Boundless: "Finance: Chapter 9, Section 2: Understanding Returns"
Reporting
Both realized and unrealized returns are reported in the firm's income statement and balance sheet respectively.
LEARNING OBJECTIVE

Differentiate between realized and unrealized returns
KEY POINTS
 Realized returns involve the sale of the asset while unrealized returns are based off of changes in the market value, but not a sale.
 Realized returns are taxed, while unrealized returns are not.
 When the value of an asset is based off of the market value at the end of the reporting period, the company is using the marktomarket valuation method.
TERMS
 unrealized return
The difference between the market price and the purchase price. The asset has not been sold.
 realized return
The difference between the price at which the asset is sold and the purchase price.
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There are two types of gains and losses on an investment: realized and unrealized.
Realized Returns
A realized return is the difference between the sale price of the asset and the purchase price. For example, if a stock is bought for $5 and sold for $7, there is a realized gain of $2. If the stock is not sold, there is an unrealized gain of $2.
The difference between realized and unrealized returns is that realized returns result from the actual sale of the asset, while unrealized returns occur when the asset is not sold and result from a change in the market price. These returns appear separately on the firm's financial statements, and have different tax implications as well.
Returns are reported each reporting period when the financial statements are created. Realized gains involve a set transaction, so determining their value is relatively straightforward. Unrealized gains/losses are usually calculated by the market value at the end of the reporting period (marktomarket). For example, unrealized gains/losses on a stock are calculated using the price at the close of the market at the end of the period. This is not the only method for calculating the change in value of an asset, however.
Unrealized Returns
Unrealized gains are also confined only to the financial period for which the statement is prepared. For example, suppose a firm purchased 1,000 shares at $10 per share on January 1, 2012. The firm uses marktomarket valuation. On December 31, 2012, the price is $12 per share. By December 31, 2013 the price has risen again to $15. The shares are not sold. There is a total unrealized gain of $5,000, but it is composed of a $2,000 unrealized gain in 2012 and a $3,000 unrealized gain reported in 2013.
In terms of taxation, only realized gains are taxed . While realized returns are reported on the income statement (and affect the cash flow statement) and unrealized returns are reported on the balance sheet, only realized returns have tax implications.
IRS
The IRS is responsible for interpreting and enforcing tax legislation passed by Congress. The IRS taxes only realized returns, though financial reports must also include unrealized returns on the balance sheet.
Dollar Returns
The dollar return is the difference between the final value and the initial value in nominal terms.
LEARNING OBJECTIVE

Calculate an investment's dollar return
KEY POINTS
 Dollar returns do not take into account things like the time value of money or the time frame of the investment.
 In security markets, the dollar return of the security is the difference in the final market price and the market price at which it was purchased.
 Dollar returns are useful for determining the nominal amount that the firm's assets will change.
TERM
 return
Gain or loss from an investment.
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The dollar return of a security is the difference between the initial and ending value.
Finding the dollar return for securities that trade in open markets is a matter of finding the difference in price from year to year. For example, consider in which a $100 security earns a stated return of 5% per year. At the end of year 1, it is worth $105, which is $5 more than $100 (its value at the beginning of year 1), so the dollar return is $5. The capital value at the end of year 2 is $110.25, which is $5.25 more than at the end of year 1, and $10.25 more than at the beginning of year 1. Therefore, the dollar gain is $10.25. This continues for each successive year.
Dollar Profit/(Loss)
The dollar return is the difference in value from year to year, plus the previous dollar return.
The dollar return does not take into account things like the time value of money or how the amount of return earned per year; it is simply the difference in nominal values. This means that dollar returns can provide an incomplete picture if used incorrectly. For example, suppose an investor has two investment options, both of which promise a dollar return of $1,000,000. S/he cannot tell which option is better without knowing additional details such as the risk or how long it will take to realize the returns. If the first option has a $1,000,000 return over two years and the other has a $1,000,000 return over 10 years, the first option is clearly more attractive.
Dollar returns are valuable for comparing the nominal differences in investments. If two investments have similar profiles (risk, duration, etc.), than dollar returns is a useful way to compare them. The investor will always choose the option with the higher dollar return. Furthermore, the dollar return is useful because it provides an idea about how the assets of a firm will change. If a firm is looking for an additional $50,000 from investment, they will only accept investments with a $50,000 dollar return, regardless of the percent return.
Percentage Returns
Percentage returns show how much the value of the investment has changed in proportion to the size of the initial investment.
LEARNING OBJECTIVE

Calculate an investment's percentage return using CAGR
KEY POINTS
 Total percentage returns divide the dollar returns by the initial value of the investment. This is also the return on investment (ROI).
 Annual returns show the percentage by which the value of the asset changes in each individual year.
 Average annual percentage returns can be calculated by dividing ROI by the number of years, or by other methods such as the compound annual growth rate (CAGR) or internal rate of return (IRR).
TERMS
 compound annual growth rate
CAGR. A method for finding the average annual return of an investment.
 return on investment
ROI. The dollar return of the investment divided by the initial value.
 internal rate of return
IRR. The rate of return on an investment which causes the net present value of all future cash flows to be zero.
FULL TEXT
The conventional way to express the return on a security (and investments in general) is in percentage terms. This is because it does not only matter how much money was earned on the investment, it matters how much was earned in proportion to the cost.
There are two types of percentage returns: total and annual. Total returns calculate how much the value of the investment has changed since it was first purchased, while annual returns calculate how much the value changed each year. When the length of time of the investment is one year, the total and annual returns are equivalent.
Total Returns
The total percentage return is based off of the final value (V_{f}), the initial value (V_{i}), and all dividend payments or additional incomes (D). If the investment is a security such as a stock, the final value is the sales price, the initial value is the purchase price, and D is the sum of all dividends received.
Return=Vf−Vi+DVi
This type of return is also called the return on investment (ROI), where the numerator is the dollar return.
Annual Returns
In , the ROI is calculated for each individual year by dividing the dollar return by the initial value of $1,000. To find the return for the security overall, simply sum the dollar returns and divide by the initial value. The ROI can be annualized by dividing by the number of years between the purchase and sale of the security. This is the arithmetic mean of the return.
Cash Flow Return
The ROI is the percentage return, and is calculated by dividing the dollar return by the initial value of the investment ($1,000).
However, this does not fully take into consideration compounding. To do so, analysts use other formulas, like the compound annual growth rate (CAGR):
CAGR=(VfVi)1t−1
In this case, the only variable that differs from the previous formula is t, which is the number of years between the beginning and end of the investment. CAGR is a way of measuring the return per year. It is widely used because it allows for the easy comparison of the growth rates of multiple investments.
Another common method for finding the annual return is to calculate the internal rate of return (IRR). Recall that the IRR is the discount rate at which the net present value(NPV) equals 0.
Historical Returns: Market Variability and Volatility
Markets and securities may follow general trends, but exogenous factors (such as macroeconomic changes) cause variability and volatility.
LEARNING OBJECTIVE

Describe how historical returns can be used to predict future performance
KEY POINTS
 Historical returns do not guarantee future returns.
 All markets have a degree of systemic risk which means that they have a risk of collapsing due to external factors. Companies are also interconnected, so the failure of one company can have farreaching effects.
 "Animal spirits" describes general investor sentiment which can affect markets, even without changes in the underlying financials.
TERMS
 systemic risk
The risk of collapse of an entire financial system or entire market.
 animal spirits
After Keynes (citation 1936, above), the emotional and intuitive factors that drive business decisions whether to make investment gambles.
 volatility
A quantification of the degree of uncertainty about the future price of a commodity, share, or other financial product.
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Historical analysis of markets and of specific securities is a useful tool for investors, but it does not predict the future of the market. There are general trends and expectations of future behavior, but they are just generalizations. For example, the Dow Jones Industrial Average (DJIA) has generally followed an upward trend from 19002009 . However, an investor who looked at this graph in early 1929 and made the decision to invest because s/he would be guaranteed to make money was in for a shock when the market crashed in October 29, 1929. Past performance is not a guarantee of future performance.
DJIA 19002009
The Dow Jones Industrial Average has generally increased overall since 1900, but its past performance is not a guarantee of future performance.
Inherent in all markets is something called "systemic risk. " Systemic risk is the risk of collapse of an entire financial system or entire market, as opposed to risk associated with any one individual entity, group, or component of a system. Macroeconomic forces, such as the Great Depression, affect the entire stock market and can't be predicted from past market performance. The failure of one company affects all the companies who purchase from it or sell to it, which in turn affects all the companies that rely on them. These types of interlinkages are a cause of the overall market variability and volatility.
Furthermore, market variability and volatility can be the cause of what John Maynard Keynes called animal spirits. Animal spirits are the emotions felt by investors who affect markets. Expectations of investors affect how they act, which in turn affects the markets. If investors are feeling optimistic, for example, the market may go up, even without an improvement in the financials of the underlying companies.
Markets and stocks are affected by many factors beyond the information in their financial statements and past performance. Historical returns may provide an idea of the overall trend, but certainly are not enough to accurately predict future performance.
Calculating and Understanding Average Returns
Average returns are commonly found using average ROI, CAGR, or IRR.
LEARNING OBJECTIVE]

Differentiate between the different methods for calculating the average return of an investment
KEY POINTS
 Average return on investment (ROI) is the arithmetic average of the total cash returns divided by the initial investment. It is useful for quick calculations and specific securities (such as bonds purchased at par), but does not account for compounding returns.
 Compound annual growth rate (CAGR) is derived from the future value formula with compounding interest. It accounts for compounding returns.
 Internal rate of return (IRR) is the discount rate at which the NPV equals 0. It is used because it allows for easy comparison between investment options and is easy to understand.
 For all three methods, the higher the average rate of return, the more attractive the investment is.
TERM
 compounding returns
Returns earned on previous returns. Akin to compounding interest.
FULL TEXT
The average return of an investment can be calculated a number of ways. The three main methods are
 Return on investment (ROI)
 Compound annual growth rate (CAGR)
 Internal rate of return (IRR).
For all three methods, the higher the rate, the more desirable the investment.
ROI
To calculate the total ROI of an investment, simply divide the total dollar returns of the investment by the initial value. The average ROI is the arithmetic average: divide the total ROI by the number of periods. If the purchase of a stock led to an ROI of 15% over 5 years, the average ROI is 3% per year. This is a simple way to calculate the average return. It is useful for certain securities such as bonds. If the only source of return on a bond is the coupon payments, then this is an accurate method. A bond purchased at par that pays a 5% coupon per year, will have a return of 25% over 5 years. However, this is a very special case. Average ROI generally does not calculate the actual average rate of return, because it does not incorporate compounding returns. A stock that appreciates by 3% per year would not actually be worth 15% more over 5 years, because the gains compound.
CAGR
CAGR stands for compound annual growth rate. It is calculated by the following formula where V_{f} is the future value, V_{i} is the initial value, and t is the number of years:
CAGR, unlike average ROI, does consider compounding returns. CAGR is derived from the compounding interest formula, FV=PV(1+i)^{t}, where PV is the initial value, FV is the future value, i is the interest rate, and t is the number of periods. The CAGR formula is what results when solving for i: the interest rate becomes CAGR, FV becomes V_{f}, PV becomes V_{i}, and the number of periods is generally assumed to be in years.
CAGR is very useful for finding the rate of return that the investment would have to earn every year for the life of the investment to turn the initial value into the future value over the given time frame.
IRR
The internal rate of return (IRR) is another commonly used method for calculating the average return . IRR is the discount rate at which the net present value (NPV) is equal to 0. Using IRR allows for easy comparison between investment options. It is also known as the effective interest rate.
Internal Rate of Return
The IRR is calculated by finding the discount rate at which the NPV of the investment equals 0.