on 15-Jul-2017 (Sat)

When analyzing rates of return, our starting point is the total return, or holding period return (HPR). HPR measures the total return for holding an investment over a certain period of time, and can be calculated using the following formula:

• P_t = price per share at the end of time period t
• P_(t-1) = price per share at the end of time period t-1, the time period immediately preceding time period t
• P_t - P_t-1 = price appreciation of the investment
• D_t = cash distributions received during time period t: for common stock, cash distribution is the dividend; for bonds, cash distribution is the coupon payment.

It has two important characteristics:

• It has an element of time attached to it: monthly, quarterly or annual returns. HPR can be computed for any time period.
• It has no currency unit attached to it; the result holds regardless of the currency in which prices are denominated.

Example

A stock is currently worth $60. If you purchased the stock exactly one year ago for $50 and received a $2 dividend over the course of the year, what is your holding period return?

R_t = ($60 - $50 + $2)/$50 = 0.24 or 24%

The return for time period t is the capital gain (or loss) plus distributions divided by the beginning-of-period price (dividend yield). Note that for common stocks the distribution is the dividend; for bonds, the distribution is the coupon payment.

The holding period return for any asset can be calculated for any time period (day, week, month, or year) simply by changing the interpretation of the time interval.

Return can be expressed in decimals (0.05), fractions (5/100), or as a percent (5%). These are all equivalent.

The dollar-weighted rate of return is essentially the internal rate of return (IRR) on a portfolio. This approach considers the timing and amountof cash flows. It is affected by the timing of cash flows. If funds are added to a portfolio when the portfolio is performing well (poorly), the dollar-weighted rate of return will be inflated (depressed).

The time-weighted rate of return measures the compound growth rate of $1 initial investment over the measurement period. Time-weightedmeans that returns are averaged over time. This approach is not affected by the timing of cash flows; therefore, it is the preferred method of performance measurement.

Example

Jayson bought a share of IBM stock for $100 on December 31, 2000. On December 31, 2001, he bought another share for $150. On December 31, 2002, he sold both shares for $140 each. The stock paid a dividend of $10 per share at the end of each year.

To calculate the dollar-weighted rate of return, you need to determine the timing and amount of cash flows for each year, and then set the present value of net cash flows to be 0: - 100 - 140/(1 + r) + 300/(1 + r)² = 0. You can use the IRR function on a financial calculator to solve for r to get the dollar-weighted rate of return: r = 17%.

To calculate the time-weighted rate of return:

• Split the overall measurement period into equal sub-periods on the dates of cash flows. For the first year:
  • beginning price: $100
  • dividends: $10
  • ending price: $150
  For the second year:
  • beginning price: $300 (150 x 2)
  • dividends: $20 (10 x 2)
  • ending price: $280 (140 x 2)

• Calculate the holding period return (HPR) on the portfolio for each sub-period: HPR = (Dividends + Ending Price)/Beginning Price - 1. For the first year, HPR₁: (150 + 10)/100 - 1 = 0.60. For the second year, HPR₂: (280 + 20)/300 - 1 = 0.

• Calculate the time-weighted rate of return:
  • If the measurement period < 1 year, compound holding period returns to get an annualized rate of return for the year.
  • If the measurement period > 1 year, take the geometric mean of the annual returns.

    

Money market instruments are low-risk, highly liquid debt instruments with a maturity of one year or less. There are two types of money market instruments: interest-bearing instruments (e.g., bank certificates of deposit), and pure discount instruments (e.g., U.S. Treasury bills).

Pure discount instruments such as T-bills are quoted differently than U.S. government bonds. They are quoted on a bank discount basis rather than on a price basis:

• r_BD = the annualized yield on a bank discount basis
• D = the dollar discount, which is equal to the difference between the face value of the bill, F, and its purchase price, P
• t = the number of days remaining to maturity
• 360 = the bank convention of the number of days in a year.

Bank discount yield is not a meaningful measure of the return on investment because:

• It is based on the face value, not on the purchase price. Instead, return on investment should be measured based on cost of investment.
• It is annualized using a 360-day year, not a 365-day year.
• It annualizes with simple interest and ignores the effect of interest on interest (compound interest).

Holding period yield (HPY) is the return earned by an investor if the money market instrument is held until maturity:

• P₀ = the initial price of the instrument
• P₁ = the price received for the instrument at its maturity
• D₁ = the cash distribution paid by the instrument at its maturity (that is, interest).

Since a pure discount instrument (e.g., a T-bill) makes no interest payment, its HPY is (P₁ - P₀)/P₀.

Note that HPY is computed on the basis of purchase price, not face value. It is not an annualized yield.

The effective annual yield is the annualized HPY on the basis of a 365-day year. It incorporates the effect of compounding interest.

Money market yield (also known as CD equivalent yield) is the annualized HPY on the basis of a 360-day year using simple interest.

Example

An investor buys a $1,000 face-value T-bill due in 60 days at a price of $990.

• Bank discount yield: (1000 - 990)/1000 x 360/60 = 6%
• Holding period yield: (1000 - 990)/990 = 1.0101%
• Effective annual yield: (1 + 1.0101%)^365/60 - 1 = 6.3047%
• Money market yield: (360 x 6%)/(360 - 60 x 6%) = 6.0606%

If we know HPY, then:

• EAY = (1 + HPY)^365/t - 1
• r_MM = HPY x 360/t

If we know EAY, then:

• HPY = ( 1 + EAY)^t/365 - 1
• r_MM = [(1 + EAY)^t/365 - 1] x (360/t)

If we know r_MM, then:

• HPY = r_MM x (t/360)
• EAY = (1 + r_MM x t/360)^365/t - 1

Periodic bond yields for both straight and zero-coupon bonds are conventionally computed based on semi-annual periods, as U.S. bonds typically make two coupon payments per year. For example, a zero-coupon bond with a maturity of five years will mature in 10 6-month periods. The periodic yield for that bond, r, is indicated by the equation Price = Maturity value x (1 + r)^-10. This yield is an internal rate of return with semi-annual compounding. How do we annualize it?

The convention is to double it and call the result the bond's yield to maturity. This method ignores the effect of compounding semi-annual YTM, and the YTM calculated in this way is called a bond-equivalent yield (BEY).

However, yields of a semi-annual-pay and an annual-pay bond cannot be compared directly without conversion. This conversion can be done in one of the two ways:

• Convert the bond-equivalent yield of a semi-annual-pay bond to an annual-pay bond.
  

• Convert the equivalent annual yield of an annual-pay bond to a bond-equivalent yield.
  

Example

• A Eurobond pays coupon annually. It has an annual-pay YTM of 8%.
• A U.S. corporate bond pays coupon semi-annually. It has a bond equivalent YTM of 7.8%.
• Which bond is more attractive, if all other factors are equal?

Solution 1

• Convert the U.S. corporate bond's bond equivalent yield to an annual-pay yield:
• Annual-pay yield = [1 + 0.078/2]² - 1 = 7.95% < 8%
• The Eurobond is more attractive since it offers a higher annual-pay yield.

Solution 2

• Convert the Eurobond's annual-pay yield to a bond equivalent yield (BEY):
• BEY = 2 x [(1 + 0.08)^0.5 - 1] = 7.85% > 7.8%
• The Eurobond is more attractive since it offers a higher bond equivalent yield.

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The money-weighted rate of return has a serious drawback. Generally, clients determine when money is given to the investment manager and those decisions may significantly influence the money-weighted rate of return.