Time Value of Money

Discounting

Reflecting on our school days, we were introduced to the principles of interest rates and compounding, which are essential in financial modelling. Interest is defined as a charge for borrowing money, expressed as a percentage of the amount borrowed over a specified period. When we deposit money in a bank, the institution pays us interest on the deposited amount, leading to an increase in our funds if left in the account. At that time, we might not have fully grasped the significance of these concepts. Understanding interest rates and compounding is fundamental to the financial sector, influencing financial markets and underpinning the global economy.

Interest can be calculated using two methods: simple interest or compound interest. Simple interest is computed solely on the original principal amount, representing the return on that principal for a single time period. In contrast, compound interest is calculated each period on the original principal amount plus all accrued interest to date, meaning that with compound interest, we earn interest not only on the original principal but also on the accumulated interest.

Let us consider an example of simple interest. Assume you deposit a principal amount of $1,000 in a bank at the beginning of year one with the intention of maintaining this deposit for five years. The bank offers a simple interest rate of 5% per annum. By the end of the fifth year, your total amount in the bank will be $1,250. The table below summarizes your $1,000 deposit (investment) for each year.

Applying the simple interest formula, $1,000 grows to $1,100 (= $1,000 + $1,000 x 5% x 2) by the end of the second year, illustrating the basic principles of simple interest.

Next, let us explore the concept of compound interest, often referred to as interest on interest. This method shows that the growth of an investment from one period to another is influenced not only by the interest earned on the initial principal but also by the interest accrued on previous periods' earnings. This concept is crucial in financial modelling, as it demonstrates how investments can grow over time.

Suppose the bank where you deposited $1,000 offers an annual compound interest rate of 5%, as opposed to a simple annual interest rate of 5%. By the end of the fifth year, your account balance will amount to $1,276. The table below illustrates the variation in interest earned during each period under this scenario.

An initial deposit of $1,000 grows to $1,276 over a period of 5 years, resulting in an interest earning of $276 through the power of compound interest.

At the end of the first year, the initial deposit increases to $1,050 ($1,000 + 5% x $1,000), demonstrating how financial modelling can effectively project future savings. By the end of the second year, the deposit further grows to $1,103, which is calculated as follows:

$1,103 = $1,000 principal

+ $50 first-year interest (simple interest)

+ $50 x 5% interest on the first-year interest

+ $1,000 x 5% interest on the principal for the second year.

You can also calculate the values in the table above using the compound interest formula provided (P = $1,000, r = 5%, n = 1, 2, …5 years). The example above illustrates how simple interest and compound interest work in financial modelling. In financial management and investment decisions, compound interest is more prevalent. This is because banks and financial institutions typically lend based on compound interest. With an understanding of both simple interest and compound interest, we can now explore the concept of the "Time Value of Money."

Refer to the compound interest rate calculation table above at a 5% compound interest rate for 5 years. Assume you are offered $1,000 today or $1,300 after 5 years. The option of receiving $1,300 after 5 years is preferred because receiving $1,000 today will grow to $1,276 in 5 years, which is less than the second option. Thus, the value of receiving $1,300 at the end of 5 years is greater than receiving $1,000 today.

If you are offered $1,276 after 5 years versus $1,000 today, one should be indifferent between the options. If $1,000 is received today, it can earn 5% compound interest for 5 years, resulting in $1,276. Therefore, both options result in the same amount. This means that the value of receiving $1,276 at the end of 5 years is equal to $1,000 today.

Consider another scenario: receiving $1,200 at the end of 5 years versus $1,000 today. Given the same interest rate assumptions, it is better to receive $1,000 today. By keeping it in the bank, it will grow to $1,276 in 5 years, which is more than $1,200. Hence, the value of receiving $1,200 at the end of 5 years is less than receiving $1,000 today.

PV $X = present value of $X as of today

PV $1,300 > $1,000

PV $1,276 = $1,000

PV $1,200 < $1,000

Note that these equations are true under the assumption of a 5% annual interest rate. If the interest rate changes, the equations no longer apply.

The value of $X today is higher than its value in the future because $X today can earn additional money through interest or investment. This principle is known as the "time value of money" and is fundamental to investment analysis.

Present value (PV) refers to the value today of an amount to be received in the future. For instance, the present value of $1,276 today is $1,000. Conversely, the future value of $1,000 today is $1,276 in 5 years, assuming a 5% annual interest rate.

If this concept is still unclear, please review the previous page carefully.

Time value problems often involve calculating the future value of an investment due to compounding. This requires projecting cash flows into the future based on an appropriate interest rate (compounding).

Finding the present value involves bringing future cash flows to today or a reference date, a process known as “discounting.” Discounting and compounding are inverse processes. Compounding projects present cash into the future, involving multiplication by the interest rate, while discounting brings future cash into the present, involving division by the interest rate.

Before discussing discounting and compounding formulas, let’s introduce the concept of a timeline. A timeline visually represents cash flows over different periods. Payments or cash outflows are shown as negative, reducing the business's cash balance, while cash inflows are shown as positive.

To visualize this, a straight line with equal intervals represents time periods. The reference date (present) is marked as ZERO (t = 0). Cash flows occurring today are placed at time ZERO. Future cash flows are placed at their respective time marks (e.g., t = 3 for year 3).

For example, if you start a car rental business by purchasing a car for $1,000, generating cash flow over 5 years, and sell the car for $100 at the end, the cash flows can be shown on a timeline. All figures are post-tax for simplicity.

Once we have delineated the cash flows on the timeline, we can effectively convert them to their present values or future values, depending on our analytical objectives in financial modelling. Converting the cash flows does not imply altering their original positions on the timeline. Instead, it involves discounting future cash flows, often using simple interest or compound interest methods, to ascertain their present value, or aggregating period-specific cash flows into a final year future value metric, which represents the cumulative future value over the duration of the project or investment.

An essential consideration in this context is whether the cash flows occur at the beginning or end of the period. This assumption is crucial for accurately assessing the time value of the cash flows.

For the purpose of this section, we assume that all cash flows occur at the end of the period. For instance, on the timeline provided, a cash inflow of $300 is assumed to occur at the end of year 1, and another cash inflow of $500 at the end of year 2. It should be noted that the end of one period coincides with the start of the subsequent period. Thus, a cash inflow of $600 can be interpreted as occurring at the end of year 3 or at the beginning of year 4. The markings on the timeline denote the end of each year.

The interest rate quantifies the time value of money. A higher interest rate corresponds to a higher future value. Conversely, a higher interest rate results in a lower present value when discounting future cash flows, whether through simple interest or compound interest.

FUTURE VALUE OF SINGLE SUM

Future value (FV) refers to the amount a current deposit will grow to over time in a compound interest account, which is a crucial concept in financial modelling, as it illustrates the benefits of compound interest compared to simple interest.

The formula involves an initial investment or cash outflow (PV) at time zero (t=0). The sum FV represents the value of the investment at the end of 'n' compounding periods, factoring in compound interest earned at the rate 'r' during each period. This approach is essential in financial modelling, where understanding the differences between compound interest and simple interest can significantly impact investment decisions.

Present VALUE OF SINGLE SUM

Let's reverse the thought process. When we knew the present value (PV), interest rate (r), and the investment horizon (n), we could calculate the future value (FV) using principles of financial modelling. Therefore, if we know the future value, the applicable interest rate, and the investment horizon, we should be able to derive the present value (PV) of the future investment, whether it involves compound interest or simple interest calculations.

The present value of a single sum is today’s value of cash that will be received in a future period. It represents the amount of money that must be invested today at a given interest rate (rate of return) for a specified period so that it grows to become equivalent to the future value (FV), factoring in the effects of compound interest over time.

The process of computing the present value of a cash flow is known as discounting. This involves discounting the future cash flows back to their present value. The interest rate used for this purpose is called the discount rate. Additionally, terms such as opportunity cost of capital, cost of capital, hurdle rate, or weighted average cost of capital (WACC) are often used interchangeably with discount rate in financial modelling contexts.

For practical purposes, we will treat these terms as synonymous. Since we already know the formula for future value (FV), we can apply basic algebraic principles to derive the formula for present value (PV), whether we are considering simple interest or compound interest.

In the above formula, the term 1/ (1 + r) n is also known as the present value factor or simply the discount factor. This formula is a fundamental concept in financial modelling and investment analysis. Understanding the concept of present value is crucial for evaluating Petroleum Economics, especially when considering the effects of compound interest and simple interest on investment outcomes. This formula influences significant investment decisions, making it important to grasp the concept of time value of money, as other details will follow from this understanding.

ADDITIVITY OF PV & FV

Using the formula above for present value (PV) and future value (FV), one can easily determine the PV or FV of a single amount. When engaging in financial modelling with multiple amounts of cash flows occurring at different times, the same formula can be applied. By computing the PV of individual components and then adding them together, the total PV can be determined. This method also applies to finding the FV, showcasing the additive property of PV or FV. Additionally, understanding concepts like compound interest and simple interest is essential in these calculations. The following example demonstrates how to calculate the PV for a series of cash flows as shown in the table below.

Assume the cash flow occurs at the end of the period. Use a discount rate r = 10% per period. To compute the future value (FV) of these cash flows at the end of period 3, sum the FV of individual cash flows for each period, taking into consideration the principles of compound interest and financial modelling.

FV = FV 1 + FV 2 + FV 3

FV = 1000 x (1 + 10%)^2 + 3000 x (1 + 10%)^1 + 5000

FV = 9,510

Note that the cash flow of period 1 is compounded by only 2 periods, not 3, because the cash flow occurred at the end of period 1 or the beginning of period 2, leaving only 2 periods to grow through compound interest.

To compute the present value (PV) of these same cash flows, sum the PV of individual cash flows for each period, which is essential for understanding simple interest versus compound interest calculations.

PV = PV 1 + PV 2 + PV 3

= 1000 / (1 + 10%)^1 + 3000 / (1 + 10%)^2 + 5000 / (1 + 10%)^3

= 1000 / 1.1 + 3000 / 1.21 + 5000 / 1.33

PV = 7,145

For further reading, let’s use the notation NCF for the net cash flow (sum of all the cash flows over all time periods). We will use CF to denote periodic cash flow.

ANNUITIES

An annuity is defined as a series of equal cash flows occurring at regular intervals. For instance, if you are depositing $1,000 every month into your savings account and plan to continue this for the next 10 or 20 years, this sequence of cash flows constitutes an annuity. The calculation of the present value (PV) or future value (FV) of an annuity is similar to calculating the PV or FV of a single sum, yet it requires financial modelling to consider multiple cash flows. In this case, we must compute the PV or FV of several such sums, which involves summing the PV or FV of all individual amounts. Additionally, understanding how compound interest affects these calculations can further improve your financial strategy, unlike simple interest, which does not take into account the potential growth of earnings on previously accrued interest.

FV OF ANNUITIES

Assume an investment made by you generates a cash flow for five years, as illustrated in the table below, which can be analyzed using financial modelling techniques. This approach allows you to consider both simple interest and compound interest to better understand the potential returns.

This method is similar to the one used for uneven cash flow series in the previous example. The distinguishing factor here is that the cash flows (CF) for each period are identical. This allows us to streamline the calculation using a shortcut formula for annuities, which is essential in financial modelling.

Observe that the sum of individual future values (FV) forms a geometric progression, making it easier to apply the formula for summing a geometric progression. It’s important to note that the FV of an annuity incorporates the effects of compound interest rather than simple interest, leading to a more accurate financial analysis.

The FV of an annuity can be expressed as follows:

PV OF ANNUITIES

To compute the present value (PV) of the annuity at the beginning of period 1 (or the end of period 0), we sum the PV of the individual cash flows for each period, which is a key aspect of financial modelling.

PV = PV 1 + PV 2 + PV 3 + PV 4 + PV 5

PV = 1000 / (1 + 10%)^1 + 1000 / (1 + 10%)^2 + 1000 / (1 + 10%)^3 + 1000 / (1 + 10%)^4 + 1000 / (1 + 10%)^5

PV = 3,791

This process is similar to what was done for uneven cash flow series in the previous example. The only difference is that the cash flows for each period are the same. Just like calculating future value (FV) using compound interest, we can apply the summation formula of a geometric series to compute the PV of an annuity, rather than relying on simple interest.