## Present Value Calculation: Formula

The present calculation formula is:

\(\hbox{Equation 2:}\)

\(C_0= \frac {C_t} {(1+i)^t}\)

But where does it come from? To understand it, we must first introduce two concepts: the time value of money and compound interest.

The **time value of money** is the opportunity cost of receiving money in the future as opposed to today. Money is more valuable the sooner it is received because it can then be invested and earn compound interest.

The** time value of money** is the opportunity cost of receiving money later rather than sooner.

Now that we understand the concept of the time value of money, we introduce the concept of compound interest. **Compound interest** is the interest earned on the original investment and the interest already received. This is why it is called **compound** interest, because the investment is earning interest on interest...it is compounding over time. The interest rate and the frequency at which it compounds (daily, monthly, quarterly, yearly) determines how fast and how much an investment's value increases over time.

**Compound interest **is interest earned on the original amount invested and the interest already received.

The following formula illustrates the concept of compound interest:

\(\hbox{Equation 1:}\)

\(\hbox{Ending value} = \hbox{Beginning Value} \times (1 + \hbox{interest rate})^t \)

\(\hbox{If} \ C_0=\hbox{Beginning Value,}\ C_1=\hbox{Ending Value, and} \ i=\hbox{interest rate, then:} \)

\(C_1=C_0\times(1+i)^t\)

\(\hbox{For 1 year}\ t=1\ \hbox{, but t can be any number of years or periods}\)

Thus, if we know the investment's beginning value, the interest rate earned, and the number of compounding periods, we can use Equation 1 to calculate the investment's ending value.

To get a better understanding of how compound interest works, let's take a look at an example.

\(\hbox{If} \ C_0=\hbox{Beginning Value,} \ C_t=\hbox{Ending Value, and} \ i=\hbox{interest rate, then:} \)

\(C_t=C_0 \times (1 + i)^t \)

\(\hbox{If} \ C_0=$1,000, \ i=8\%, \hbox{and} \ t=20 \hbox{ years, what is the value of the investment} \)\(\hbox{after 20 years if interest compounds annually?} \)

\(C_{20}=$1,000 \times (1 + 0.08)^{20}=$4,660.96 \)

Now that we understand the concepts of the time value of money and compound interest, we can finally introduce the present value calculation formula.

By rearranging Equation 1, we can calculate \(C_0\) if we know \(C_1\):

\(C_0= \frac {C_1} {(1+i)^t}\)

More generally, for any given number of periods t, the equation is:

\(\hbox{Equation 2:}\)

\(C_0= \frac {C_t} {(1+i)^t}\)

This is the present value calculation formula.

**Present value** is the present-day value of future cash flows of an investment.

By applying this formula to all expected future cash flows of an investment and summing them up, investors can accurately price assets in the market.

## Present Value Calculation: Example

Let's take a look at a present value calculation example.

Suppose you just got a $1,000 bonus at work and you are planning to put it in the bank where it can earn interest. Suddenly your friend calls you and says he is putting a little money into an investment that pays out $1,000 after 8 years. If you put the money into the bank today you will earn 6% interest annually. If you put the money into this investment, you will have to forgo the interest from the bank for the next 8 years. In order to get a fair deal, how much money should you put into this investment today? In other words, what is the present value of this investment?

\(\hbox{The present value calculation formula is:} \)

\(C_0=\frac{C_t} {(1 + i)^t} \)

\(\hbox{If} \ C_t=$1,000, i=6\%, \hbox{and} \ t=8 \hbox{ years, what is the present value of this investment?} \)

\(C_0=\frac{$1,000} {(1 + 0.06)^8}=$627.41 \)

The logic behind this calculation is two-fold. First, you want to make sure you would get at least as good of a return on this investment as you would if you put it in the bank. That, however, assumes that this investment carries about the same risk as putting the money in the bank.

Second, with that in mind, you want to figure out how much is a fair value to invest to realize that return. If you invested more than $627.41, you would receive a smaller return than 6%. On the other hand, if you invested less than $627.41, you may get a larger return, but that would likely only happen if the investment is riskier than putting your money in the bank. If, say, you invested $200 today and received $1,000 in 8 years, you would realize a much larger return, but the risk would also be much higher.

Thus, the $627.41 equates the two alternatives such that the returns for similarly risky investments are equal.

Now let's take a look at a more complicated present value calculation example.

Suppose you are looking to buy a corporate bond that currently yields 8% annually and matures in 3 years. The coupon payments are $40 per year and the bond pays the $1,000 principle at maturity. How much should you pay for this bond?

\(\hbox{The present value calculation formula can also be used to price an asset} \) \(\hbox{with multiple cash flows.} \)

\(\hbox{If} \ C_1 = $40, C_2 = $40, C_3 = $1,040, \hbox{and} \ i = 8\%, \hbox{then:} \)

\(C_0=\frac{C_1} {(1 + i)^1} + \frac{C_2} {(1 + i)^2} + \frac{C_3} {(1 + i)^3} \)

\(C_0= \frac{$40} {(1.08)} + \frac{$40} {(1.08)^2} + \frac{$1,040} {(1.08)^3} = $896.92 \)

Paying $896.92 for this bond ensures that your return over the next 3 years will be 8%.

The first example only required us to calculate the present value of one cash flow. The second example, however, required us to calculate the present value of multiple cash flows and then add up those present values to obtain the overall present value. A few periods aren't so bad, but when you are talking about 20 or 30 periods or more, this can get very tedious and time-consuming. Therefore, financial professionals use computers, computer programs, or financial calculators to carry out these more complex calculations.

## Net Present Value Calculation

A net present value calculation is used to determine whether or not an investment is a wise decision. The idea is that the present value of future cash flows must be greater than the investment made. It is the sum of the initial investment (which is a negative cash flow) and the present value of all future cash flows. If the net present value (NPV) is positive, the investment is generally considered a wise decision.

**Net present value** is the sum of the initial investment and the present value of all future cash flows.

To get a better understanding of net present value, let's take a look at an example.

Suppose XYZ Corporation wants to buy a new machine that will increase productivity and, thereby, revenue. The cost of the machine is $1,000. Revenue is expected to increase by $200 in the first year, $500 in the second year, and $800 in the third year. After the third year, the company plans to replace the machine with an even better one. Also suppose that, if the company doesn't buy the machine, the $1,000 will be invested in risky corporate bonds that currently yield 10% annually. Is buying this machine a wise investment? We can use the NPV formula to find out.

\(\hbox{If the initial investment} \ C_0 = -$1,000 \)

\(\hbox{and } C_1 = $200, C_2 = $500, C_3 = $800, \hbox{and} \ i = 10\%, \hbox{then:} \)

\(NPV = C_0 + \frac{C_1} {(1 + i)^1} + \frac{C_2} {(1 + i)^2} + \frac{C_3} {(1 + i)^3} \)

\(NPV = -$1,000 + \frac{$200} {(1.1)} + \frac{$500} {(1.1)^2} + \frac{$800} {(1.1)^3} = $196.09 \)

\(\hbox{The expected return on this investment is: } \frac{$196} {$1,000} = 19.6\% \)

Since NPV is positive, this investment is generally considered a wise investment. However, we say generally because there are other metrics used to determine whether or not to take on an investment, which are beyond the scope of this article.

In addition, the 19.6% expected return on buying the machine is far greater than the 10% yield on the risky corporate bonds. Since similarly risky investments must have similar returns, with such a difference, one of two things must be true. Either the company's revenue growth forecasts due to buying the machine are quite optimistic, or buying the machine is far riskier than buying the risky corporate bonds. If the company reduced its revenue growth forecasts or discounted the cash flows with a higher interest rate, the return on buying the machine would be closer to that of the risky corporate bonds.

If the company feels comfortable with both its revenue growth forecasts and the interest rate used to discount the cash flows, the company should buy the machine, but they should not be surprised if revenue doesn't grow as strongly as predicted, or if something goes wrong with the machine in the next three years.

## Interest Rate for Present Value Calculation

The interest rate for present value calculation is the interest rate that is expected to be earned on a given alternative use of the money. Generally, this is the interest rate earned on bank deposits, the expected return on an investment project, the interest rate on a loan, the required return on a stock, or the yield on a bond. In each case, it can be thought of as the opportunity cost of an investment that results in a future return.

For example, if we want to determine the present value of $1,000 we would receive one year from now, we would divide it by 1 plus the interest rate. What interest rate shall we choose?

If the alternative to receiving $1,000 one year from now is to put the money into a bank, we would use the interest rate earned on bank deposits.

If, however, the alternative to receiving $1,000 one year from now is to invest the money in a project that is expected to pay out $1,000 one year from now, then we would use the expected return on that project as the interest rate.

If the alternative to receiving $1,000 one year from now is to lend the money out, we would use the interest rate on the loan as the interest rate.

If the alternative to receiving $1,000 one year from now is to invest it in buying shares of a company, we would use the required return of the shares as the interest rate.

Finally, if the alternative to receiving $1,000 one year from now is to buy a bond, we would use the yield of the bond as the interest rate.

The bottom line is that the interest rate used for present value calculation is the return on an alternative use of the money. It is the return you give up now in the expectation of receiving that return in the future.

Think of it this way. If person A has a piece of paper that says Person B owes Person A $1,000 one year from now, how much is that piece of paper worth today? It depends on how person B is going to raise the cash to pay off the $1,000 one year from now.

If Person B is a bank, then the interest rate is the interest rate on bank deposits. Person A will put the present value of $1,000 one year from now in the bank today and receive $1,000 one year from now.

If person B is a company taking on a project, then the interest rate is the return on the project. Person A will give Person B the present value of $1,000 one year from now and expect to be paid back $1,000 one year from now with the returns on the project.

Similar analyses can be conducted for loans, stocks, and bonds.

If you would like to learn more, read our explanations about Banking and Types of Financial Assets!

It is important to note that the riskier the way in which the money is to be raised to pay back the investment, the higher is the interest rate, and the lower is the present value. Since putting money in the bank is very low risk, the interest rate is low, so the present value of $1,000 received one year from now is not very much less than $1,000. On the other hand, putting money in the stock market is very risky, so the interest rate is much higher, and the present value of $1,000 received one year from now is much lower than $1,000.

If you would like to learn more about risk, read our explanation about Risk!

Generally speaking, when you are given present value problems in economics, you are given an interest rate, but rarely do they tell you what interest rate is being used. You just get the interest rate and proceed on to your calculations.

## Present Value Calculation: Equity Shares

Calculating the price of equity shares is basically a present value calculation. The price is simply the sum of the present value of all future cash flows. For a stock, the future cash flows in most instances are the dividends per share paid out over time and the sale price of the stock at some future date.

Let's look at an example of using a present value calculation to price equity shares.

\(\hbox{The present value calculation formula can be used to price a stock} \) \(\hbox{with dividends per share and the sale price as cash flows.} \)

\(\hbox{Let's look at a stock with dividends paid out over 3 years.} \)

\(\hbox{Suppose} \ D_1 = $2, D_2 = $3, D_3 = $4, P_3 = $100, \hbox{and} \ i = 10\% \)

\(\hbox{Where:}\)

\(D_t = \hbox{The dividend per share in year t}\)

\(P_t = \hbox{The expected sale price of the stock in year t}\)

\(\hbox{Then: } P_0, \hbox{the current price of the stock, is:}\)

\(P_0=\frac{D_1} {(1 + i)^1} + \frac{D_2} {(1 + i)^2} + \frac{D_3} {(1 + i)^3} + \frac{P_3} {(1 + i)^3}\)

\(P_0=\frac{$2} {(1 + 0.1)^1} + \frac{$3} {(1 + 0.1)^2} + \frac{$4} {(1 + 0.1)^3} + \frac{$100} {(1 + 0.1)^3} = $82.43\)

As you can see, using this method, known as the dividend discount model, an investor can determine the price of a stock today based on expected dividends per share and the expected sale price at some future date.

One question remains. How is the future sale price determined? In year 3, we simply do this same calculation again, with year three being the current year and the expected dividends in the following years and the expected sale price of the stock in some future year being the cash flows. Once we do that, we ask the same question again and do the same calculation again. Since the number of years can, in theory, be infinite, the calculation of the final sale price requires another method that is beyond the scope of this article.

If you would like to learn more about expected returns on assets, read our explanation about the Security Market Line!

## Present Value Calculation - Key takeaways

- The time value of money is the opportunity cost of receiving money later rather than sooner.
- Compound interest is interest earned on the original amount invested and the interest already received.
- Present value is the present-day value of future cash flows.
- Net present value is the sum of the initial investment and the present value of all future cash flows.
- The interest rate used for present value calculation is the return on an alternative use of the money.

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##### Frequently Asked Questions about Present Value Calculation

How do you calculate present value in economics?

Present value in economics is calculated by dividing the future cash flows of an investment by 1 + the interest rate.

In equation form, it is:

Present Value = Future Value / (1 + interest rate)^{t}

Where t = number of periods

How is the present value formula derived?

The present value formula is derived by rearranging the equation for future value, which is:

Future Value = Present Value X (1 + interest rate)^{t}

Rearranging this equation, we get:

Present Value = Future Value / (1 + interest rate)^{t}

Where t = number of periods

How do you determine present value?

You determine present value by dividing the future cash flows of an investment by 1 + the interest rate to the power of the number of periods.

The equation is:

Present Value = Future Value / (1 + interest rate)^{t}

Where t = number of periods

What are the steps in calculating present value?

The steps in calculating present value are knowing the future cash flows, knowing the interest rate, knowing the number of periods of cash flows, calculating the present value of all cash flows, and summing up all of those present values to get the overall present value.

How do you calculate present value with multiple discount rates?

You calculate present value with multiple discount rates by discounting each future cash flow by the discount rate for that year. You then sum up all of the present values to get the overall present value.

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