Quantitative Methods - Calculating Yield

Calculating Yield for a U.S. Treasury Bill
A U.S. Treasury bill is the classic example of a pure discount instrument, where the interest the government pays is the difference between the amount it promises to pay back at maturity (the face value) and the amount it borrowed when issuing the T-bill (the discount). T-bills are short-term debt instruments (by definition, they have less than one year to maturity), and there is zero default risk with a U.S. government guarantee. After being issued, T-bills are widely traded in the secondary market, and are quoted based on the bank discount yield (i.e. the approximate annualized return the buyer should expect if holding until maturity). A bank discount yield (RBD) can be computed as follows:

Formula 2.10
RBD = D/F * 360/t

Where: D = dollar discount from face value, F = face value,
T = days until maturity, 360 = days in a year

By bank convention, years are 360 days long, not 365. If you recall the joke about banker's hours being shorter than regular business hours, you should remember that banker's years are also shorter.

For example, if a T-bill has a face value of $50,000, a current market price of $49,700 and a maturity in 100 days, we have:

RBD = D/F * 360/t = ($50,000-$49,700)/$50000 * 360/100 = 300/50000 * 3.6 = 2.16%

On the exam, you may be asked to compute the market price, given a quoted yield, which can be accomplish by using the same formula and solving for D:

Formula 2.11
D = RBD*F * t/360

Example:
Using the previous example, if we have a bank discount yield of 2.16%, a face value of $50,000 and days to maturity of 100, then we calculate D as follows:

D = (0.0216)*(50000)*(100/360) = 300

Market price = F - D = 50,000 - 300 = $49,700

Holding-Period Yield (HPY)
HPY refers to the un-annualized rate of return one receives for holding a debt instrument until maturity. The formula is essentially the same as the concept of holding-period return needed to compute time-weighted performance. The HPY computation provides for one cash distribution or interest payment to be made at the time of maturity, a term that can be omitted for U.S. T-bills.

Formula 2.12
HPY = (P1 - P0 + D1)/P0
Where: P0 = purchase price, P1 = price at maturity, and D1= cash distribution at maturity

Example:
Taking the data from the previous example, we illustrate the calculation of HPY:

HPY = (P1 - P0 + D1)/P0 = (50000 - 49700 + 0)/49700 = 300/49700 = 0.006036 or 0.6036%

Effective annual yield (EAY)
EAY takes the HPY and annualizes the number to facilitate comparability with other investments. It uses the same logic presented earlier when describing how to annualize a compounded return number: (1) add 1 to the HPY return, (2) compound forward to one year by carrying to the 365/t power, where t is days to maturity, and (3) subtract 1.

Here it is expressed as a formula:

Formula 2.13
EAY = (1 + HPY)365/t - 1

Example:
Continuing with our example T-bill, we have:

EAY = (1 + HPY)365/t - 1 = (1 + 0.006036)365/100 - 1 = 2.22 percent.

Remember that EAY > bank discount yield, for three reasons: (a) yield is based on purchase price, not face value, (b) it is annualized with compound interest (interest on interest), not simple interest, and (c) it is based on a 365-day year rather than 360 days. Be prepared to compare these two measures of yield and use these three reasons to explain why EAY is preferable.

The third measure of yield is the money market yield, also known as the CD equivalent yield, and is denoted by rMM. This yield measure can be calculated in two ways:

1. When the HPY is given, rMM is the annualized yield based on a 360-day year:

Formula 2.14
rMM = (HPY)*(360/t)
Where: t = days to maturity

For our example, we computed HPY = 0.6036%, thus the money market yield is:

rMM = (HPY)*(360/t) = (0.6036)*(360/100) = 2.173%.

2. When bond price is unknown, bank discount yield can be used to compute the money market yield, using this expression:

Formula 2.15

rMM = (360* rBD)/(360 - (t* rBD)

Using our case:

rMM = (360* rBD)/(360 - (t* rBD) = (360*0.0216)/(360 - (100*0.0216)) = 2.1735%, which is identical to the result at which we arrived using HPY.

Interpreting Yield
This involves essentially nothing more than algebra: solve for the unknown and plug in the known quantities. You must be able to use these formulas to find yields expressed one way when the provided yield number is expressed another way.

Since HPY is common to the two others (EAY and MM yield), know how to solve for HPY to answer a question.

Effective Annual Yield EAY = (1 + HPY)365/t - 1 HPY = (1 + EAY)t/365 - 1
Money Market Yield rMM = (HPY)*(360/t) HPY = rMM * (t/360)

Bond Equivalent Yield
The bond equivalent yield is simply the yield stated on a semiannual basis multiplied by 2. Thus, if you are given a semiannual yield of 3% and asked for the bond equivalent yield, the answer is 6%.

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