Investment Planning - Measures of Investment Returns
Consider a $10,000 bond paying a 3.5% annual coupon rate for 10 years. The bond's present value is $8,140.32, if its cash flows are discounted at the YTM of 6% per year (or 3.0% per 6-month period). The present value of $175 coupons per 6-month period for 20 periods is:
Solving for YTM: 10000 FV 20 n 175 PMT 8140.32 CHS PV i 2 x = 6%
Assume the bond has a 7% call premium, which means it is callable at 107% of its par in 5 years (10 semi-annual periods). In this case, the bond's yield-to-call would be (4.686% per 6-month period) x (two 6-month periods per year) = 9.373% at an annual rate. To compute the YTC set T = 10 semi-annual periods, FV = $10,700 (equals 107% of $10,000 par), and solve for the unknown YTC value. The solution is YTC = 4.686%. When doubled to be restated as a annual rate, YTC = 9.373%.
10700 FV 10 n 175 PMT 8140.32 CHS PV i 2 x = 9.37%
Jean's portfolio of stocks in her IRA had a value of $55,460 in the beginning of the year. At the end of the year, it was valued at $67,467. At the middle of the year, she made a contribution of $2,000 and her confirmation statement reflected an account value of $58,578 before the $2,000 contribution. What was her time-weighted portfolio return for the year?
17.63% Return for first half of year = ($58,578 - $55,460)/ $55,460 = 5.622%. Return for second half of year = ($67,467 - $58,578 - $2,000)/ ($58,578 + $2,000) = 11.37%. Time-weighted portfolio return = (1.05622)(1.11372) - 1 = 17.63%
The quarterly returns for Fred's portfolio are 2.5%, 1.7%, -3.6% and 2%. What was the annualized return of his portfolio using the compounding method?
2.5% Return = [(1+.025)(1+.017)(1-.036)(1+.02)]-1 = 2.5%.
Fred had $20,000 in his portfolio at the beginning of the year. He added $5,000 to the portfolio in the middle of the year. The account value before he added the $5,000 was $19,178 and at the year-end, it was $25,998. What was the annual time-weighted return for Fred's portfolio?
3.108% The return for first half year = ($19,178 - $20,000)/$20,000 = -.0411. The return for second half year = ($25,998 - $19,178 - $5,000) /($19,178 + $5,000) = .075275. The time-weighted return = [(1-.0411)(1+.075275)-1] = .03108 or 3.108%.
Jed was in the 31% tax bracket and his taxable investments had a total return of 45%. What was his after tax return?
31.05% Jed's after tax return is equal to his taxable return times reciprocal of his tax bracket. 45%(1 - 0.31) = 31.05%
What is the Bond Equivalent Yield (BEY)?
- used to make pure-discount bond yield similar to regular semi-annual coupon-paying bond yields for comparison purposes.
Kerry held an S&P Index Fund for three years. Her returns during that time were 20%, -10%, and 5% respectively. What was her arithmetic mean return?
(1/3)(.20 -.10+ .05) = .05 or 5%
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. What is the dollar-weighted return?
$100 million = -$5 million/ (1+r) + $103 million/(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%
The total return for XYZ fund for one year is 2%. For the year, interest income was 6%, and there was a 4% loss in principal. What is true?
- Assuming that the fund's interest is not reinvested, the price at the end of the year was lower than the beginning - The interest was assumed to have been reinvested The total return for the fund was 2%, which was comprised of a 4% loss of principal, and a 6% interest payment. Using the HPR formula, and assuming a starting value of $1,000, we would calculate the total return as ($960 + $60 - $1,000) / $1,000 = 2%.
What is true about measuring returns for investments?
- Choose investments with similar risks - Comparing to historic performance can identify extraordinary performances When comparing returns, you should be sure that the investments are of similar risk and the returns are reflecting an identical holding period. Extraordinary performances can be averaged out or identified through a historic study of the investment's returns. Returns are historic by nature, they are not indicative of future performances.
What are useful to compare with investment return figures?
- Personal required rate of return - Returns of investments with similar risk - Average returns of a relevant market - Average returns of company's industry index - Historic returns of the investment
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. What is the time-weighted return for the first half and second half year?
- Return for first half year = ($96 million - $100 million)/$100 million = -4% - Return for second half year = ($103 million - $101 million)/$101 million = 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) x (1 + .0198)] - 1 = -2.1%
Melissa is comparing the total returns of three Growth and Income Funds. What should she consider about the figures?
- Same holding periods - Three, five and ten year annualized total returns - Income portion versus change in price Total return figures are only useful for comparison when they are of the same holding periods. One-year returns are subject to extraordinary performance from short-term fluctuations. Reviewing longer-term annualized returns would reveal a better picture of the fund's performance over time. If Melissa was interested in receiving the dividends from the fund, it would be important for her to note what percentage of the total return is generated from dividends versus price change. All dividends are assumed to be reinvested for total return figures.
What scenarios would lead to realization of YTM for a bond?
- The bond is held to maturity; coupons are reinvested immediately at YTM; all payments are received on time. - The bond is a zero coupon bond and is held to maturity; par payment is received on time. For YTM to be realized, the bond must be held to maturity, the issuer must make full payments of coupon and par on time, and the cash flows must be reinvested immediately at YTM. Pure-discount (zero coupon) bonds do not have cash flows to reinvest, so if held to maturity, the holder earns YTM.
What reasons make the dollar-weighted return inappropriate for evaluating a portfolio?
- The return is strongly influenced by the size and timing of the cash flows. - The investment manager typically has no control over the deposits and withdrawals. In general, the dollar-weighted return method of measuring a portfolio's return for purposes of evaluation is regarded as inappropriate. The reason behind this view is that the return is strongly influenced by the size and timing of the cash flows (namely, deposits and withdrawals), over which the investment manager typically has no control.
What statements are true about YTM and YTC?
- YTM is based on maturity date of the bond - YTC is based on first callable date YTM is the yield to the maturity date of the bond. The YTC is the yield up to the first date that the yield can be called. The appropriate yield to choose for callable bonds may be the lower of the the YTC and YTM. YTC cannot be determined for a non-callable bond.
What is true about bond yields?
- YTM is less for a premium bond because the premium is amortized over the remaining years to maturity, so less money is available for compounding. - If a bond is selling at a premium, then the nominal yield is divided by a larger price to obtain the current yield, therefore the current yield would be lower. - If the bondholder reinvests cash flows at a rate below YTM, there will be less interest to compound - Bond equivalent yield makes a pure discount bond's yield equivalent to a coupon paying bond.
Consider a $10,000 bond paying a 3.5% annual coupon rate for 10 years. The bond's present value is $8,140.32, if its cash flows are discounted at the YTM of 6% per year (or 3.0% per 6-month period). The bond's yield-to-call is 9.373% at an annual rate. What will you consider to make the buy/sell decision for the bond?
- Yield to maturity = 6% After computing the two different yields, you must select the lower yield for investment decision-making purposes, because that return represents the minimum yield that the investor can expect to earn.
What is most likely to realize its YTM?
- Zero coupon bond The biggest and least likely assumption of realizing YTM is the fact that cash flows can be invested immediately at YTM. Zero coupon bonds do not have any cash flows except for the final payment. Therefore, it does not have to deal with reinvestment risk.
What is the arithmetic mean?
- an average of historical one-period rates of return
What is the geometric mean?
- compound average rate of return
What is yield to maturity?
- 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
What is the total return?
- stated holding period return used by investments such as mutual funds to communicate how they performed in the recent past - Total return is a standard of measurement that can be used to look at a fund's historic performance, as well as to compare it to similar funds
What is realized compound yield?
- the actual yield that an investor earned or will earn by reinvesting the cash flows
What is the holding period return?
- used as a measure for any investment - specifies a holding period of any length of time, and the result will be a raw number that is not adjusted to take time value of money into account
Jill Edwards purchased 1,000 shares of ABC mutual fund for $10.00 per share, for a total investment of $10,000. A short time later the fund paid a $550 dividend, which Jill decided to have reinvested back into the fund. At the time of the reinvestment, ABC fund was selling for $11.00 per share. At the present time, ABC fund is worth $13.70 per share. What is the holding period return (HPR)?
43.85% The $550 dividend was reinvested at a price of $11.00, increasing the number shares owned to 1,050. ($550 / $11.00 = 50 shares). Therefore, the value of the dividend is imbedded in the ending account value of $14,385 ($13.70 x 1,050 shares). In this case, the HPR would simply be the price at the end, minus the price at the beginning, divided by the price at the beginning. Plugging in the numbers, the calculation yields: {($14,385 - $10,000) / $10,000} = .4385 = 43.85%
The following data are available on the returns of the Smith Tinker Corporation (STC) for the past 4 years: Y1 = 10%, Y2= -1%, Y3= 15%, Y4= 12%. The arithmetic return on the STC stock is 9%. Calculate the GMR on the stock.
8.8% The geometric mean return from this 4-year investment is calculated as follows: GMR = [(1.10)(.99)(1.15) (1.12)]^.25 = 8.8%
Janet owns a 9% coupon bond that is currently selling for $988.09. Assuming the bond's par value is $1,000, what is the bond's current yield?
Current Yield = Dollars of coupon interest per year/Bond's current market price Current Yield = $90/$988.09 = 9.108%
Dan purchased a round lot of 100 shares of Dannon stock for $2,000 or $20/share. Two years later, he sold his shares for $32/share. Dan also received dividends of $4/share. What would be the holding period 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. ($3,200 + $400 - $2,000)/$2,000 = .80 or 80%
Gilbert invested $50,000 in a Capital Growth Fund. Two years later it was worth $67,900. The first year total return for the fund was -13% and the second year was 56.092%. What statements are true?
Holding period return = ($67,900/$50,000) - 1 = 35.8%. Arithmetic mean per year = (1/2)(.56092 -.13) = 21.5% Geometric mean per year = [(1.56092)(1 - .13)]^1/2 - 1 = 16.53% Or (on calculator) FV 67,900 PV (50,000) N 2 solve for I = 16.53% .
The equation for the TEY is:
TEY = Tax Free Yield/(1 - Tax Bracket)
Donald is helping his client pick between a taxable bond fund and a tax-free bond fund. Donald's client is a very conservative investor and would want to earn the best yield possible at his tax bracket of 36%. What would present the best choice for his client?
Tax Free Bond Fund currently paying a 30-day yield of 4% The tax equivalent yield for the Tax Free Bond Fund is 6.25%, which is higher than the GNMA fund, and is less aggressive than the two high yield bond funds. The high yield funds may present too much risk for Donald's client who is very conservative. If risk was not an issue, the highest yielding fund would be the High Yield Tax Free Bond Fund with a tax equivalent yield of 9.375%.
Consider a bond that has $1,000 par and is selling currently for $1,065.90, pays semi-annual coupons of $50 (10% coupon), and matures in 1 and ½ years. What is its YTM?
YTM = 5.37% 1000 FV 3 n 50 PMT 1065.90 CHS PV i 2 x You must change PV to negative to indicate outflow of money. Also, the initial answer of 2.68 represents a semi-annual rate; therefore it must be multiplied times 2 in order to annualize
A $1,000 par bond has a market price of $1,147.20. It pays an 8% annual coupon and matures in 10 years. The bond is callable in 5 years for $1,050. What statements are true?
YTM = 6% YTC = 5.45% Very good! YTM: PV = -1147.2, PMT = 80, N = 10, FV = 1000, I = YTM = 6; YTC: PV = -1147.2, PMT = 80, N = 5, FV = 1050, I = YTC = 5.45. Cash flows must be reinvested at YTM or 6% in order for realized compound yield at maturity to = YTM. If interest rates remain unfavorable for issuer to call the issue, YTM would be a better indicator of the yield from the bond.
Kerry held an S&P Index Fund for three years. Her returns during that time were 20%, -10%, and 5% respectively. What was her geometric mean return?
[(1 + .2)(1 - .1)(1 + .05)]^1/3 - 1 = [(1.2)(.9)(1.05)]^1/3 - 1 = (1.134)^1/3 - 1 = 4.28%
What is time-weighted return?
a return based on portfolio growth from one period to the next
What is dollar-weighted return?
a return based on the initial value plus investments versus ending value
What is annualized return?
gives the annual measure of return by adding or multiplying all the quarterly returns
