1 Rates and Returns

• An investor purchases a share of Redwood Alternatives stock for $50, buys another share after a year for $75, and sells both shares a year later for $100 each.
• There are no transaction costs or taxes, and the investor's required return is 35.0%.
• The stock paid a $5.00 per share dividend in year one and $7.50 per share dividend in year two.

Time-Weighted Return Calculation

• Holding Period 1 (Year 1):

  • Beginning price (P₀): $50.00
  • Dividend (D₁): $5.00
  • Ending price (P₁): $75.00
  • HPR₁ = (75 - 50 + 5) / 50 = 0.60, or 60%

• Holding Period 2 (Year 2):

  • Beginning price (P₁): $75.00
  • Dividend (D₂): $7.50
  • Ending price (P₂): $100.00
  • HPR₂ = (100 – 75 + 7.50) / 75 = 0.433, or 43.3%

• Geometric Mean Return Calculation:

  • Return = [(1 + HPR₁) × (1 + HPR₂)]¹/² – 1 = [(1.60) × (1.433)]¹/² – 1 = 0.5142, or 51.4%

• If a stock decreases from $90 to $80, the continuously compounded rate of return for the period is -0.1178.

• Calculated as: ln(80 / 90) = -0.1178.

• An investor expects a stock selling for $20 to increase to $25 by year-end.

• The dividend last year was $1, and this year's dividend is expected to be $1.25.

• The expected holding period return on this stock is 31.25%.

• Return = [1.25 + (25 – 20)] / 20 = 6.25 / 20 = 0.3125.

• Vega research indicates investors need at least 1.0% (1-year CD) or 1.5% (2-year CD) more than a savings account to tie up their money.

• The savings account rate is 3%.

• To raise funds with 2-year CDs, the yield must be at least 4.5%.

• 4.5% represents a required rate of return (= 3.0 + 1.5).

• Wei Zhang has funds earning 6% interest.

• The 6% interest best represents as the opportunity cost if $15,000 of these funds are withdrawn to purchase an automobile.

• Selmer Jones wants to set aside money inherited for a vacation in Hawaii one year from today.

• The bank will pay 5% interest on any funds deposited.

• The 5% should be used as a discount rate to determine how much money must be set aside.

• An investor purchases a share of stock for $50, buys another after a year for $75, and sells both shares after one more year for $100 each.

• There are no transaction costs or taxes.

• The investor's required return is 35%.

• The stock paid a $5 per share dividend in year one and 7.50 per share in year 2.

• The investor's money-weighted return is closest to 48.9%.

• A 10% coupon bond was purchased for $1,000.

• One year later, the bond was sold for $915 to yield 11%.

• The investor's holding period yield on this bond is closest to 1.5%.

• HPY = [(interest + ending value) / beginning value] – 1 = [(100 + 915) / 1,000] - 1 = 1.015 - 1 = 1.5%

• An investor buys a non-dividend paying stock for $100 with 50% initial margin.

• At the end of the year, the stock price is $95.

• Deflation of 2% occurred during the year.

• The real return measure for this investment will be greatest.

• The continuously compounded rate of return that will generate a one-year holding period return of -6.5% is closest to -6.7%.

• Continuously compounded rate of return = ln(1 – 0.065) = -6.72%.

• Time-weighted returns are used by the investment management industry because they are not affected by the timing of cash flows.

• The money-weighted rate of return will tend to be elevated if funds are contributed to a portfolio just before a period of favorable performance.

• Computing the internal rate of return of the inflows and outflows of a portfolio would give the money-weighted return.

• A stock that pays no dividend is currently priced at €42.00.

• One year ago, the stock was €44.23.

• The continuously compounded rate of return is closest to -5.17%.

  • ln (S₁ / S₀) = ln (42.00 / 44.23) = ln (0.9496) = - 0.0517 = - 5.17%.

• Over a year, an investor's portfolio declined in value from 127,350 to 108,427.

• The continuously compounded rate of return is -16.09%.

• The continuously compounded rate of return = ln(S₁ / S₀) = ln(108,427 / 127,350) = – 16.09%.

• An investor buys a stock on March 24 for $63.25.

• The stock pays quarterly dividends of $0.54 on May 1 and August 1.

• On September 27, the investor sells the stock for $62.80.

• The investor's holding period return is closest to 1.0%.

  • [(62.80+0.54+0.54) / 63.25] - 1 = 0.01 = 1%.

• The effective rate of return with continuous compounding will be higher compared to the effective rate of return with discrete compounding, for a given stated annual rate of return.

• Stock XYZ is purchased on January 2 at $12 per share.

• The investor receives a quarterly dividend of $0.60 per share on April 1, and the stock closes on June 30 at $13 per share.

• The holding period return is closest to 13.33%.

• The real risk-free rate can be thought of as approximately the nominal risk-free rate reduced by the expected inflation rate.

• An investor begins with a $100,000 portfolio.

• At the end of the first period, it generates $5,000 of income, which he does not reinvest.

• At the end of the second period, he contributes $25,000 to the portfolio.

• At the end of the third period, the portfolio is valued at $123,000.

• The portfolio's money-weighted return per period is closest to 0.94%.

• An asset manager's portfolio had the following annual rates of return:

  • 20X7: +6%
  • 20X8: -37%
  • 20X9: +27%

• The manager states that the return for the period is -5.34%.

• The manager has reported the geometric mean return.

Geometric Mean Return Formula

• Geometric Mean Return = [(1 + 0.06) (1 – 0.37) (1 + 0.27)]^(1/3) – 1 = -5.34%

  • Holding period return = (1 + 0.06)(1 – 0.37)(1 + 0.27) – 1 = -15.2%
  • Arithmetic mean return = (6% - 37% + 27%) / 3 = -1.33%.

• An investor buys a share of stock for $100.

• At the end of year one, she buys three more shares at $89 per share.

• At the end of year two, she sells all four shares for $98 each.

• The stock paid a dividend of $1.00 per share at the end of year one and year two.

• The investor's money-weighted rate of return is 6.35%.

• A stated interest rate of 9% compounded continuously results in an effective annual rate closest to 9.42%.

  • The effective annual rate with continuous compounding = e^r – 1 = e^0.09 – 1 = 0.09417, or 9.42%.

• A security portfolio earns a gross return of 7.0% and a net return of 6.5%.

• The difference of 0.5% most likely results from fees.

• When a stock increases in value, the holding period return will be greater than its continuously compounded return.

• The components of the required interest rate on a security are the real risk-free rate, the expected inflation rate, the default risk premium, a liquidity premium, and a premium to reflect the risk associated with the maturity of the security.

• An investor buys one share of stock for $100. At the end of year one, she buys three more shares at $89 per share.

• At the end of year two, she sells all four shares for $98 each.

• The stock paid a dividend of $1.00 per share at the end of year one and year two.

• The investor's time-weighted rate of return is 0.06%.

• A client that invests $2,000 each month into a blue-chip stock, based on advice, regarding dollar cost averaging.

• The stock price on the date of purchase each month over a four-month stretch was $12, $14, $11, and $9.

• Using the harmonic mean, the average cost per share of the stock is closest to $11.20.

• Over the last four years, an investor's portfolio has the following returns: 5.26%, -2.10%, 3.86%, and 8.18%.

• The arithmetic mean return is closest to 3.80%.

  • The arithmetic mean is equal to the average of the four data points, calculated by summing all four returns and dividing by the number of returns:[(R1 + R2 + R3 + R4) / 4] [(0.0526 – 0.0210 + 0.0386 + 0.0818) / 4] = .0380 or 3.80%

• A dataset contains six values, none of which are equal.

• The arithmetic mean of the data is 13.25, and the geometric mean of the data is 12.75.

• The harmonic mean will be less than 12.75.

• The product of the arithmetic mean and the harmonic mean is the square of the geometric mean.

• The real risk-free interest rate is best described as purely representing time preference.

• A bond was purchased a year ago for $910 and was sold today for $1,020.

• During the year, the bond made two semi-annual coupon payments of $30.

• The holding period return is 18.7%.

  • HPY = (1,020 + 30 + 30 - 910) / 910 = 0.1868 or 18.7%.

• An investor sold a 30-year bond at a price of $850 after purchasing it at $800 a year ago.

• They received $50 of interest at the time of the sale.

• The annualized holding period return is 12.5%.

• A stock is currently worth $75.

• If the stock was purchased one year ago for $60 and the stock paid a $1.50 dividend during the year, the holding period return is 27.5%.

• T-bill yields can be thought of as nominal risk-free rates because they contain an inflation premium.

• A given a holding period return of R, the continuously compounded rate of return is ln(1 + R).

• An investor bought a stock for $32 and sold it nine months later for $37.50 after receiving $2 in dividends.

• The holding period return on this investment is 23.44%.

• Assuming at least some variations in a set of data, the arithmetic mean is greater than the geometric mean, which is greater than the harmonic mean.

• An investor buys a share of stock for $200 at time t = 0 and another share for $225 at time t = 1.

• At time t = 2 the investor sells both shares for $235.

• During both years, the stock paid a per share dividend of $5.

• The approximate time-weighted and money-weighted returns are, respectively, 10.8%; 9.4%.

• An analyst evaluates a dataset with eight values.

• The geometric mean calculated is 8.50, and the arithmetic mean is 8.90.

• The harmonic mean is closest to 8.12. [(8.50)^2 / 8.90] = 8.12

• On January 1, Jonathan Wood invests $50,000.

• At the end of March, the investment is worth $51,000.

• On April 1, Wood deposits $10,000, and by the end of June, the account is worth $60,000.

• Wood withdraws $30,000 on July 1 and makes no more transactions.

• At the end of the year, the account is worth $33,000.

• The time-weighted return is closest to 10.4%.

• [(1 + 0.02)(1 – 0.0164)(1 + 0.10)] - 1 = 0.1036 or 10.36%

• An investor with a buy-and-hold strategy who makes quarterly deposits into an account should most appropriately evaluate portfolio performance using the geometric mean return.

• The most appropriate measure of the increase in the purchasing power of a portfolio's value over a given span of time is the real return.