Time Value Of Money – Discounting Projected Cashflows

You would rather have $1 today than $1 in a year. If you got the $1 today, you could invest it and earn a return so that in a year it is more than $1. Or, you could purchase an item today for $1 and inflation will likely make that item more than $1 in a year. The concept of money today being more valuable than money in the future is formally known as the time value of money. It is a fundamental component of finance and used extensively in both the personal and corporate finance space.

However, at its core, the time value of money is just the concept that money today is worth more to you than the same money in the future.

Time value of money comes in handy when you are projecting cashflows. If you know that a dollar is ‘worth less’ in the future, then what is the equivalent value today? Said another way, if you need to pay $1 in the future, how much do you need to have today?

TVM is a fundamental concept to learn when starting your personal finance journey. After you have read this post, you will be able to answer this question and calculate equivalent values of any cashflow at any point in time.

What is Time Value of Money?

Time value of money is the concept that $1 today is worth more to you than receiving $1 in the future. This is largely due to 2 concepts:

1. Opportunity Cost: Opportunity cost is a measure of what you give up from not choosing an alternative option. If you choose to receive $1 in the future, your opportunity cost is the earnings you could have earned on that dollar if you received it today and invested it.
2. Inflation: Inflation is the increase in prices of items over time. This decreases your future purchasing power. If you receive $1 in the future, you will be able to purchase less due to the cost of items increasing over time. If inflation is 5%, then something that cost $1 today will cost $1.05 in a year.

Both concepts work in similar ways. If you have $100 today and you put it in a savings account at 5%, you would have $105 in a year. Therefore, your opportunity cost of receiving the money in a year is $5. (IE – you missed out on $5 or returns).

Similarly, if you have $100 today and purchase items with it. If inflation is 5% per year, you would only be able to buy the equivalent of ~$95 of items a year from now.

The utility (aka value) you get out of receiving money today is more than the same dollar amount received in the future.

How Do You Calculate Time Value of Money?

There are 2 fundamental formulas for time value of money. When looking at a stream of cashflows you can calculate the present value and future value of that stream. The present value is the equivalent value of the money if received as a lump sum today. Similarly, the future value is the equivalent value of the money if received as a lump sum at some future point.

To calculate the present and future values of the cashflow, you need to have a rate and the number of periods.

Calculating the Future Value of Money

The future value (FV) of money is the lump-sum value in the future that is equivalent to the cashflows.

If you had $100 today and you invested it at a 5% rate of return, it would grow to $105 in a year. ($100 x (1 + 5%) = $105).

Similarly, if you had $100 and invested it at a 5% growth rate for 2 years you would have $110.25. ($100 x (1 + 5%)² = $110.25).

You can repeat this for any growth rate and any number of periods. This means that if you received $100 today, $105 in a year, or $110.25 in 2 years, you would be indifferent as each has the same value to you.

You may have noticed the increase from time 0 to time 1 was $5, but from time 1 to time 2 is was $5.25. This is called compounding. Initially you only had $100 invested. But after 1 year you had $105 invested. You earned the 5% return on that extra $5, which resulted in the extra $0.25 growth in year 2 vs year 1.

Calculating the Present Value of Money

The present value of money is the lump-sum value you would receive today that is equivalent to receiving money in the future. You can rearrange the above future value formula to get the formula for present value.

If you were going to receive $105 in a year and the discount rate was 5%, then the present value equivalent would be $100 today. ($105 / ( 1+ 5%) = $100).

Similarly, if you were going to receive $110.25 in 2 years, the equivalent would be $100 today. ($110.25 / (1 + 5%)² = $100).

You reach the same conclusions as in the future value since our interest, time, and cashflows used are the same.

This highlights an important aspect of time value of money. You can calculate the value of money at any point in time forwards and backwards. This gives you have the ability to value any cashflow stream at any point in time.

Calculating Time Value of Money With Different Factors

The above 2 formulas assumed you had a constant rate and did the calculation for a single time step. What happens if you have different rates, numerous cashflows, and different time steps?

We can apply the same principles from the simple examples above and extend it out to handle any situation.

[Professor B.T. Effer Note – Interest rates, discount rates, growth rates, compounding rates, rates of return, etc. What is the difference between all the rates? In short, they are all different names for the same concept. They tend to be used interchangeably. Although, there is some difference between each.

If you are calculating a present value, you are discounting future cashflow to today. Therefore, the rate used is a discount rate.

If you are calculating a future value, you are growing or compounding cashflows. Therefore, the rate used is a growth rate or compounding rate.

Interest rates are often used as a rate of return, although you could choose to use returns on stocks or other assets. Additionally, you can use the APR on a loan for your discount/growth rate. The rate of return is used when growing or discounting cashflows.]

Example 1: Future Value of a Cashflow Using the Same Rate

If you get $100 today and assume a 5% growth rate, that money grows to $105 in one year and $110.25 in 2 years. We used this example earlier, but visually this is how it would look.

You can either grow the $100 at 5% in two steps or use (1.05)² to do it in one step.

In this example, you would be indifferent to receiving $100 today, $105 in a year, or $110.25 in 2 years. You can repeat this rolling the money forward as many years in the future as you want.

Example 2: Present Value of a Cashflow Using the Same Rate

This example replicates example 1, but working in reverse to bring a lump sum in the future back to the present.

Instead of growing $100 at 5%, you discount the future amounts back to today at 5%.

This shows that you can move a lump sum forwards and backwards in time and get the same values.

In this example, if someone owed you $110.25 in 2 years, they could pay you $100 today to settle the debt.

But what if the rate used isn’t the same for both time periods?

Example 3: Future Value & Present Value of a Cashflow Using the Different Rates

You don’t need to use the same rate for each period when calculating future values and present values.

In the next 2 examples we assume a 5% rate in the first year, and a 10% rate in the second year. In this case, $100 today grows to $105 in one year and $115.50 after 2 years.

If you thought you could get a 5% then 10% return, you would be indifferent between those 3 values.

When you have different rates, you need to have do the growth & discounting in steps.

Similarly, you can take the present value of $115.50 out 2 years from now and discount it back by 10% and 5% to get to $100 today.

Time value of money principles hold up over multiple rate changes.

This is powerful when doing projections over the long-term if you think the rate environment will be different. For example, if you are using US Treasury rates for your calculations, this allows you to reflect changing rates.

Example 4: Future Value & Present Value of a Multiple Cashflows

So far we have used 1 lump-sum cashflow to calculate different future & present values. But you can follow the same principles when you have a stream of cashflows.

In this example we have cashflows of $100 in year 1 and another $100 in year 2. If we assume a 5% growth rate, what is the future value of the cashflows in year 4?

We can take the $100 in year 1 and grow it for 3 years. And we can take the $100 in year 2 and grow it for 2 years. The result is $226.01.

You can replicate the process to get the present value of multiple cashflows. Using the same $100 in year 1 and year 2, if you wanted to find the present value of the cashflows, you discount them by 1 and 2 years respectively. The result is $185.94.

If you are familiar with annuities or lump-sum payments, this is the way those are calculated.

For example those “need cash now” commercials that will take an annuity or structured settlement and pay you a lump sum. This is the service they provide, taking all your future cashflows and discounting them to today. Albeit, they use a very high discount rate so the present value is relatively lower and they can earn a profit.

The beauty of time value of money is that you can combine all the examples above and find a lump-sum in any given year for any cashflow and rate combinations.

If you have multiple cashflows you can calculate a lump-sum future or present value. And then you can take that value and move it forward or backwards any number of years.

Using the same example as previously, we showed that taking the two $100 cashflows you would wind up with $226.01 in year 4. You can then take the $226.01 and discount it 4 years at a 5% rate and the result is $185.94.

This was the same result as discounting the 2 cashflows separately to get the present value.

You could continue to expand this example and use different interest rates at different periods. You would just need to apply the principle in example 3 for using different rates.

The concept is simple when you understand it, but extremely powerful.

Periodic Compounding – How to Handle Multiple Compounding in a Year

Thus far we have been assuming annual compounding. But most assets compound more frequently than annually. You can easily adjust the above formulas to account for semi-annual (2x a year), quarterly (4x a year), monthly (12x a year), or daily (365x a year) compounding.

To reflect different compounding periods, you use the expanded formulas for future value and present value shown here. (Note – if you have annual compounding, n=1 and you get the formulas above).

If we assume a 10% rate, the more often it gets compounded, the larger future value the money grows to. This is because each time you compound, you start earning a return on that additional dollar amount.

If you invest $1,000 in an asset that returns 10% and compounds once a year, you have $1,010. ($1,000 * (1 + 10%) = $1,100).

However, if you compound twice a year, after 6 months you have earned $50 and invest $1,050 for the next 6 months. Your $1,000 earns another $50, but that extra $50 you reinvested also earns 6 months of interest ($2.50). As you can see, the more often you are able to reinvest at the same rate, the more you can earn.

[Professor B.T. Effer Note – Most accounts will show an effective APR which takes into account the compounding. Explaining all the nuances on rates, compounding periods, and differences in calculations would be a separate post. For now, we assume simple interest.]

How Does Time Value of Money Relate to Opportunity Costs?

Opportunity cost is closely related to the concepts of time value of money. Opportunity cost is the return you give up by choosing one option over another.

For example if you have the ability to earn 3% in a savings account or you can spend the money on an item. If you spend the money, you give up the 3% return of a savings account. Therefore, the cost of the item is not only the price you pay, but includes the cost of the 3% return you gave up by not putting the money in your savings account.

Similarly, if you could get $100 today or $100 in the future and you can earn 3% on the money you get today. The opportunity cost of receiving the money at a future point of time is the 3% return you missed out on. Time value of money deals with money today being more valuable than money in the future.

How Does Time Value of Money Relate to Inflation?

Inflation is very similar to the concept of opportunity cost. Inflation raises the price of goods and services each year. Therefore, receiving $100 in the future allows you to purchase less than receiving it today.

The logic is the same, you would need to discount the $100 you receive in the future at the rate of inflation to figure out the equivalent present value today.

Why is the Time Value of Money Important?

Time value of money is fundamental to both personal and corporate finance.

In corporate finance, if you have 2 projects with different cashflows, you need to be able to compare the relative value. One of the most powerful ways to do this is by finding the present value of each and seeing which is most profitable. This is the central concept behind discounted cash flow (DCF) analysis used in business planning and stock analysis.

In personal finance, you can use time value of money to determine where to invest and to value future earnings.

One of the best examples is when looking at college. You can project out 2 paths; one with college and the income of your planned career and the other where you forgo college and work in a career without college. If you skip college, you may make less money per year, but you start earning income 4 years earlier and avoid any debts. You can take the present value of the 2 options. The one that has a higher present value is the mathematically optimal choice.

The Final Word – Time Value of Money

Time value of money is a fundamental concept in finance. It allows you to make more informed and optimal decisions by comparing different cashflow streams and options. TVM reflects the reality that a dollar today is more valuable than a dollar in the future. This principle is due to both opportunity costs and inflation.

Learning how to value different cashflows you are well on your way to making optimal choices for your personal financial needs.

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