Time Value of Money

Discounting

Reflecting on our school days, we were introduced to the principles of interest rates and compounding. 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. Consequently, our funds increase if left in the account. At that time, we might not have fully grasped the significance of this concept. Interest rates and compounding are 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. It represents the return on that principal for a single time period. Compound interest, on the other hand, is calculated each period on the original principal amount plus all accrued interest to date. In essence, compound interest involves interest on the original principal as well as interest 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 (= $1000 + $1,000 x 5% x 2) by the end of the second year.

Next, let us consider compound interest. The concept of compound interest, also known as interest on interest, denotes 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.

Suppose the bank, where you deposited $1,000, offers an annual compound interest rate of 5%, instead of 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.

At the end of the first year, the initial deposit increases to $1,050 ($1,000 + 5% x $1,000). 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

+ $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 and compound interest work.

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 simple 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 an offer of receiving $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 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. Converting the cash flows does not imply altering their original positions on the timeline. Instead, it involves discounting future cash flows 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.

FUTURE VALUE OF SINGLE SUM

Future value (FV) is the amount a current deposit will grow to in a compound interest account over time.

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, given that it earns interest at the rate 'r' in each period.

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). 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.

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).

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.

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).

In the above formula, the term 1/ (1 + r) n is also known as the present value factor or simply the discount factor.

The above formula is a fundamental concept in investment analysis. Understanding the concept of present value is crucial for evaluating Petroleum Economics. This formula influences significant investment decisions. Therefore, it is 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 dealing 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. The same method applies to finding the FV, which is referred to as the additive property of PV or FV.

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 FV of these cash flows at the end of period 3, sum the FV of individual cash flows for each period.

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.

To compute the PV of these same cash flows, sum the PV of individual cash flows for each period.

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 intend to continue this for the next 10 or 20 years, this sequence of cash flows constitutes an annuity. Calculating the present value (PV) or future value (FV) of an annuity is analogous to calculating the PV or FV of a single sum. However, in the case of an annuity, we must compute the PV or FV of multiple such sums instead of just one. Therefore, mathematically, it involves summing the PV or FV of all individual sums.

FV OF ANNUITIES

Assume an investment made by you generates a cash flow for five years as presented in the table below

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.

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. As with finding the future value (FV), we can use the summation formula of a geometric series to compute the PV of an annuity.