The Investor's Dilemma Decoded E-Book

Roger D. Silk

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Veröffentlichungsjahr: 2024

Beschreibung

Few aspects of life are as important as personal finance, as subject to your control, and as suffused with misinformation, noise, and confusion. Now, authors Dr. Roger D. Silk and Katherine A. Silk cut through that confusion and share with you the fruits of their knowledge and experience developed over the last 43 years. After completing a Ph.D. at Stanford where he studied at the cutting edge of finance theory, Dr. Silk's experience includes managing billions of dollars at the World Bank and running a family office for one of the nation's wealthiest families. For the last 26 years as CEO of the nation's leading firm which advises high net worth individuals on financial and other aspects of their philanthropy, Dr. Silk has worked with countless individual investors and financial professionals. Katherine Silk, who holds a master's in history from Stanford, adds a valuable and often-missing historical perspective. Their weekly blog, dealing in depth with a variety of financial, economic, and planning issues, is read by thousands. Unlike many authors in the Personal Finance space, the Silks have the deep technical expertise (it's hard to get a graduate degree from Stanford without it), decades of experience, and the rare ability to express complex ideas in clear, easy-to-understand prose. When Gary Taubes wrote The Case for Keto, he considered calling it "How to Think About How to Eat." Similarly, The Investor's Dilemma Decoded could be titled "How to Think about How to Invest." Investor's Dilemma gives you the tools that 99.9% of investors never master -- these tools allow you to understand how to think about almost any category of investment, and almost any investment product or program. In addition, the authors take a deep dive into topics including * What actually generates investment returns (it's probably not what you think) * Is owning a home an investment (you'll learn why the answer is sometimes yes, and sometimes no) * Should you own gold (clue: the largest gold holders in the world are central banks) * What is a hedge, and are commodity funds an inflation hedge * What many well-known investment personalities get wrong on about returns (they tell the truth, but it's the wrong truth) * What risk is, and isn't, and why the "safe" course might be the riskiest (but the government says it's safe). * How professional financial advisors can add huge value to their individual clients (it's not by picking the best stocks) Should you read this book? If you want to understand how professionals think about investing, about what is realistic and unrealistic, and learn to spot the difference between a Bull Market and Bull-xxxx, the answer is yes.

Leseprobe

Additional Praise for The Investor's Dilemma Decoded

“This excellent book is a must-read for investors interested in acquiring a better understanding of the theory and practice of finance. It is exceptionally well-written. Complex ideas are explained thoroughly but in a simple and easy-to-understand and engaging fashion. Investors will find the numerous practical examples and advice invaluable in managing their portfolios.”

—Professor Robert I. Webb, University of Virginia

“The Silks' book, The Investor's Dilemma Decoded, should be required reading for every American citizen, including college students of any background, who wants a better understanding of investments and the real-world economy. As a dually licensed attorney and CPA, the information contained in this book will help me to better communicate with my clients on the legal, tax, and business consequences associated with investments-related activities and issues which they confront on a regular basis. The book is also a good refresher for those of us who need to go back to basics to understand and address new problems and innovations on behalf of our clients.”

—Clinton McGrath

“Roger has a gift of bringing clarity to virtually every item in the investor lexicon and makes it easy for the non-math layperson to understand while technical enough to challenge the most analytical professional investor's thinking. Armed with this book, any investor will be able to better able to understand, analyze and ask insightful questions about their portfolio and challenge their advisors. It is really a masterclass covering all of the important concepts and tools for analyzing asset classes, risk, returns and valuations that underly portfolio allocation and is a must read for serious investors, and a serious reference book that will still be relevant in 10 years.”

—Reg Wilson, Pres EPIC Financial Consulting, Inc.

“I find the Intelligent Layman's Guide to Personal Investing to be very readable and surprisingly comprehensive given its length. I recommend it for a novice and for anyone looking to broaden their existing investment knowledge. The book discusses a number of different types of investment for someone who wants to shape their own investment strategy or who simply wants to understand what a given fund is trying to accomplish with a given investment strategy.”

—Tom Arnold, PhD., CFA, CIPM Joseph A. Jennings Chair in Business, Robins School of Business, University of Richmond

“Roger and Katherine have taken a subject that can require years to master, and they have condensed the key areas into a single book that goes beyond the basics. With the information in this book, most people will be in a great position to work with their wealth manager much more effectively. For those without an advisor, this information will help them to make much better decisions.”

—Greg Freeman, Advisor Serve

“With his latest publication of “The Investor's Dilemma Decoded” Roger Silk once again provides the reader with a timely, helpful & informative roadmap for individual investors across the wealth spectrum. Silk's detailed review and condensed reminder of investment fundamentals pairs well along with recently updated research on what specific value is attributed to receiving professional advice from reputable financial planners in today's often confusing marketplace. Whether you are already an informed investor or wish to become one, perhaps this book's most impactful and practical value is to help today's investor ‘avoid making avoidable mistakes!”

—Michael S. Millman, CFP®, ChFC®, CLU, AEP®, CASL, RICP, ChSNC

Independent Private Wealth Advisor

“Well-rounded and well-written book not only for the layman but a refresher for professional advisor alike. Asset classes, expected returns and risk, portfolio construction and even math! Yes, software is great but understanding the underlying math is essential and made accessible in The Investor's Dilemma Decoded. I found a gift for my team of CFPs.”

—Kevin Kroskey, CFP®, MBA, Managing Partner, True Wealth Design

The Investor’s Dilemma Decoded

Roger D. Silk, PhD

Katherine Silk, MA

Copyright © 2024 by John Wiley & Sons, Inc. All rights reserved.

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Synopsis of The Investor's Dilemma Decoded

The Investor's Dilemma Decoded provides readers with the background to make informed, effective decisions about their personal investing. The book explains, in a correct, thoroughly documented manner, the relevant economic and financial theory, as well as the historical performance of a wide variety of asset classes that an individual investor might consider.

The book offers some surprising insights, such as the idea that investing in 100% equities may well be the least risky option for young investors; that financial planners have been found in many studies to add significant value without “beating the market” and without exposing the client to excessive market risk; that a home is not a great investment; why Nobel Prize–winner Harry Markowitz did not use his own Nobel formula in his personal investing; and the truth about many supposed yields on futures contracts.

Chapter 1 explains the Time Value of Money. Everything else equal, a dollar today is worth more than a dollar tomorrow. Readers will learn how to compare the value of a dollar today to the value of a dollar in the future.

Chapter 2 models investments as a series of cash flows. The value of an investment is the present value of the sum of its future cash flows. Readers will learn how to value bonds using the present value of cash flows framework. We explain the difference between arithmetic and geometric returns, and the significance of the difference.

Chapter 3 focuses on bonds in more depth. Bonds generate returns primarily through their coupons. We review the risks to bonds, including credit risk, interest rate risk, inflation risk, and default risk. After explaining how bonds produce return, and examining how bonds have performed in the past, we show that a bond's current yield to maturity is the best predictor of its future return.

Chapter 4 focuses on equities. Many stock investors have no real understanding of how stocks generate returns. We explain that over the very long run, earnings are by far the dominant source of equities' returns. We review several useful measures for evaluating whether a market is cheap, expensive, or neither. Focusing on the Price-to-Earnings (P/E) ratio and Price-to-Book (P/B) ratio, we explain why, everything else being equal, expected returns are higher when a market has a lower valuation. Risks to equities include the risks of a decrease in valuation multiple or a decrease in earnings.

Chapter 5 examines real estate, the biggest asset class in the world (as measured by market value). We model the value of farmland as the value of a perpetual stream of cash flows. Farmland runs the risk of decreasing in value if its cash flow decreases or if interest rates rise. Homes, contrary to many perceptions, have not generally been a great investment. Over the past 120 years, the real compound annual growth rate of home prices is only about half a percent. The expected returns to apartment buildings are the net rents plus inflation. Overall, returns to income real estate are likely to equal the current yield plus some adjustment for inflation.

Chapter 6 explains that gold is not an investment, but is likely to hold its purchasing power over long periods of time. Over short periods, the price of gold is volatile. The returns to gold have been uncorrelated with the returns to equities, meaning that gold can play a diversifying role in portfolios.

Chapter 7 corrects some common misconceptions regarding futures contracts (sometimes referred to as “commodities”). We explain the arbitrage model of the pricing of futures contracts, and argue that the expected real return to commodity indices is negative. Furthermore, so-called “spot yields,” “roll yields,” and “collateral yields” are not true yields, and may not be positive.

Chapter 8 examines mutual funds, a vehicle used to invest in assets. We review the advantages of mutual funds, such as the fact that they allow for diversification and may offer reduced transactions costs, especially for bonds. We discuss the difference between open-ended funds, exchange-traded funds, hedge funds, and sector funds.

Chapter 9 discusses financial theories including the Markowitz Model, Capital Asset Pricing Model, Random Walk Theory, and Efficient Market Hypothesis. We examine the trade-off between risk and return in the Capital Asset Pricing Model and discuss that stock prices often follow (or seem to follow) a “random walk with drift” from day to day. We explain that while the Efficient Market Hypothesis is true in many cases, value (i.e. low market valuations) has historically outperformed the broad market over a wide range of markets and long time spans.

Chapter 10 explains the concept of financial leverage. Leveraging a portfolio can lead to greater returns and also carries greater risk. We argue that for the individual investor, investing in the already leveraged S&P 500 index or similar indices carries sufficient leverage.

Chapter 11 reevaluates the ideas of risk and uncertainty. We define risk as calculable statistical probabilities, whereas uncertainty refers to unknown possibilities. We review statistical approaches to calculating risk through traditional means, including standard deviation and variance. We revisit the importance of understanding the difference between the geometric and arithmetic mean when considering expected returns.

Chapter 12 evaluates portfolio allocation strategies, examining expected returns for equities, bonds, cash, gold, futures, and managed futures.

Chapter 13 reframes the idea of risk: Rather than focus on the risk of losing money, many investors may find that the most important risk they face is the risk of running out of money. We present 11 portfolio simulations in which investments are allocated between equities and cash. We find that for young investors, investing in 100% equities yields the lowest risk of running out of money, while the “riskless” alternative of 100% in cash (or equivalents) virtually guarantees disaster.

Chapter 14 explains the value of financial advisors. We report the results of a number of academic studies that find financial advisors add value in a number of ways, most of which don't involve “beating the market.” Advisors help clients get appropriately invested, stay invested in the market during downturns, provide asset allocation strategies, help clients take advantage of opportunities to save taxes, and help clients rebalance their portfolios. Advisors can vastly increase a client's ability to earn the returns that are available to the informed investor who avoids avoidable mistakes. People with greater financial literacy tend to benefit more from professional advice.

Chapter 15 discusses the practical side of investing using the concepts we have introduced.

The appendix explains some of the math used in the book. You should feel absolutely free to skip it. If you took a statistics course, or if you have read finance texts, and not fully understood some of the concepts, this appendix will, we hope, help. We try to clarify some of the concepts that are often seen (or hidden) in investment contexts. In our experience, in classes and in books, the professor or author often maddeningly skips steps, so that it is a great struggle for people who don't already understand to follow. If we've done that here, please email Roger at [email protected] and let us know. We'll try to fix it in the next edition.

Acknowledgments

We would like to thank everyone who has contributed to the creation of this book. Writing a book is a collaborative effort, and we are fortunate to have received support from many people. Unfortunately, space doesn't allow us to name everyone who has, in one way or another, contributed to this book. Among those we can thank are the following.

We wish to thank particularly Professor Tom Arnold, of the Robins School of Business, and Jade Lintott, PhD candidate at Georgie Institute of Technology, for generously reviewing the math in this book. Of course, any errors that remain (which, please point out to us if you find them – we'll give you gift card as a thank-you) remain solely our responsibility.

In addition, we thank Shannon Swery of Sterling Foundation Management for her careful review of the final manuscript. Again, any errors that remain are our responsibility, and if you spot them and point them out to us, we'll send you a gift card as a thank-you.

For Roger, some of the ideas and understandings presented in this book represent the evolution of ideas and concepts I learned decades ago from teachers. Among them are many of my professors at Stanford, including Anne Peck, Roger Gray, Jeffrey Williams (now at UC Davis), Walter P. Falcon, V. “Seenu” Srinivasan, Bruce Grundy, Scott Pearson, Harry Paarsh, and Carl Gotsch.

In addition, we thank all the readers who read our book and gave us comments that helped us improve the book. Tyler Cowen's feedback and comments were very valuable, and we are grateful for his insights. We also thank Kevin Kroskey, Clinton McGrath, Jimmy Jacobs, Tom Myers, Bruce Popper, Reg Wilson, Michael Millman, Robert Webb, and Tom Arnold for their review of the draft.

Again, we thank everyone who contributed to the writing and publication of this book.

Introduction

Successful investing is hard, even for professionals. As Warren Buffet's partner, Charlie Munger (himself a billionaire investor) told Howard Marks (another billionaire investor), investing is “not supposed to be easy. Anyone who finds it is easy is stupid.”1 Nevertheless, according to Buffet, “You don't need to be a rocket scientist,”2 either.

The purpose of this book is to help you learn how to think about investing. Then, if you choose not to do all your investing yourself (few non-professionals do, or should), you will be able to be an informed, careful consumer of professional investment services. And evidence shows (see chapter 14) that the more you know, the more value you are likely to gain from professional advice.

We provide a framework for thinking about investing in a consistent, careful way. If you develop a thoughtful approach to your personal investing, and implement it consistently, you'll be ahead of most people, and well on your way to investment success.

This is not a get-rich-quick book. But if you want a get-rich-quick scheme, you need look no farther than Will Rogers's advice from almost a hundred years ago. “Take all your savings and buy some good stock, and hold it till it goes up, then sell it. If it don't go up, don't buy it.”3 Most get-rich-quick schemes are scams, and we won't have any more to say about them.

Instead, we will explain the basics, and build up to a systematic approach to personal investing that, while not guaranteed, should give you about the best odds that are available.

We begin with the “time value of money,” and explain how that concept is used in investment analysis. We then devote chapters to bonds, stocks, real estate, gold, and commodity futures, explaining how each of these types of assets generates returns (or doesn't), and we look at long-run historical performances.

In short, we try to explain in theoretical terms how each asset produces returns, and in historical terms what those returns have been. These discussions are intended to help you understand what future returns might reasonably be. We also discuss historical volatility of returns.

We then turn to a discussion of funds as useful vehicles through which to invest, followed by four chapters that focus on portfolio theory. Portfolio theory is mainly about how to get as much return as possible for each “unit” of risk that you accept. We discuss risk in some depth, and also leverage, which can affect risk greatly.

The next two chapters discuss how to assemble a portfolio, and what you might reasonably expect from a variety of different portfolios.

We devote a chapter to the role of professional financial advisors, and conclude that for many people, the right professional financial advisor offers a sound value proposition.

We have tried to present the material as clearly as possible, without dumbing it down. There are zillions of dumbed-down books out there promising that it's easy. If it were that easy, you would probably already know it, and wouldn't have picked up this book.

However, if you're willing to put in a bit of effort to learn the basics, we believe that the concepts and tools provided here will give you an excellent shot at building real, life-changing wealth over time.

You may notice that there are footnotes in this book. If you don't like footnotes, please feel free to ignore them. They are there to offer additional detail that didn't quite fit into the text and to show the source of quotations or other claims.

There is some math in this book. You don't necessarily need to know the math, but we've found that there are too many people out there giving advice, including some recognized “experts” whose failure to understand the math leads them into errors that could be very costly to people who rely on their advice. So we've included the math for the people who want it.

Even if you don't understand everything, you'll still be able to gain valuable ways of thinking about different asset classes, about risk, and about where returns come from and what might be expected in the future.

We've included a mathematical appendix for those who want a refresher on the math.

Notes

1

   Oaktree Capital Management, 2021, “It's Not Supposed to Be Easy: How Oaktree Strives to Add Value in Uncertain Times,”

https://www.oaktreecapital.com/insights/oaktree-insights/special-editions/it-s-not-supposed-to-be-easy-how-oaktree-strives-to-add-value-in-uncertain-times

2

   Carol J. Loomis, “The Wit and Wisdom of Warren Buffett,”

Fortune

(November 19, 2012),

https://fortune.com/2012/11/19/the-wit-and-wisdom-of-warren-buffett/

3

   Will Rogers, “Thoughts of Will Rogers on the Late Slump in Stocks,”

New York Times

(November 1, 1929),

https://www.nytimes.com/1929/11/01/archives/thoughts-of-will-rogers-on-the-late-slump-in-stocks.html

Chapter 1Time Value of Money

Which is worth more: one dollar today, or one dollar tomorrow?

The Standard Theory

The standard theory of the time value of money states that a dollar in your hand today is worth more than the same dollar in your hand in the future. This is true even if there is no inflation.

Here's one example. Would you rather have $100 today, or $100,000,000 in a million years? Even if you were absolutely certain that the $100,000,000 would be “yours” in a million years, it would be worth nothing to you, because you would not expect to be around to enjoy it. Further, even if your great, great, great, etc., grandchildren could be assured of receiving it, it would probably mean little to you.

But what about more realistic waiting periods? For example, suppose you have the choice between $100 now, and $100 in a year from now. The standard theory says that the rational person will always choose the $100 now, because you could do everything with $100 now that you could do with it in a year from now, and you could invest it to earn interest between now and then.

This standard theory is right, but it has a few important assumptions buried in it. The most important of these assumptions is that you can save money at no cost and at no risk between now and the future. In thinking about the standard theory, or any economic theory, it is important to remember the assumption of “all other things equal” (from the Latin, ceteris paribus).

Assumption: You Can Save Money at No Cost

The assumption that you can save money at no cost means that if you have a certain amount of money today, you can save that money for any period at no cost. No cost means that you don't have to pay any storage charges, or insurance charges, or handling charges, or taxes, or charges of any kind.

In the last hundred years or so, citizens in developed economies have gotten used to the idea that they can save money, in a bank for example, and not have to pay for the service. We will not be surprised if financial historians of the future look back and see this as a sort of financial magical thinking, which has indirectly had catastrophic costs by making the entire banking system unstable.

In general, if you want to store any valuable commodity, you would have to do it yourself – perhaps in your home, in a rented vault, or perhaps even burying it in the ground. You would not expect that you can store it securely for free.

In principle, money is no different. But the way banking has developed over the past several hundred years, bankers, with the willing cooperation of their depositors, have purported to “save” money for free for their depositors. Of course, nothing is free. The bankers have been willing to offer this service for free so that they could get their hands on the money. Do they “save” it for you? Absolutely not. They lend it out. This lending puts your money at risk, and generates income for the bank, provided that the bank doesn't suffer too many loan losses. If the bank suffers too many loan losses, they may not even be able to pay you back the money that you deposited.1

Despite the inherent instability of this practice, it has become so entrenched in the modern banking system that it seems like a permanent feature. We don't think history will prove it so, but for the time being (and perhaps for the foreseeable future), it is.

Assumption: You Can Save Money at No Risk

The assumption that you can save money at no risk of not getting it back (in other words, that you can save money with the guarantee that you'll receive 100% of it back) is easy to understand in principle. It is a bit surprising that so many people believe it to be literally true. There is no truly riskless proposition in the material world. It may be the case that some things carry extremely low risk, but in our world, that risk is never actually zero.

Nevertheless, most people in the developed world behave as though they can put money in the bank, and have no risk of not getting it back in the future.

As long as you can put money in the bank and expect to receive it back in the future, and the bank doesn't charge you for the service, a dollar now is worth more than a dollar in the future, because you can turn it into a dollar in the future by storing it. Anything you can do with a dollar in a year, you could do with a dollar now because you could just wait. But you can do things with the dollar now, such as spend it now, that you cannot do with a dollar in a year. So the dollar now is worth at least as much as the dollar in the future.2

Time Value of Money and Compound Growth

Economists, financial professionals, and talking heads often conflate the time value of money with the phenomenon of compound growth. You don't need to worry about the finer points of theory as discussed earlier, but you do need to understand the math of exponential or compound growth.

Perhaps the easiest way to understand compound growth is to think in terms of “interest on interest.” Suppose you have $100 and you can earn 10% per year. (For the purpose of illustration, we assume that the 10% earnings are risk-free, but remember the real world is never risk-free.) Table 1.1 shows how your money would grow if you get paid interest at the end of every year, and reinvest the interest.

Tabelle 1.1 Compound Growth Assuming Interest Payments Are Reinvested

Year

Beginning Principal

Interest

Ending Principal

 1

100.00

10.00

110.00

 2

110.00

11.00

121.00

 3

121.00

12.10

133.10

 4

133.10

13.31

146.41

 5

146.41

14.64

161.05

 6

161.05

16.11

177.16

 7

177.16

17.72

194.87

 8

194.87

19.49

214.36

 9

214.36

21.44

235.79

10

235.79

23.58

259.37

11

259.37

25.94

285.31

12

285.31

28.53

313.84

13

313.84

31.38

345.23

14

345.23

34.52

379.75

15

379.75

37.97

417.72

16

417.72

41.77

459.50

17

459.50

45.95

505.45

18

505.45

50.54

555.99

19

555.99

55.60

611.59

20

611.59

61.16

672.75

In the first year, you earn $10 of interest, which is 10% of your $100. But in the second year, you earn $11, because you earned your 10% on your original $100, but you also earned 10% on your $10 of interest. That “interest on interest” earned you $1 in year two. You earned a total of $11, which could be thought of as $10 on your original $100, plus $1 on the interest you earned the first year.

Each year, you still earn the $10 on your original $100, but the “interest on interest” gets bigger every year. By the ninth year, you are earning more “interest on interest” than on your original $100!

This compound growth is the phenomenon that people get so excited about. Almost everyone who has ever amassed significant wealth legitimately (i.e. other than by stealing it) has done it, at least in part, by putting compound interest, or exponential growth (another term for the same phenomenon), to work. Just in case you're not convinced, here is a bit of argumentum ad verecundiam (just a fancy sounding way of saying argument from authority).

“Compounding is the magic of investing.”

—Jim Rogers

“The effects of compounding even moderate returns over many years are compelling, if not downright mind boggling.”

—Seth Klarman

“Understanding both the power of compound interest and the difficulty of getting it is the heart and soul of understanding a lot of things.”

—Charlie Munger

“The sooner you start, the more compounding can do for you. If, beginning at the age of twenty, you sock away just $100 a month in stocks, and your portfolio compounds at 10%, which is what stocks have provided historically, you will be a millionaire when you retire at sixty-five.”

—Ralph Wanger3

Comparing Values Across Time

The ability to compare values across time, a process called present value analysis, is useful to understand the process of building wealth.

Suppose I told you that a gallon of regular gas is $3.50 at the Exxon station and $3.65 at the Chevron station down the block. Assuming both stations are equally convenient, safe, busy, etc., you would have no difficulty telling me that the Exxon station was a better deal.

But what if I told you I was going to buy an item you're selling, and I'll pay you either $65 for it now, or $68 in a year. Now how easy it is for you to say which is a better deal? As we explained in the previous section, you know that a dollar today is worth more than a dollar in a year. But is $65 today worth more than $68 in a year?

For most people, thinking about money across time does not come naturally. You have to work at it, just as you do for most skills. It's not hard, but you do have to practice.

It's great to have the tools of present value in your toolbox, but it's even better if you know how to use them. You learn by practice. You might keep your eye out for retail offers such as per month pricing versus subscription discounts for longer terms, for investment advertisements, and for claims made by politicians, and apply the tools of present value analysis where you can.

Real-World Compounding Versus “Pure” Theory of Time Value of Money

In the basic models discussed in this chapter, we always assumed a known and constant rate of return. However, in the real world, there is always risk, and rates of return are rarely, if ever, stable over time. Even so, these models can give you a rough idea of what something is worth now compared to some time in the future. As long as you don't believe that your models represent reality, you will be well served by being as comfortable as possible with making present value calculations, future value calculations, and in general using the concepts of compounding in a wide variety of situations.

In order to compare the value of $65 today to $68 in a year, you may use the following formula:

(1)

In this formula, FV represents the future value of your investment in the future, while PV represents the present value, or value today, of your investment. The length of time, in years, is represented by t while the rate of return is represented by r. (This formula is explained in the Appendix to this chapter.)

For the purposes of the above example, $65 is the present value, $68 is the future value, and t is one year.

Whether $65 now is worth more than $68 in a year, depends on your own personal discount rate. A rate of 4.6% per year makes $65 now equal to $68 in a year. There are several ways to think about this, but the most useful one for investing is that if the market rate of interest (e.g. the rate on one-year loans of the same risk category as the loan in question) is 4.6%, then the market value of $65 now is the same as the market value of $68 in a year.

Appendix

The Math of Compounding

In the example in this chapter, interest was compounded each year. It is possible to compound more often, for example monthly, weekly, or daily. It turns out that you can even compound continuously and still get a meaningful answer. If you can use a calculator, you can make all the time value of money calculations you will need. It is often easier, however, to use a computer spreadsheet. You can also do the calculations using natural logs and the exponential function, but in my experience the need or usefulness of those in the real world has been limited.

There is one equation that you really should know your way around backwards and forwards, because that equation is at the heart of the miracle of compounding. Here it is:

(1)

Where FV is the Future Value (the value you'll have in the future), PV is the Present Value (the value you have now), r is the growth rate (or interest rate) per period, and t is the number of periods. Often, r is expressed as an annual interest rate, and t is measured in years. So, for example, if the PV is $100, r is 10%, and t is 7, the future value will be $194.87.

Here is the calculation:

(2)

You might have noticed that you could have looked this up in Table 1.1, because that table used $100 as the beginning value, 10% annual interest, and compounded annually.

More Frequent Compounding

What if we used the same 10% annual interest rate, but compounded twice a year? The formula is equation (1), but now r is 5% because interest is paid twice, so we divide 10% by 2 and t is 14, because there are now twice as many periods. So the calculation is:

(3)

We can generalize the compounding formula to

(4)

where r is the annual interest rate, t is the number of years, and n is the number of times we're compounding during the year.

As we make n larger and larger, we are compounding over shorter and shorter periods of time. We can use calculus to show that as n goes to infinity, the limit of the function becomes:

(5)

where e is the base of the natural log function, approximately 2.71828.

You don't have to understand where equation (4) or (5) comes from to use it. For most purposes, either one is fine.

A Little Algebra

Equations (1), (2), and (3) are good if we know the value now, the interest rate, and the number of years. But what if we know how much we're going to get in the future, when we're going to get it, and the interest rate, and we want to calculate what that is worth today? We want the present value, and we can get it by just rearranging one of our equations. Let's take equation (1) and solve it for PV. We get:

(6)

Rearranging equation (5) gives us

(7)

Given any three of PV, FV, r and t, it is possible to solve for the fourth one. Above we gave equations solving for FV and PV. As an exercise, you can try solving for r, and then for t. Following are the answers.

(8)

For example, 55 years ago a man bought a building for $200,000. Now the building is worth $10 million. His annual rate of return is:

(9)

The man earned a 7.37% compound annual rate of return.

Fortunately, calculators and computers make it simple to perform these calculations.

Now we will solve for t. The equation is:

(10)

where ln is the natural logarithm function.

Again, it is easy to perform these calculations with a calculator or computer. For example, suppose you put $1,000 into an investment that you expect will earn 5% each year indefinitely. How long would it take to grow to $10,000?

(11)

Exercises

Here are 40 sample problems, consisting of 10 each where you are to solve for PV, FV, r, and t. The answers are on the following page.

Tabelle 1.2 Chapter 1 Exercises

Exercise #

FV

r

t

PV

 1

795

14

76.51

 2

734

0.068

210.30

 3

 17

0.094

7.57

 4

246

0.122

 7

 5

0.018

14

303.03

 6

752

19

59.34

 7

0.023

15

396.02

 8

875

 2

685.25

 9

0.168

15

6.91

10

972

0.09

206.06

11

723

0.147

19

12

 56

0.055

47.69

13

542

0.175

10

14

0.082

 4

227.64

15

412

0.05

338.95

16

669

0.086

15

17

0.076

19

98.71

18

325

0.178

16

19

438

0.009

372.76

20

417

 9

171.16

21

848

0.049

20

22

490

18

282.84

23

704

18

329.96

24

0.012

 5

77.25

25

0.159

 0

321.00

26

505

0.084

18

27

592

0.042

319.38

28

754

13

95.79

29

422

20

25.78

30

933

0.04

498.14

31

667

0.127

19

32

 91

10

16.95

33

166

0.179

 2

34

875

0.046

426.09

35

293

0.017

278.55

36

0.033

13

9.84

37

0.167

 1

419.88

38

530

12

61.26

39

391

0.07

 5

40

0.117

 9

237.17

Answers

Tabelle 1.2 Chapter 1 Answers

Exercise #

FV

r

t

PV

 1

795

0.182

14

76.5107

 2

734

0.068

19

210.3014

 3

17

0.094

9

7.5734

 4

246

0.122

7

109.8968

 5

389

0.018

14

303.0266

 6

752

0.143

19

59.3389

 7

557

0.023

15

396.0233

 8

875

0.13

2

685.2533

 9

71

0.168

15

6.9122

10

972

0.09

18

206.0579

11

723

0.147

19

53.3867

12

56

0.055

3

47.6904

13

542

0.175

10

108.0490

14

312

0.082

4

227.6384

15

412

0.05

4

338.9534

16

669

0.086

15

194.0791

17

397

0.076

19

98.7091

18

325

0.178

16

23.6352

19

438

0.009

18

372.7633

20

417

0.104

9

171.1648

21

848

0.049

20

325.7512

22

490

0.031

18

282.8395

23

704

0.043

18

329.9553

24

82

0.012

5

77.2523

25

321

0.159

0

321.0000

26

505

0.084

18

118.2400

27

592

0.042

15

319.3787

28

754

0.172

13

95.7883

29

422

0.15

20

25.7843

30

933

0.04

16

498.1363

31

667

0.127

19

68.7973

32

91

0.183

10

16.9509

33

166

0.179

2

119.4209

34

875

0.046

16

426.0898

35

293

0.017

3

278.5510

36

15

0.033

13

9.8353

37

490

0.167

1

419.8800

38

530

0.197

12

61.2556

39

391

0.07

5

278.7776

40

642

0.117

9

237.1682

Notes

1

   In the current system, deposits may be “insured” by government guarantees. These are not true insurance, because it is not optional, it is not truly risk-based, and it is ultimately backed by taxpayer funds. Furthermore, it is very likely the case that deposit insurance results in bankers taking greater risks with depositors' funds than they would without it, and thereby make the entire system less stable. The financial crisis of 2008 was a case in point, and ended up costing taxpayers at least half a trillion dollars. (See Deborah Lucas, 2019, “Measuring the Costs of Bailouts,”

Annual Review of Financial Economics

11: 85–100.) For a discussion of how deposit insurance causes instability, see Asli Demirgüc-Kunt and Enrica Detragiache, 2002, “Does Deposit Insurance Increase Banking System Stability? An Empirical Investigation,”

Journal of Monetary Economics

49: 1373–1406.

2

   The pure time value of money analysis does not depend on the assumption of there being no risk of not getting your money back, but the assumption simplifies the analysis.

3

   All these quotations were sourced from

http://www.valuewalk.com/2016/10/compounding-quotes/

. We have not independently verified them.

Chapter 2Basic Investment Analysis

From a financial point of view, all investments may be viewed in terms of cash flows. When you make an investment, cash goes out. Some time (or times) in the future (we hope) cash flows back in. In this chapter, we look at some of the basic ways of analyzing such cash flows.

Basic Terminology

In financial terms, an investment is the exchange of cash for some other asset (which could be a business, a share in a business, or a loan) with the expectation of receiving a greater value of cash in return at some future date.

Many sophisticated investors think about investment returns almost exclusively in terms of cash flows. A cash flow is money going out or money coming in. The return is then calculated from the cash flows, using the time value of money concepts. In everyday language, people tend to use a few different terms when speaking of cash flows. Common terms include interest, principal, dividends, and capital gains. A few examples will help illustrate these terms.

A simple investment might consist of you depositing $100 into a bank account, and receiving back $101 after one year. In this case, the $100 you deposit, and the $100 you get back, are the principal. The additional $1 cash flow you receive is referred to as interest.

A stock purchase is another example of an investment. An investor might purchase a share of stock for $100, and receive a payment each quarter, called a dividend, of $.50, for a total of $2 in a year. If the investment were then sold for $103 after a year, the $3, the difference between the selling price and the purchase price, would be a capital gain. The investor's total return in this case would be the sum of all the dividends received, $2, plus the capital gain of $3, for a total of $5. Again, the $100 in this example is the principal.

Modeling an Investment as a Series of Cash Flows

Bonds are a common form of investment, in which you invest a certain amount, such as $1,000, for a certain period, such as five years. Most bonds pay interest on a regular basis. In the United States, most bonds pay a coupon every six months. For example, if the bond mentioned earlier in this paragraph (which we'll call bond 1, and refer to later) has a 5% coupon, then you'd receive payments corresponding to 5% of $1,000 every year.

Suppose you paid $1,000 for the bond now. If you held the bond to maturity (meaning that you held it for the entire length of the period the bond was supposed to last; in this case, five years), you would receive $50 of interest in a year ($25 twice a year), another $50 in two years, and so on, until at the end of the fifth year you receive the final $25 interest payment, as well as your $1,000 principal.

The heart of all investment analysis involves the projection of the cash flows involved, and then the adjustment of the expected cash flows for the time value of money. You can make allowances for risk in the way you adjust the expected cash flows for time value. You might also build more complicated models (which we don't do here) by explicitly applying a probability to the future receipt of any specific cash flow.

Net Present Value

In the previous chapter, we explained how to calculate the present value of an amount of money at one future period. When there are multiple future periods, we can apply the same tools multiple times.

In theory, and in practice, you can calculate the present value of a stream of cash flows by applying the present value formula to each cash flow individually, and then summing them all up. This is easy to do with a spreadsheet. For example, the bond discussed earlier would be analyzed as follows, assuming that the discount rate, which we called r in the previous chapter, was 5% per year. When people refer to a model similar to this model, they may refer to r as the discount rate, the interest rate, or the rate of return.

First, we map out the cash flows in Table 2.1 as follows:

Tabelle 2.1 Cash Flows from Bond

Time 0

Time 1

Time 2

Time 3

Time

Time 5

−$1,000

+$50

+$50

+$50

+$50

+$1,050

We now calculate the present value of each cash flow, using the present value formula. We show the calculation in Table 2.2 below.

Tabelle 2.2 Present Value of Cash Flows from Bond with Discount Rate of 5%

Time 0

Time 1

Time 2

Time 3

Time

Time 5

Cash Flow

−$1,000

+$50

+$50

+$50

+$50

+$1,050

PV Formula

PV

−1,000

47.62

45.35

43.19

41.14

822.70

Summing up all the numbers in the bottom row, we get a number called the net present value (“NPV”) of the stream of cash flows. It is zero. Why? Because the present values of all the cash flows, calculated at 5%, exactly equals the amount we invested at time zero. That is not coincidence. It is a direct result of the fact that we set the cash flows on the basis of a 5% annual return.

The net present value of a stream of cash flows does not have to be zero. In fact, it will only be zero when the discount rate and the original rate of return are exactly equal.