# Understanding the Time Value of Money

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In January, Darrow wrote a post sharing his portfolio moves and performance in 2021. A reader left this comment in response:

“I read (in another recent post) that your rough “best case” for a 50/50 portfolio moving forward was about 3.8% real rate of return. Yet, this recent post calls the quote for a 4.7% immediate annuity return ‘in the gutter.’ Can the apparent contradiction be explained by the different time frames involved?”

This is a great question. We will all face a similar situation at some point as we make decisions between different financial options.

Making good decisions in these scenarios requires understanding the concept of time value of money and the related concepts of compounding and discounting. This question provides an opportunity to explore these concepts and how to perform time value of money calculations. Let’s dive in…..

## What is Time Value of Money (TVM)?

The concept of the time value of money is that a dollar received today is different from the value of a dollar received tomorrow. Typically, a dollar received today is assumed to be more valuable than a dollar received in the future. This is because you can invest your dollar today putting it to work creating more future dollars.

Conversely, your dollar today could be less valuable in the future if it is not invested at a rate greater than inflation. Let’s demonstrate this point with the example of college education.

Imagine you had a baby yesterday and put a lump sum large enough to cover the first year of college tuition at today’s prices into a savings account earning 1% interest. Assume the rate of inflation of college tuition is 5% over the ensuing 18 years.

Your money will collect interest for nearly two decades. The nominal value will be larger in the future when your child is ready to start college.

A time value of money calculation would demonstrate a considerable loss of purchasing power. The future value of your money would have lost about 50% of its present value due to inflation!

## Variables Required for TVM Calculations

The variables that impact TVM calculations are:

• Present Value (PV)
• Future Value (FV)
• Number of compounding periods (n)
• Rate of interest (i)
• Payments (PMT)

TVM calculations utilize at least the first four variables. If you know any 3 of them, you can calculate the other one.

In some scenarios, you may also have to account for a fifth variable, payments. This is necessary if there is a stream of cash flows into or out of the investment over time. In this case, if we know any four variables, we can solve for the fifth.

## Compounding vs. Discounting

Compounding is the process involved when calculating forward. You start with a present value (the value at the beginning of a period). The rate of return over the number of compounding periods, accounting for any cash flowing in or out along the way, determines a future value.

Discounting is simply the reverse of compounding. You start with a known future value (the value at the end of a period), then calculate back to the beginning of the period to determine a present value.

## Making Assumptions and Identifying the Variables

The first step to being able to utilize TVM concepts is being able to accurately identify the variables to plug into the TVM equation.

In the case of the reader’s question, we are comparing apples to oranges. This makes identifying the variables more challenging.

The reader’s comment compares the 4.7% immediate annuity “return” with an assumed 3.8% real rate of return on the investment portfolio. This is not accurate.

An immediate annuity is an insurance product to protect against longevity risk. It involves paying a lump sum of money to an insurance company in exchange for a guaranteed stream of income (i.e. payments) until you die. Payments are the same each year and continue until death.

We are comparing that to keeping money invested in a portfolio with an assumed real (inflation adjusted) rate of return. You decide how much, if any, payment to create from the portfolio each year. You also have to determine how to create income from dividends, interest, selling off a portion of principal, or any combination of the above.

In the portfolio scenario, there is a risk that due to low returns or poor management you may run out of money before you run out of life. There is also a potential upside that you could meet your lifetime spending needs and still have money left over to leave an inheritance.

To make any type of meaningful comparison, we have to identify our starting variables and assumptions. We can then make subsequent assumptions to compare options.

## Our Starting Assumptions

Let’s assume that we’re starting with the decision of taking a portion of our portfolio equal to \$500,000 at 60 years of age. We have two options:

1. Purchase a \$500,000 immediate annuity paying 4.7% of the purchase amount annually to create an income floor that (assuming the insurance company is in good standing and  remains solvent) is guaranteed to produce this same payment indefinitely until death. At death, payments stop and no money is returned to your estate.
2. Keep our \$500,000 invested in our portfolio that we will continue to manage with an assumed real (after-inflation) rate of return of 3.8%.

### Annuity Variables

Let’s start with the 4.7% payout. The reader’s question compared this to the 3.8% real rate of return on the investment portfolio. This is incorrect.

The 4.7% is used to determine the payment (PMT) of the annuity. Thus the variables we know are:

• Present Value (PV) = -\$500,000
• Future Value (FV) = \$0
• Payments (PMT) = \$23,500/year or \$1,958/month
• Rate of return (i) = ????
• Number of Compounding Periods (n) = ????

The PV is expressed as a negative number for calculations because it is money flowing away from you (cash outflow). The FV is zero by definition with an immediate annuity, because when you die payments stop and the insurance company keeps any remaining proceeds.

The payment is simply the 4.7% annuity rate multiplied by the \$500,000 purchase price. Since you would likely elect to receive monthly payments, you could divide this by 12 to get the monthly amount. It is important to keep track of whether you are using monthly or annual payments.

The number of compounding periods and the rate of return on investment are unknowns until death. To determine either the rate of return or the number of compounding periods, we would need to make assumptions about the other.

Using the annual PMT amount will produce the number of years of compounding and the annual rate of return for calculations. Using the monthly PMT number, would require you to multiply 12Xn and divide i/12 to reflect the accurate number of months and monthly rate for compounding.

### Portfolio Variables

We also have incomplete information regarding the variables required to perform a TVM calculation of the investment portfolio. Here is what we know:

• PV = -\$500,000
• FV = ????
• PMT = ????
• i = 3.8% annual or .31666% per month real (after inflation) return
• n = ????

The present value again is negative because it is cash flowing away from us into the investment portfolio. The rate of return is defined in the assumptions.

Without making any further assumptions, we can not identify the other three variables.

## Use Time Value of Money to Make Better Investment Decisions

Remember that we need any three variables in a TVM scenario to calculate the fourth variable if there are no payments. If there are payments, we will need four variables to solve for the fifth.

So how do we use this concept to make a meaningful comparison with the information provided?

We can make some assumptions about the unknown variables to better understand our investment options. I’ll include an example of a compounding calculation and a discounting calculation.

## Compounding Example

We know that an immediate annuity provides longevity insurance. It will produce income indefinitely in the event that you live an unusually long life.

The risk of keeping your money invested is that you could run out of money later in life. So one thing we can do is to see how long our invested portfolio could last to determine at what point an annuity would be advantageous. To do this, we need to solve for n for the portfolio.

We know our present value and assumed real rate of return for the investment portfolio. An annuity would produce a payment of \$23,500/year and would leave you with a future value of \$0 at death. We can use those numbers to see how long we could produce an equivalent annual income from our portfolio before running out of money.

So to solve for n, we can plug in:

• PV = -\$500,000
• FV = \$0
• PMT = \$23,500
• i = 3.8%
• n = ????

In this scenario, n = 41 years. This tells us that if we get our assumed real rate of return and take the same payment that the annuity would provide it would take 41 years to run out of money.

Given our starting assumptions, this means that we would not run out of money until we are 101 years old. Using 2022 IRS life expectancy tables, a 60 year old would have a life expectancy of 25.2 additional years.

There is a very small chance of outliving your money, and thus little justification for buying the annuity. This favors the decision to not buy the annuity.

## Discounting Example

The primary reason we would buy an annuity is to protect against longevity risk. However, we also have to consider how much our annual annuity payment would be worth decades into the future when we would most need it. We could do this by discounting the future payment to see what it would be worth in today’s dollars.

In this scenario we will solve for the present value of a future payment of \$23,500. Our variables to plug into the TVM equation are:

• PV= ????
• FV= \$23,500
• PMT= 0
• i= 2.25
• n= 30

We are simply discounting the future value of the annual annuity payment to find its present value in today’s dollars. There is no payment in this equation.

I chose to discount by the average U.S. inflation rate, which is 2.25%. I chose 30 years to represent a time decades into the future when an annuity would theoretically be most important to protect against longevity risk.

After plugging our variables into the TVM equation, we learn that our \$23,500 annual payment 30 years in the future is worth only \$12,055 in today’s dollars. At an average inflation rate, our money loses about half of its value over 30 years.

What if average inflation over the next three decades is higher than it has been historically? If we change our assumption to a 3% inflation rate, our \$23,500 payment 30 years in the future will be worth only \$9,681. It would lose approximately 60% of its purchasing power.

The higher the inflation rate and the longer we live, the less valuable the future annuity payment is. This loss of future purchasing power could be another reason to hold off on an annuity purchase when rates are low and you are relatively young.

## Time Value of Money Take Home Points

TVM is the concept that a dollar today has a different value than it did at a time in the past or will have in the future.

Compounding starts with a present value and calculates forward to determine a future value. Discounting takes this process in reverse to find the present value of a future value of money.

Variables that impact compounding and discounting are the rate of return, the number of compounding periods, and any payments made or received.

## Performing Your Own Time Value of Money Calculations

Once you understand TVM concepts, you can perform your own TVM calculations. The key is to be able to properly identify the variables to be plugged into the TVM equation.

The easiest way to do the calculations, after a small investment of time to learn how, is to use Excel, a financial calculator, or dedicated online TVM calculators like the ones at CalculatorSoup

Find a method that works for you and practice until you are comfortable with these calculations. Then utilize the time value of money concept to help make better financial decisions.

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[Chris Mamula used principles of traditional retirement planning, combined with creative lifestyle design, to retire from a career as a physical therapist at age 41. After poor experiences with the financial industry early in his professional life, he educated himself on investing and tax planning. After achieving financial independence, Chris began writing about wealth building, DIY investing, financial planning, early retirement, and lifestyle design at Can I Retire Yet? He is also the primary author of the book Choose FI: Your Blueprint to Financial Independence. Chris also does financial planning with individuals and couples at Abundo Wealth, a low-cost, advice-only financial planning firm with the mission of making quality financial advice available to populations for whom it was previously inaccessible. Chris has been featured on MarketWatch, Morningstar, U.S. News & World Report, and Business Insider. He has spoken at events including the Bogleheads and the American Institute of Certified Public Accountants annual conferences. Blog inquiries can be sent to chris@caniretireyet.com. Financial planning inquiries can be sent to chris@abundowealth.com]

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1. Phillip says:

These same principals can be used to calculate the value of pensions and social security if you want to. IMO, the PV of these annunities (discounted … I ballpark around 30% discount from what insurnace companies sell) can be legitimately added to you NW if you’re into those calcs. And not to get policitcal, but i’ve also used these type methods to estimate the pension value of govenment employees to see (in my own mind) how much these civil servants are really getting paid.

2. Sand Dollar Investor says:

Hello Chris – Long time reader here and fantastic article again. Regarding time value of money and rates, I had invested in a GNMA fund for income, but as you know those funds along with other bonds have crashed. I was thinking on dollar cost averaging the GNMA (buying at the lower prices now) so my cost basis goes down. Hopefully, once GNMA prices go up, my balance will go up faster than if I just wait it out. What are your thoughts on DCA for bond funds, GNMA etc.? Thank you.

1. Chris Mamula says:

SDI,

If you’re still in accumulation mode, DCA makes sense as you’ll be gradually be buying more income with each dollar as rates go up and also as you reinvest bonds that mature.

I don’t know a ton about GNMAs specifically, as I don’t invest in them other that what is in a total bond fund. They share a common risk with most other marketable bonds where if rates go up the bond value decreases if you need to sell, but they also have some risk if rates go down, people refinance, and you have to reinvest at a new lower ROR.

Best,
Chris