Measures of Investment Returns Flashcards
Holding-Period Return
A measure that can be used for any investment is its holding-period return. Holding period is defined as the length of time over which an investor is assumed to invest a given sum of money. The holding period return has a major weakness because it does not consider the time or how long it took to earn the return. When this procedure is applied, the performance of a security can be measured by comparing the value obtained in this manner at the end of the holding period with the value at the beginning.
Please note that in the formula below, any coupon (from a bond), interest, dividend, or any other cash flow received from the investment does not assume reinvestment. Any reinvestment (like capital gains distributions and dividends from a mutual fund) would be embedded in the ending value (P1) and the separate addition of these payments would overstate the return.
HPR=(P1+D−P0) / P0
Where P0 = price in the beginning of the period, P1 = price at the end of the period, and D = any dividend, interest, or cash flow paid.
Arithmetic Mean
The arithmetic average rate of return is a summary of a great deal of information and provides a good way to compare the performance of different investments. The arithmetic mean return (AMR), an average of historical one-period rates of return, is computed as follows:
AMR=r¯=(1 / T) r∑t−1
rt=(1 / T)(r1+r2+…+rt)
Or more commonly seen as:
AM=a1+a2+a3+⋯+an / n
Note: This is the given formula format on the CFP Formula table.
Geometric Mean
The compound average rate of return (geometric return) is similar to the arithmetic mean return, except the geometric return, because it does take compounding into account, will always be less than the arithmetic return. The compound average rate of return is also called the geometric mean return (GMR), where the GMR is computed over T successive time periods.
The GMR formula is restated equivalently as follows:
GMR= r√(1+r1)(1+r2)(1+r3)…(1+rT) - 1
After-Tax Return
Taxes can take away from returns as well. It is important to consider how much is paid in taxes on investment gains when evaluating how much you have really earned. For example, if your investments yield long-term capital gains of $1,000, then 15% of the long-term gains, or $150, is due to the IRS as Federal tax on the gains. Depending on the state you live in, there could be an additional amount due for state taxes as well.
After-Tax Return = Total Return (1 − tax bracket)
Real (Inflation Adjusted) Return
In times of changing prices, the nominal return (dollars received) on an investment may be a poor indicator of the real return (also known as the real rate) obtained by the investor.
This adjustment is helpful because it accounts for the natural rise in prices over time. Goods and services will cost more as inflation slowly but steadily increases their cost. As a result, adjustments to the nominal return are needed to remove the effect of inflation to determine the real return. Frequently, the consumer price index (CPI) is used to measure the amount of price change in a basket of goods.
For example, assume that at the start of a given year the CPI is at a level of 150, and that at the end of the year it is at a level of 160. This means that the same amount of the CPI market basket of goods that could have been purchased for $150 at the start of the year costs $160 at the end of the year. Assuming that an investment’s nominal return is 9% for this year, the investor who started the year with $150 and invested it would have $150 x 1.09 = $163.50 at year-end.
An increase in the CPI from 150 to 160 can be translated into an inflation rate of (160/150) - 1 = .0667, or 6.67%. This inflation rate can be denoted as Rinf
or Ri
, and the real return can be calculated using the following formula, known as the Fisher model:
[(1+NR) / (1+Ri)]−1=RR
Note that for the example, RR=[(1.09)(1.0667)]−1=.0218
, or 2.18%.
Note for a quick calculation, many investors will do the following shortcut versus the Fisher model; the real return can be estimated by simply subtracting the inflation rate (Ri
) from the nominal return: (NR−Ri)≈RR
In this example, the “quick method” results in an estimate of the real return of .09 − .0667 = .0233, or 2.33%.
However, a word of advice and caution should be heeded; this is not as useful for real calculations and may result in wrong answers. The error resulting from use of this method is .0233 − .0218 = .0015, or .15%.
Yield to Maturity (YTM)
The rate of return on bonds that is most often quoted for investors is the yield to maturity (YTM), which is defined as the discount rate that equates the present value of all the bond’s future cash flows with its current market price (purchase price). The YTM is the compounded rate of return of a bond.
To determine the yield to maturity, or internal rate of return, an investor would use the bond’s coupon rate, price, par value, and term to maturity. Thus, the present-value formula is appropriate for computing the compounded YTM for a bond that makes annual coupon payments and has T years until the bond matures and repays its par value (principal, face value).
PV= (Coupon1/2) / [1+(YTM/2)]1 + (Coupon2/2) / [1+(YTM/2)]2 +…+ (CouponT+Par/2) / [1+(YTM/2)]N
=(Coupon/2)(PVIFAYTM/2,2N)+(PVIFYTM/2,2N)
PV = Present Value of bond
N = number of compounding periods until payment Coupon/2 = semi-annual payment of coupon amount YTM/2 = semi-annual value of discount rate (YTM) Par = $1,000
Bond Equivalent Yield
Most bonds in the United States pay semi-annual coupons. Semi-annual coupons are half the size of annual coupons but are paid twice as often. When the YTM of a fixed-income structure is calculated by doubling the number of time periods from T years with annual coupons, to 2T 6-month periods with semi-annual coupons used in annual coupons it is called a bond equivalent yield.
Additionally, money market instruments that are sold at a discount with no actual interest payments can be quoted as having a bank discount rate or a bond equivalent yield.
Yield to Call (YTC)
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.
Callable bonds have an embedded option that the bond issuer may or may not find profitable to exercise. The uncertainty about whether or not the bond will be called forces the investor to evaluate each of the possible future scenarios:
First scenario: If market interest rates rise, then it is safe to assume the bond issuer will not find it profitable to exercise the call option. When the bond is not called, it is customary to assume the bond remains outstanding until its maturity date. In the event of this first outcome, the traditional YTM is the appropriate yield measure.
Second scenario: If market interest rates decline, then it would be profitable for the bond’s issuer to call the bond before it matures. To evaluate this second scenario, the bond’s Yield to Call (YTC) must be calculated.
The investor never knows in advance which of the two scenarios will occur. The conservative approach is to compute both of these two different yields and then select the lower yield for investment decision-making purposes, because that return represents the minimum yield that the investor can expect to earn.
YTC is calculated the same way as YTM, except the terminal number (T) is the time to call rather than time to maturity.
PV= (Coupon1/2) / [1+(YTM/2)]1 + (Coupon2/2) / [1+(YTM/2)]2 +…+ (CouponT+Par/2) / [1+(YTM/2)]N
=(Coupon/2)(PVIFAYTM/2,2N)+(PVIFYTM/2,2N)
PV = Present Value of bond
N = number of compounding periods until payment Coupon/2 = semi-annual payment of coupon amount YTM/2 = semi-annual value of discount rate (YTM) Par = $1,000
Realized Compound Yield
The assumption of the ability to reinvest at the YTM or YTC is crucial for determining how much of a return the bond will provide. The realized compound yield allows investors to make their own reinvestment rate assumptions, or in an ex-post (historical data) context, calculate the actual yield resulting from their reinvestment decisions. Since we usually compare this statistic with YTM calculations, we will use the formula in an ex-ante (projected or expectational data) context as follows:
Take for example:
$1,000 = par
$80 = annual coupon
8% = YTM
3 = N
$1,000 = PV
If the realized compound yield = YTM, then:
1st coupon will compound at 8% for two years or 80(1 + .08)2 = $93.31
2nd coupon will compound at 8% for one year or 80(1 + .08) = $86.40
3rd coupon will be paid with par value = 80 + 1,000 = $1,080.00
Total amount from bond at T = $1,259.71
If the realized compound yield > YTM, then the total accumulation would be > $1,259.71.
If the realized compound yield < YTM, then the total accumulation would be < $1,259.71.
After-Tax Yield
When considering a taxable bond versus a municipal bond, a comparison can be drawn between the after-tax yield of the taxable bond and the tax-free yield of the municipal bond. The same comparison can be made between a taxable bond fund and a municipal bond fund. The Tax Equivalent Yield (TEY) is often used to make these comparisons.
The equation for the TEY is:
TEY=Tax Free Yield / (1 - Tax Bracket)
This yield gives the person a basis to take a tax-free yield and make it a taxable yield equivalent.
Current Yield
The current yield of a bond measures an annual cash flow relative to the bond’s current market price. Retirees who live on investment cash flows, for instance, are interested in a bond’s current yield. The current yield is an accurate representation of what income you are earning based on the price you paid. If the bond is selling at par, then the current yield is the same as the coupon rate.
However, if a bond is selling at a premium, the current yield will be lower than the yield. On the other hand, if the bond is selling at a discount, the current yield will be higher than the original yield.
Current Yield=Dollars of coupon interest per year / Bond’s current market price
Portfolio Returns
Frequently, portfolio performance is evaluated over a time interval, with returns measured for a number of periods within the interval — typically monthly or quarterly. In general, the market value of a portfolio at a point in time is determined by adding the market values of all the securities held at that particular time.
R=(Ve−Vb) / Vb
Return=(value at the end minus value in the beginning) / value in the beginning
Dollar-Weighted Return
Difficulties are encountered when deposits or withdrawals occur sometime between the beginning and end of the period. One method that has been used for calculating a portfolio’s return in this situation is the dollar-weighted return (or internal rate of return). Dollar-weighted return is what an investor should use to determine how well their investment performed.
Example
Consider a portfolio that at the beginning of the year has a market value of $100 million. In the middle of the year, the client deposits $5 million with the investment manager, and subsequently at the end of the year, the market value of the portfolio is $103 million.
$100million=−$5million / (1+r)+$103million / (1+r)2
r = -.98%
The dollar-weighted average (r) is a semi-annual rate of return. It can be converted into an annual rate of return by adding 1 to it, squaring this value, and then subtracting 1 from the square, resulting in an annual return of -1.95%.
[1+(-.0098)]2 − 1 = -1.95%
Time-Weighted Return
Alternatively, the time-weighted return on a portfolio only concerns itself with the portfolio appreciation or depreciation in value from one period to the next. That is, all cash flows into and out of the fund are totally disregarded from the return data. Time-weighted return is the only acceptable method to display the results of a portfolio manager’s performance.
From the previous example, assume that in the middle of the year the portfolio has a market value of $96 million, so that right after the $5 million deposit the market value was $96 million + $5 million = $101 million.
Return for first half year=($96million−$100million) / $100million = -4%
Return for second half year=($103million−$101million) / $101million = 1.98%
Next, these two semi-annual returns can be converted into an annual return by adding 1 to each return, multiplying the sums, and then subtracting 1 from the product.
Annual return of [(1 - .04) × (1 + .0198)] - 1 = -2.1%
