Time Value of Money

What is the Time Value of Money (TVM)?

The time value of money (TVM) is a fundamental concept in finance. The central idea is that money now is worth more than money in the future.

This principle is essential in finance because it reflects two fundamental properties of money: First, the potential earning capacity of money, whether through investments, interest, or other financial opportunities. Second is the gradual loss of purchasing power that comes with inflation.

These key drivers behind TVM are so ingrained into the modern financial system that it will affect nearly all financial decision-making over a long enough period, from the everyday person to the most important institutions.

Understanding The Time Value Of Money

There are three main reasons money is worth more now than later: 

1. Interest: Allows money to earn returns when invested or saved. This could be small amounts of interest from a bank account or risk-free investments in government bonds. It could also amount to larger returns from riskier investments or, in the case of a company, internal investments.
2. Inflation: Inflation decreases the purchasing power of money over time. As a result, cash now can buy more than the same cash in the future. How much it affects your money varies over time and geography.
3. Uncertainty: In practice, when you are due to receive money in the future, issues can arise before the time it’s due. Not receiving the cash you’re owed is the worst-case scenario, and this risk is eliminated by paying upfront.

These risks all become more pronounced the more time passes. If you’re looking to make financial decisions spanning multiple years, calculating the effect interest and inflation will have on your money should be a high priority. Thankfully, this is quite simple.

Calculating Present Value

Present Value is a pretty self-explanatory term. If you were to receive money in the future, how much would it be worth now? What would be its present value? If we assume we know how much your investment will return, the calculation is as follows: 

PV = FV / ((1+r) ^ n)​

Where:

• PV is the present value
• FV is the future value (money you will receive in the future)
• r is the interest rate per period
• n is the number of periods (this will almost always be years, but check to be sure)

Present Value Example

Suppose you expect to receive $1,000 five years from now, and the annual interest rate is 5%. To find the present value of this future amount, you would use the formula:

PV = 1000 / ((1+0.05) ^ 5)​

Calculating this gives:

PV = 1000 / 1.27628 ≈ 783.53

So, the present value of $1,000 received five years from now at a 5% interest rate is approximately $783.53. This means $783.53 today is equivalent to $1,000 in five years, considering the time value of money.

Calculating Future Value

Let’s consider the opposite situation; we have money now, how much is it going to be worth in the future? What will its future value be? We can rearrange our equation from earlier to get:

FV = PV * ((1+r) ^ n)

Intuitively, this version makes a lot more sense. Future value is your present value multiplied by your returns on your investment to the power of the amount of time you invest. 

Future Value Example

Let’s use the same example from earlier, but instead, you receive $1000 now. Our annual interest rate is still 5%. We would use the formula:

FV = 1000 * (1+0.05) ^ 5

Calculating this gives:

FV = 1000 × 1.27628 ≈ 1276.28

So, the future value of a $1,000 investment today at a 5% annual interest rate over 5 years is approximately $1,276.28. This means that after 5 years, your initial $1,000 investment will grow to $1,276.28, considering the time value of money.

From these calculations, you may be wondering how we account for inflation. It’s quite simple, you apply the following formula:

Real (or inflation-adjusted) interest rate = ((1+ interest rate) / (1 + inflation rate)) - 1

Depending on the context, you may or may not choose to adjust for inflation when calculating FV and PV. 

Time Value of Money — Personal Finance

When making personal economic decisions, you have to factor in how the value of your money will change as time goes on. In fact, most financial planning decisions you will make in your life span such long periods of time that if you don’t factor in TVM, you will arrive at completely incorrect conclusions. 

Below are some prominent examples of personal finance decisions that are heavily swayed by TVM:

Retirement Planning

TVM is crucial for planning retirement. Individuals need to estimate how much they need to save today to achieve a certain amount of money in the future. This involves calculating the present value of future retirement needs and determining how much to contribute. 

Example

Suppose Alice starts saving $200 per month into a retirement account with an annual interest rate of 6%. By using the TVM concept and the formula for the future value of an annuity, Alice can calculate how much she will have saved by retirement in 30 years:

Calculate the Real Interest Rate:

• Nominal interest rate = 6% or 0.06
• Inflation rate = 2% or 0.02
• The real interest rate can be calculated using the Fisher equation:

First, add 1 to both the nominal interest rate and the inflation rate:

1 + Nominal interest rate = 1.06

1 + Inflation rate = 1.02

Divide the adjusted nominal interest rate by the adjusted inflation rate:

1.06 / 1.02 ≈ 1.0392

Subtract 1 from this result to get the real interest rate:

Real interest rate = 1.0392 - 1 = 0.0392 or 3.92%

Calculate the Future Value Using the Real Interest Rate:

• Monthly savings = $200
• Real annual interest rate = 3.92% or 0.0392
• Monthly real interest rate = 0.0392 / 12
• Number of months = 30 years * 12 months/year = 360 months

Using the future value of an annuity formula with the real interest rate:

First, calculate the monthly real interest rate:

Monthly real interest rate = 0.0392 / 12 ≈ 0.003267

Then, calculate (1 + Monthly real interest rate) raised to the power of the number of months:

(1 + 0.003267)^360 ≈ 3.267

Subtract 1 from this result:

3.267 - 1 ≈ 2.267

Divide this result by the monthly real interest rate:

2.267 / 0.003267 ≈ 695.82

Multiply this result by the monthly savings:

Future Value = 200 * 695.82 ≈ 139,164

Loan Repayments

TVM helps individuals understand the cost of borrowing money. When taking out a loan, knowing the present value of loan repayments allows borrowers to compare different loan offers and understand the total interest paid over the loan period.

Example

Suppose Bob takes out a loan of $50,000 with an annual nominal interest rate of 5% and plans to repay it over 10 years. We will adjust for an inflation rate of 2% to determine the real cost of the loan repayments.

Calculate the Real Interest Rate:

• The nominal interest rate is 5% or 0.05.
• The inflation rate is 2% or 0.02.

First, add 1 to both the nominal interest rate and the inflation rate:

1 + 0.05 = 1.05

1 + 0.02 = 1.02

Next, divide 1.05 by 1.02:

1.05 / 1.02 ≈ 1.0294

Notice how we pay a lower actual rate if there is inflation. This is because the money we pay back in the future is worth less than the money we pay back now.

Subtract 1 from this result to get the real interest rate:

1.0294 - 1 = 0.0294, or 2.94%

Calculate the Monthly Loan Repayment Using the Real Interest Rate:

• The loan amount (Principal) is $50,000.
• The real annual interest rate is 2.94% or 0.0294.
• To find the monthly real interest rate, divide the annual rate by 12:

0.0294 / 12 ≈ 0.00245

Bob will repay the loan over 10 years, which equals 120 months (10 years * 12 months/year).

Using the formula for monthly loan repayments, we need to calculate:

First, calculate the numerator (Loan amount * Monthly real interest rate):

$50,000 * 0.00245 ≈ $122.50

Next, calculate the denominator. To do this, first find the value of (1 + Monthly real interest rate) raised to the power of the number of months:

(1 + 0.00245)^120 ≈ 0.7513

Subtract this value from 1:

1 - 0.7513 ≈ 0.2487

Finally, divide the numerator by the denominator to find the monthly payment:

$122.50 / 0.2487 ≈ $492.58

Time Value of Money — Business Finance

Businesses often have to plan out their finances years in advance to be as efficient as possible. In some cases, businesses will know exactly how money will be moving in the future, and in others, they’re making the best projections they can.

In both cases, having an accurate TVM will lead to significantly better decision-making. Below are some examples of businesses using TVM to understand their future finances accurately.

Discounted Cash Flow (DCF) Modeling

• DCF modeling is a valuation method used to estimate the value of an investment based on its expected future cash flows.
• The future cash flows are discounted using the future value equation we mentioned earlier.
• By discounting these cash flows back to their present value using a discount rate called the weighted average cost of capital (WACC), investors can determine the intrinsic value of the investment, the net present value (NPV).
• It’s important to note that most companies don’t account for inflation in many items in their financial statements. The cost of capital is also calculated without accounting for inflation. Hence, it is very uncommon to account for inflation when discounting cash flows.
• DCF modeling is a complex process, and if you want to learn more about it, click here.

Capital Budgeting

Businesses use TVM in capital budgeting to evaluate long-term investments such as new projects, equipment, or facilities. By calculating the present value of expected future cash flows from these investments, businesses can assess their profitability and make informed decisions.

Example

A company is considering a new project that requires an initial investment of $50,000. The project is expected to generate $15,000 annually for 5 years. 

Once again, in the business world, it is not standard practice to account for inflation. Using a discount rate of 10%, the Net Present Value (NPV) of the project is calculated as follows:

1. Discounting Cash Flows:

Year 1: $15,000 / (1 + 0.10)^1 ≈ $13,636

Year 2: $15,000 / (1 + 0.10)^2 ≈ $12,397

Year 3: $15,000 / (1 + 0.10)^3 ≈ $11,270

Year 4: $15,000 / (1 + 0.10)^4 ≈ $10,245

Year 5: $15,000 / (1 + 0.10)^5 ≈ $9,314

2. Summing Present Values:

Total Present Value of inflows = Approximately $56,862 (sum of all discounted cash flows)

3. Calculating NPV:

NPV = Total Present Value of inflows - Initial Investment

NPV = $56,862 - $50,000

NPV ≈ $6,862

Since the NPV is positive, amounting to approximately $6,862, the project is expected to generate more value than its initial cost. Therefore, based on this NPV analysis, the project appears to be a viable investment opportunity for the company.

Free Resources

To continue learning and advancing your career, check out these additional helpful WSO resources:

• Ballpark Figure
• Cap Rate (REIT)
• Corporate Bond Valuation
• Debt Free Cash Free Valuation
• EBITDA Multiple