The Concept of Inflation

We are all intuitively aware of the concept of inflation. We know that money loses its value every year — the same amount of money will purchase less and less over time. Let’s say $100 is required to purchase a certain basket of goods today. If there is inflation of 10%, the same goods will cost $110 next year.

Nominal Value of Money

If we made an investment yielding a 9% return this year, we would have a total of $109 next year from the $100 we invested. In accounting terms, we would show a profit of $9. This is because we are only considering nominal values. Nominal values do not account for inflation, the opportunity cost of capital, and other forces that cause the value of money to decrease over a given time period.

The Problem with Using Nominal Values to Measure a Firm’s Performance: Nominal values can present a distorted picture of a firm’s performance to its shareholders. Consider the case above: with 10% inflation and only a 9% nominal return, the firm has actually lost about 1% of purchasing power. The firm would have been better off spending the $100 in Year 1, when it had higher purchasing power, rather than investing it and ending up with $109 a year later—an amount that buys fewer goods than the original $100 did. If only nominal values are considered, firms can end up eroding their capital by investing in projects whose rate of return is below the firm’s cost of capital.

Real Value of Money

To address this problem, corporate finance uses the concept of the real value of money. The real value of money accounts for inflation, the opportunity cost of capital, and similar forces. Firms that base their decisions on these inflation-adjusted values tend to make better financial decisions than those that rely on nominal values alone. Real and nominal values can be converted using the following formula:

Real Value = Nominal Value / (1 + (i / 100))

Where i is the prevailing inflation rate in the market.

Subjectivity in Real Value of Money: It’s important to understand that calculated real values are subjective because they depend on which inflation rate is used. There is no single, universally agreed measure of inflation — governments themselves produce multiple estimates, and for a company’s own purposes these broad measures may not be precise enough. A company may therefore build its own inflation index and calculate real values against that instead. As a result, different companies can arrive at different real values for the same nominal cash flow, depending on the inflation assumptions they use.

The biggest takeaway from the concept of nominal and real values is that money in one time period is not directly comparable to money in another time period. This is exactly why we need to calculate present values and future values, which the rest of this article covers.

Value of Money Depends Upon Time

Money today is more valuable than the same sum received in the future because immediate funds carry no uncertainty and have not yet lost purchasing power to inflation.

The simple implication of this is that we cannot directly compare the dollars we have on hand today to dollars we’ve been promised at a future date. In corporate finance, we call the value of money we have on hand today the present value, and the value of an amount of money we will receive at a future date the future value.

We often come across complex schedules of payments and receipts — cash may have to be paid today or at a later date, and receipts may arrive today or later still. To compare all of these, we must first convert every value to a common basis, usually present value.

Calculating Future Values

Let’s work through an example. Say we have $1,000 today, our cost of capital is 10%, and we want to invest this money for 3 years.

The formula for calculating future value is:

Future Value = Present Value x (1 + (cost of capital / 100))^{number of years}

i.e. Future Value = $1,000 x (1.10)³

i.e. Future Value = $1,331

This means the equivalent sum of money we should expect in 3 years, given our cost of capital, is $1,331. We should accept proposals whose future value is more than $1,331, reject proposals whose future value is less than $1,331, and be indifferent to proposals whose future value equals exactly $1,331.

From here on, we’ll express this by saying that the future value of $1,000, at our given cost of capital, for a period of 3 years, is $1,331. It should also be noted that future values are nominal in nature.

Present Values

Present values are the mirror image of future values. When calculating future value, we compound a present sum forward at a given rate. In present value calculations, we instead discount a future value — which is nominal in nature — back to the present, at the given cost of capital, for the given number of periods.

Say we have a proposal that offers to pay us $1,000, 3 years from now, and our cost of capital is 10%.

The formula for calculating present value is:

Present Value = Future Value / (1 + (cost of capital / 100))^{number of years}

i.e. Present Value = $1,000 / (1.10)³

i.e. Present Value = $751.31

This means the equivalent sum of money we should expect today, given our cost of capital, is $751.31. We should accept proposals whose present value is more than $751.31, reject proposals whose present value is less than $751.31, and be indifferent to proposals whose present value equals exactly $751.31.

When finance professionals use the term “present value,” they are referring to this kind of discounted, present-day value, which is equivalent to a nominal future value.

The concepts of present value and future value form the basis of corporate finance. Any student of finance should be well versed in them, since variations of these concepts are used throughout the rest of corporate finance — including, as the next section shows, in how compounding itself is calculated.

Compounding Intervals and Interest Rate

Theoretically, there are two types of interest: simple and compound. In finance, however, the word “interest” usually refers to compound interest — simple interest almost never factors into financial calculations, and compound interest is used throughout present value and future value work. As a student of corporate finance, it’s also essential to know the effect that compounding intervals have on the effective interest rate paid on an investment.

Simple Interest vs. Compound Interest

With simple interest, the principal amount never changes. So if $100 is lent for 3 years at 10% simple interest, the interest paid in each of the 3 years is $10.

But if $100 is lent at 10% for 3 years with annual compounding, the interest earned would be $10, $11, and $12.10 in years 1, 2, and 3 respectively. This is because at the end of each period the accrued interest is added to the principal, so the interest earned in the next period is calculated on a slightly larger base.

Annual vs. Semi-Annual Compounding

10% compounded annually and 10% compounded semi-annually (twice a year) do not mean the same thing. Consider this example:

Annual Compounding: $100 @ 10% — interest = $10.

Semi-Annual Compounding: $100 @ 10% — interest of $5 after 6 months, and $5.25 after the next 6 months. The total interest for the year is $10.25, as opposed to $10 on an annual basis.

Rates Increase as Compounding Intervals Grow Smaller: As the example above shows, semi-annual compounding produces more interest than annual compounding for the same stated rate. Extending this logic, monthly compounding produces more interest than semi-annual compounding, and weekly compounding produces more than monthly. As a rule of thumb, the smaller the compounding interval, the higher the effective interest actually paid. In practice, most investments are compounded annually or semi-annually; smaller compounding frequencies are uncommon, with credit cards being one of the few everyday cases where rates are commonly expressed on a monthly compounding basis.

Continuous Compounding

So far we’ve considered discrete compounding intervals — we could shrink those intervals down to hours, minutes, or even seconds, and they would still be discrete. Theoretically, it’s possible for interest to be paid continuously over a period of time, which isn’t achievable in reality but is a useful simplification for mathematical purposes, which is why continuously compounded rates are often used in finance. The future value under continuous compounding is calculated as FV = PV x e^rt, where:

• e = 2.718 (Euler’s number)
• r = the annually compounded rate of interest
• t = number of time periods

Frequently Asked Questions

1. What’s the difference between nominal and real value of money?

   Nominal value is the plain dollar amount, unadjusted for inflation. Real value adjusts that amount for inflation so it reflects actual purchasing power.

2. Why does compounding more frequently increase the effective interest earned or paid?

   Because interest is added to the principal more often, so each new interest calculation is based on a slightly larger balance sooner than it would be under less frequent compounding.

3. Is continuous compounding used in practice?

   Not directly for everyday accounts, but it’s widely used in financial theory and modeling because it simplifies certain calculations, even though real-world compounding always happens at discrete intervals.