Chapter 5
Measuring Investment Returns
- Individual Security Measures of Return
- Components of Return and Yield to Maturity
Realized return is what the term implies; it is ex post (after the fact) return, or return that was or could have been earned. Realized return has occurred and can be measured with the proper data. Expected return, on the other hand, is the estimated return from an asset that investors anticipate (expect) they will earn over some future period. As an estimated return, it is subject to uncertainty and may or may not occur.
The objective of investors is to maximize expected returns, although they are subject to constraints, primarily risk. Return is the motivating force in the investment process. It is the reward for undertaking the investment.
Returns from investing are crucial to investors; they are what the game of investments is all about. An assessment of return is the only rational way (after allowing for risk) for investors to compare alternative investments that differ in what they promise. The measurement of realized (historical) returns is necessary for investors to assess how well they have done or how well investment managers have done on their behalf. Furthermore, the historical return plays a large part in estimating future, unknown returns.
Return on a typical investment consists of two components:
- Yield: The basic component that usually comes to mind when discussing investing returns is the periodic cash flows (or income) on the investment, either interest or dividends. The distinguishing feature of these payments is that the issuer makes the payments in cash to the holder of the asset. Yield measures relate these cash flows to a price for the security, such as the purchase price or the current market price.
- Capital gain (loss): The second component is also important, particularly for common stocks but also for long-term bonds and other fixed-income securities. This component is the appreciation (or depreciation) in the price of the asset, commonly called the capital gain (loss).It is the difference between the purchase price and the price at which the asset can be, or is, sold.
Given the two components of a security's return, we need to add them together (algebraically) to form the total return, which for any security is defined as
Total return = Yield + Price change where:
the yield component can be 0 or +
the price change component can be 0, +, or -
This is a conceptual statement for the total return for any security. The important point here is that a security's total return consists of the sum of two components, yield and price change. Investors' returns from assets can come only from these two components-an income component (the yield) and/or a price change component, regardless of the asset.
The rate of return on bonds most often quoted for investors is the yield to maturity (YTM), which is defined as the promised compounded rate of return an investor will receive from a bond purchased at the current market price and held to maturity. It captures the coupon income to be received on the bond as well as any capital gains and losses realized by purchasing the bond for a price different from face value and holding to maturity. Similar to the Internal Rate of Return (IRR), in financial management, the yield to maturity is the periodic interest rate that equates the present value of the expected future cash flows (both coupons and maturity value) to be received on the bond to the initial investment in the bond, which is its current price.
An investor would use the bond’s coupon rate, price, par value and term to maturity to determine the yield to maturity, or internal rate of return. For a bond selling at $1,000 and expected to be redeemed by the issuer at $1,000, the current yield and the yield to maturity are identical. However, the yield to maturity will differ from the current yield if the bond sells at a discount or a premium. Yield to Maturity Calculator
- Yield to Call
Most corporate bonds, as well as some government bonds, are callable by the issuers, typically after some deferred call period. For bonds likely to be called, the yield-to-maturity calculation is unrealistic. A better calculation is the promised yield to call. The end of the deferred call period, when a bond can first be called, is often used for the yield-to-call calculation. This is particularly appropriate for bonds selling at a premium (i.e., high-coupon bonds with market prices above par value).
Bond prices are calculated on the basis of the lowest yield measure. Therefore, for premium bonds selling above a certain level, yield to call replaces yield to maturity, because it produces the lowest measure of yield.
- Compounding vs. Discounting
The concept of compounding, that is interest on interest, is an important concept, as is its complement, discounting. Compounding involves future value resulting from compound interest. Present value (discounting) is the value today of a dollar to be received in the future. Such dollars are not comparable, because of the time value of money. In order to be comparable, they must be discounted back to the present. Tables exist for both compounding and discounting, and calculators and computers make these calculations simple. Future Value Calculator
- After Tax Yield
As an investor you learn very quickly that taxes can take a big bite out of your investment returns. After all, it’s not what you make, but what you keep that really counts. If an investment is made in a taxable vehicle within a taxable account, one should really look at the net after tax return. Tax Planning
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Example
Mr. Jones has an investment with a 6% taxable yield. Jones is in the 28% tax bracket. What is Jones' true after-tax yield?
The after-tax yield is found by multiplying the return by the reciprocal of the tax bracket. In this case:
1.00 - 0.28 = 0.72.
0.72 x 6% = 4.32%
Therefore Jones' after-tax return is 4.32%, which barely beats inflation. |
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Example
Mr. Jones has an tax-free investment with a 5% yield. Jones is in the 28% tax bracket. What is Jones' taxable equivalent yield?
To find the taxable equivalent yield of a tax-free investment, you simply divide the return by the reciprocal of your tax bracket:
| Taxable Equivalent Yield |
= |
Tax free yield |
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100% - Your tax Rate |
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| Taxable Equivalent Yield |
= |
5% |
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100% - 28% |
So, if a tax free yield is 5%, the taxable equivalent yield would be 5% divided by 0.72 or 6.94%.
For this reason, it makes sense for many investors to take advantage of tax-deferred accounts such as retirement accounts or annuities. Another consideration could be the tax-free investment vehicle, municipal bonds. |
- Annualized Return
An annualized return is a rate of return over a full calendar year on an investment that is held for less than a full calendar year. For example, if an investment produced a return of 5% in 182 days, the annualized yield would be approximately 10%. It is important to note that this return measurement assumes the return could be duplicated over the full year, which may or may not actually be achievable.
- Real (inflation adjusted) Return
All of the returns discussed above are nominal returns, or money returns. They measure dollar amounts or changes but say nothing about the purchasing power of these dollars. To capture this dimension, we need to consider real returns, or inflation-adjusted returns.
To calculate inflation-adjusted returns, we divide 1 + nominal total return by 1 + the inflation rate as shown in the following equation. This calculation is sometimes simplified by subtracting rather than dividing, producing a close approximation.
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Example
The total return for large common stocks in 1995 was 37.43%. The rate of inflation was 2.74%.
What was the real (inflation-adjusted) total return for large common stocks in 1995?
| TR |
= |
(1 + TR) |
- 1 |
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1 + IF |
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| TR |
= |
1 + .3743 |
- 1 |
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1 + .0274 |
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| TR |
= |
1.3743 |
- 1 |
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1.0274 |
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| TR |
= |
1.3376 |
- 1 |
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| TR |
= |
0.3376 or 33.76% |
Merely subtracting 2.74 from 37.43 would give you a simpler, yet less accurate, return of 34.69 percent. |
- Total Return
A correct returns measure must incorporate the two components of return, yield and price change. Returns across time of from different securities can be measured and compared using the total return concept. Formally, the total return for a given holding period is a decimal(or percentage) number relating all the cash flows received by an investor during any designated time period to the purchase price of the asset. Total return is defined as:
| TR |
= |
Any cash payments received + Price changes over the period |
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Price at which the asset is purchased |
Keep in mind the dollar price change over the period, defined as the difference between the beginning (or purchase) price and the ending (or sale) price, can be either positive (sales price exceeds purchase price), negative (purchase price exceeds sales price) or zero. The cash payments can be either positive or zero. Netting the two numbers in the numerator together and dividing by the purchase price results in a decimal return figure that can easily be converted into percentage form.
The cash flow from a bond comes from the interest payments received, and that for a stock comes from the dividends received. Although one year is often used for convenience, this calculation can be applied to periods of any length.
In summary, the total return concept is valuable as a measure of return because it is all-inclusive, measuring the total return per dollar of original investment. It facilitates the comparison of asset returns over a specified period, whether the comparison is of different assets, such as stocks versus bonds, or different securities within the same type, such as several common stocks. Remember that using this concept does not mean that the securities have to be sold and the gains or losses actually realized.
- Holding Period Return
A holding period return is the total return actually realized or expected from holding a specific asset for a specified period of time (not necessarily one year). The return is
measured the same as total return, that is, adding dividends and capital gains and dividing the resulting figure by the purchase price.
- Portfolio Performance Measurement
- Benchmark Portfolios
Evaluation of portfolio performance, the bottom line of the investing process, is an important aspect of interest to all investors and money managers. The framework for evaluating portfolio performance consists of measuring both the realized return and the differential risk of the portfolio to use to compare a portfolio’s performance, and recognizing any constraints that the portfolio manager may face. A 12% return, by itself, is a fairly meaningless figure. It must be viewed in comparison to the performance, over the same timeframe, of alternative investments bearing a similar level of risk.
Remember, one can only measure return in relation to the risk taken. Investing is always a two-dimensional process based on return and risk. These two factors are opposite sides of the same coin, and both must be evaluated if intelligent decisions are to be made. Therefore, if we know nothing about the risk of and investment, there is little we can say about its performance. Given the risk that all investors face, it is totally inadequate to consider only the returns from various investment alternatives. Although all investors prefer higher returns, they are also risk averse. To evaluate portfolio performance properly, we must determine whether the returns are large enough given the risk involved. If we are to assess performance carefully, we must evaluate performance on a risk-adjusted basis.
We must make relative comparisons in performance measurement, and an important related issue is the benchmark to be used in evaluating the performance of a portfolio. The essence of performance evaluation in investments is to compare the returns obtained on some portfolio with the returns that could have been obtained from a comparable alternative. The measurement process must involve relevant and obtainable alternatives; that is, the benchmark portfolio must be a legitimate alternative that accurately reflects the objectives of the portfolio owners.
An equity portfolio consisting of S&P 500 stocks should be evaluated relative to the S&P 500 Index or other equity portfolios that could be constructed from the Index, after adjusting for the risk involved. On the other hand, a portfolio of small capitalization stocks should not be judged against that same benchmark. If a bond portfolio manager’s objective is to invest in bonds rate A or higher, it would be inappropriate to compare his or her performance with that of a junk bond manager. Even more difficult to evaluate are equity funds that hold some madcap and small stocks while holding many S&P 500 stocks. Comparisons for such a widely diversified group can be quite difficult.
The S&P 500 has been the most frequently used benchmark for evaluating the performance of institutional portfolios such as those of pension funds and mutual funds. The S&P 100 and the Wilshire 5000 index are also popular. Many observers now agree that multiple benchmarks can be more appropriate to use when evaluating portfolio returns. All investors should understand that even in today’s investment world of computers and databases, exact, precise, universally agreed upon methods of portfolio evaluation remain an elusive goal. An evaluation is imperative, though, and it is unfortunate that some studies have indicated that most investors don’t have a good idea how well their portfolios are actually performing.
One warning about published performance is warranted. When investors are selecting money managers to turn their money over to, they typically evaluate these managers only on the basis of their published performance statistics. If the published track record looks good, that is typically enough to convince many investors to invest in a particular mutual fund. However, the past is no guarantee of an investment manager’s future. Short-term results may be particularly misleading.
- Risk-adjusted Returns
Recognizing the necessity to incorporate both return and risk into the analysis of portfolio return, three researchers- William Sharpe, Jack Treynor, and Michael Jensen- developed measures of portfolio performance in the 1960s. These measures are often referred to as the composite (risk-adjusted) measures of portfolio performance, meaning that they incorporate both realized return and risk into the evaluation. These measures are still used by mutual funds and money managers. The formulas created by these three men are rather complex and rather than showing the mathematical equations, a brief description of them should be of more benefit.
William Sharpe introduced a risk-adjusted measure of portfolio performance called the reward-to-variability ratio (RVAR).This measure uses a benchmark based on the ex post capital market line. Sharpe used RVAR to do the following:
- Measure the excess return per unit of total risk ( as measured by standard deviation)
- Rank portfolios by RVAR ( the higher the RAVR, the better the portfolio performance)
Jack Treynor presented a similar measure called the reward-to=volatility ratio (RVOL). Like Sharpe, Treynor sought to relate the return on a portfolio to its risk. Treynor, however, distinguished between total risk and systematic risk, implicitly assuming that portfolios are well diversified; that is, he ignores any diversifiable risk. He used as a benchmark the ex post security market line.
Michael Jensen’s measure of portfolio performance was differential return measure (Alpha). It calculated the difference between what the portfolio actually earned and what it was expected to earn given its level of systematic risk. Basically, it attempts to measure the constant return that the portfolio manager earned above, or below, the return of an unmanaged portfolio with the same market risk.
The Sharpe and Treynor measures can be used to rank portfolio performance and indicate the relative positions of the portfolios being evaluated. Jensen’s measure is an absolute measure of performance.
- Time vs. Dollar-weighted Return
When portfolio performance is evaluated, the investor should be concerned with the total change in wealth. We have seen that a proper measure of this return is the total return, which captures both the income component and the capital gains (or losses) component of return. In the simplest case, the market value of a portfolio can be measured at the beginning and ending of a period, and the rate of return can be calculated as
Where Rp is return of portfolio; Ve is the ending value of the portfolio and Vb is its beginning value.
This calculation assumes that no funds were added to or withdrawn from the portfolio by the client during the measurement period. If such transactions occur, the portfolio return as calculated, may not be an accurate measure of the portfolio’s performance. For example, if the client adds funds close to the end of the measurement period, use of this equation would produce inaccurate results because the ending value was not determined by the actions of the portfolio manager. Although a close approximation of portfolio performance might be obtained by simply adding any withdrawals or subtracting any contributions that are made very close to the end of the measurement period, timing issues are a problem. The following two means of return measurement help alleviate those problems.
- Dollar-Weighted Returns Traditionally, portfolio measurement consisted of calculating the dollar-weighted rate of return , which is equivalent to the internal rate of return (IRR) used in several financial calculations. The IRR measures the actual return earned on a beginning portfolio value and on any net contributions made during the period.
The DWR equates all cash flows, including ending market value, with the beginning market value of the portfolio. Because the DWR is affected by cash flows to the portfolio, it measures the rate of return to the portfolio owner. However, because the DRW is heavily affected by cash flows, it is inappropriate to use when making comparisons to other portfolios or to market indexes, a key factor in performance measurement.
- Time-Weighted Returns Since 1966, the time-weighted rate of return (TWR) typically is calculated for comparative purposes when cash flows occur between the beginning and the end of a period. TWRs are unaffected by any cash flows to the portfolio; therefore, they measure the actual rate of return earned by the portfolio manager.
Calculating the TWR requires information about the value of the portfolio's cash inflows and outflows. To compute the TWR, we calculate the return to the portfolio immediately prior to a cash flow occurring. We then calculate the return to the portfolio from that cash flow to the next, or to the end of the period. Finally, we link these rates of return together by computing the compound rate of return over time. In other words, we calculate the rate of return for each time period defined by a cash inflow or outflow, and then calculate a compound rate of return for the entire period. If frequent cash flows are involved, substantial calculations are necessary.
The two methods described, the dollar- weighted return and the time-weighted return, can produce different results, and at times these differences are substantial. The time-weighted return captures the rate of return actually earned by the portfolio manager, while the dollar-weighted return captures the rate of return earned by the portfolio owner.
For evaluating the performance of the portfolio manager, the time-weighted return should be used because he or she generally has no control over the deposits and withdrawals made by the clients. The objective is to measure the performance of the portfolio manager independent of the actions of the client, and this is better accomplished by using the time-weighted return.
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