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Foreign hi, I'm Andy Tempte, and welcome to the Saturday Morning Muse. Start your weekend with musings that are designed to improve financial literacy around the world. Today is August 23, 2025. Last week we explored inflation and how it erodes purchasing power over time. We also connected this concept back to our August 2nd episod where we introduced the interest rate equation, which is expected inflation plus the real risk free return plus the risk premium that all equals an interest rate. Today, we're going to dive deeper into one of the components of this formula, the concept of return. And we're going to discover why understanding different types of returns is crucial for making smart financial decisions. Decisions in that August 2nd discussion, we defined the real return as compensation beyond just maintaining purchasing power, essentially what you earn after accounting for inflation. But the world of investment returns, as we will learn, is far more nuanced than this single definition suggests. I want you to think of returns like temperature measurements. Just as we have Fahrenheit, Celsius and Kelvin to measure heat under different circumstances, we have various ways to measure investment performance, depending on what question we're trying to answer. Each return calculation that we will learn about serves a specific purpose and tells us something different about an investment's performance. So let's talk about some simple return computations. And let's start with the nominal return. This is the most straightforward calculation. It is simply what your investment gained or lost without adjusting for inflation. For example, if you bought a stock for $100 and sold it for $110, you your nominal return would be 10%. So then here's a Math alert. Math alert. The nominal return is the ending value of an investment minus its beginning value, all divided by its beginning value. Now, this computation is going to give you a decimal representation of return, so that 10% was going to be expressed as 0.1. If you want to convert any decimal return into a percentage, Simply multiply by 100. Now, as we learned from our inflation discussion, nominal returns can be misleading. If inflation ran at 4% during this investment period, your real return, your purchasing power gain, was only about 6%. Remember from that August 2nd episode that real return represents a combination of two the real risk free rate plus a risk premium that you earned. Now, over longer periods of time, even modest inflation rates compound and significantly reduce the purchasing power of what appears to be a healthy nominal return. The second type of return that we want to talk about today is holding period return, and sometimes you'll hear this phrased as hpr. This measures your total gain or loss over the period that you owned or held an investment. If you bought that same stock for $100 and you might have received $5 in dividends, sold it for 110, your holding period return percentage terms would be 15%, which is 110 minus 100, plus the $5 in dividends, all divided by 100. Now, again, this is going to give you a decimal representation of 0.15. So to convert that to 15%, simply multiply 0.15 by 100. Now, another term that you might hear thrown around is total return. This is very similar to holding period return, but is typically expressed as an annual figure that includes all sources of income. You've got your capital gains, which is the profit from selling an investment for more than you paid. You have dividends, which are cash payments that companies make to shareholders. Or you might have interest in the equation, which are payments from bonds or savings accounts. In most cases, total return and holding period return will be exactly the same. They will be interchangeable. Now, we also need to compare investments over different time periods. And for this we want to use the compound annual growth rate or, or the cagr, commonly known as cagr. This return measure smooths out year over year volatility to show what constant annual return would have produced the same end result. If an investment doubled over seven years, the CAGR or the compound annual growth rate would be approximately 10.41%. Now, how do we compute this? Well, if we took $1 and then that dollar ended at $2 over the seven year investment period, you would take $2 minus one, divide that by the dollar that you started with, and then raise that whole thing to the 1 over 7 power, and then subtract off a 1. Now that sounds really, really complicated, but compound annual growth rate is computed simply as ending value minus beginning value. That whole thing raised to the 1 over N power, where n is the number of years in the investment period, and then we subtract off a one. We are going to be talking about compounding a lot in future episodes. So this introduction to compounding, while it might sound complicated, is going to really pay off for us in future conversations. Now, the mathematical foundations for return calculations. Well, those things, they trace way back to where you might expect. If you've been listening to our other episodes, Babylonian and ancient civil Egyptian civilizations where merchants needed to calculate profits from trading expeditions and our modern understanding of investment returns, they developed alongside some of the same evolutions in financial markets that we've been talking about. Individuals of note that have helped develop return calculations. Well, they are famous names like Leonardo Fibonacci, who's famous for the Fibonacci number sequence. He contributed to early compound interest calculations. In his 1202 work called Liber Abaci, he laid the groundwork for understanding how money grows over time. And then you have the Dutch mathematician Johan de Witt, who served as the grand pensionary of Holland in the 1600s. We know that's also known as the 17th century. Well, that individual made crucial contributions to what we now call present value calculations. And we're going to talk about present value and future value a lot on the show. Also, as you might expect, Benjamin Franklin popularized compound growth concepts in American culture with his famous observation that money makes money and the money that money makes makes money. His mathematical demonstrations of compound interest help colonial Americans understand how small, consistent investments over time could grow substantially over time. Benjamin Franklin here is going to help us a lot with compound interest. Now, why does this matter to you? When evaluating the performance of your portfolio, you need to look at real returns, not just those nominal returns. Because an 8% annual return looks a lot less impressive when inflation has been running at 3%, leaving you only with 5% inflation net purchasing power gains. Also, the marketing materials from financial firms often highlight their best looking return numbers. An actively managed fund might advertise impressive short term holding period returns while glossing over poor risk adjusted performance or high fees that erode long term real returns. Next, when comparing investment options, the compound annual growth rate helps you see through the noise of year over year volatility. Making apples to apples comparisons in investments is absolutely critical. A consistent 7% annual rate of return often beats a flashy investment that might gain 20% in one year but lose 10% the next. And finally, tax implications also matter. Now, we didn't include taxes in the calculations that we talked about previously, but your after tax return is what really counts for building wealth. An investment generating 10% in a taxable account might net you less than an investment yielding 8% in a tax advantaged account. So the key insight for now is this, not all returns are created equal and apples to apples comparisons are what you really want to look look at. The number that matters most depends on your specific situation and goals. Understanding return calculations and distinctions empowers you to ask better questions, make more informed decisions, and avoid being misled by flashy marketing materials that emphasize the most favorable return calculations while ignoring others. So until next week, I wish you grace, dignity and compassion. My name is Andy Tempte. This is the Saturday morning Muse. You can find the show on all the major streaming services as well as out on YouTube. Please like subscribe, rate and most importantly, share this public good with your friends, your family, your neighbors, and maybe even a colleague at work. The show is produced by Nicholas Dempte. We'll see you next time on the Saturday Morning Museum.