia Ltd in return for a $2 dividend in perpetuity.

** Read**

Pages 169β183 of the textbook provide further examples illustrating how to calculate the future value and present value of annuities and perpetuities. Also covered is how to find the present value of a perpetuity when either the payment amount, number of periods or the interest rate is unknown.

To learn more about the Discounted Dividend Model read ‘Valuing ordinary shares using the discounted dividend model’ on pages 309-314 in Chapter 10 of the textbook.

** Optional reading**

If you want to delve even deeper, this source provides example scenarios to explain how to calculate the time value of money concepts covered in this module using Microsoft Excel. Functions to calculate net present value (πππ)NPV) (which will be introduced in the next module) are also covered:

### Summary

This module has examined the concept and application of the time value of money, one of the key principles of finance for both personal and corporate financial decision making. We have learnt how the future value of an investment can grow with the effect of compounding, and about how we can get to a present value with the concept of discounting cash flows. In applying time value of money concepts, we have also looked at annuities and perpetuities and, in particular, some practical applications for home mortgages and share valuation. In the next module, we will extend the time value of money concept further in the process of exploring techniques to value and select capital projects.