T-Bill Yields 

T-bills are purchased by investors at a weekly auction at less than face
value and are redeemed at maturity at face value. The difference between the
purchase price and the face value of the T-bill is the investor's return. 

The investor's return is used in mathematical formulas to determine the yield
on T-bills. One formula, the discount yield method, takes into account the
return as a percent of the face value of a T-bill, rather than its purchase
price. Since the purchase price is typically less than face value, the
discount method tends to understate the yield. An alternative formula, called
the investment yield method, also can be used to calculate the yield. Unlike
the discount yield formula, the investment yield method relates the investor's
return to the purchase price of the bill. 

Both the discount yield and the investment yield, as well as the high, low and
average prices of the auctioned T-bills, are made public in an official
Treasury report shortly after the auction. 

The Treasury uses the discount and investment formulas for calculating yields
on all T-bills, except the one-year bill. Yields reported by the Treasury are
precise to several decimal places. 

The Discount Yield Method 
The following formula is used to determine the discount yield for T-bills that
have three- or six-month maturities: 

Discount yield = [FV - PP/FV] * [360/M] 

FV = face value
PP = purchase price
M = maturity of bill. For a three-month T-bill (13 weeks) use 91, and for a
    six-month T-bill (26 weeks) use 182
360 = the number of days used by banks to determine short-term interest rates
      (the investment yield method is based on a calendar year: 365 days or
      366 in leap years). 

                                 Example 

What is the discount yield for a 182-day T-bill, auctioned at an average price
of $9,659.30 per $10,000 face value? 

Discount yield = [(FV - PP)/FV] * [360/M]

FV = $10,000        PP = $9,659.30           M = 182


Discount yield = [((10,000) - (9,659.30)) / (10,000)] * [360/182]
Discount yield = [340.7 / 10,000] * [1.978022]
Discount yield = .0673912 = 6.74%

For the 13-week bill, the same formula would be used, dividing 360 by a
maturity of 91 days rather than 182 days. 

The Investment Yield Method 
When comparing the return on investment in T-bills to other short-term
investment options, the investment yield method can be used. This yield is
alternatively called the bond equivalent yield, the coupon equivalent rate,
the effective yield and the interest yield. 

The following formula is used to calculate the investment yield for T-bills
that have three- or six-month maturities: 

Investment yield = [FV - PP/PP] * [365 or 366/M] 

                                 Example 

What is the investment yield of a 182-day T-bill, auctioned at an average
price of $9,659.30 per $10,000 face value? 

Investment yield =  [FV - PP/PP]  * [365/M]

FV = $10,000        PP = $9,659.30            M = 182

Investment yield = [(10,000 - 9,659.30) / (9,659.30)] * [365/182]
Investment yield = [340.70] / 9,659.30] * [2.0054945]
Investment yield = .0707372 = 7.07%

For the 13-week bill, the same formula can be used, dividing 365 (or 366) by a
maturity of 91 days. 

Yields on Treasury Notes and Bonds 
Treasury notes and bonds, fully-backed U.S. debt instruments with maturities
of more than one year, pay the investor a fixed annual rate of return or
coupon (paid semi-annually). The return on a Treasury note or bond is equal to
its face value times the coupon interest rate. 

Formulas used by Treasury to calculate the investment yield on notes and bonds
are complicated and vary depending on the maturity of the issue. 

However, the investment yield on a bond or note held to maturity can be
approximated with the following formula: 

                                    
                    {R +  [(FV - PP)/M]}
Investment yield =  --------------------   
                       [(FV + PP)/2]

R  =  coupon rate              FV =  face value
PP =  purchase price           M  =  years to maturity 

                                 Example 

What is the investment yield of a seven-year Treasury note issued at a price
of $99.709, with an annual Treasury announced coupon of 7 7/8, payable semi-
annually? 

R = 7  7/8  (7.875)  FV = $100   PP = $99.709   M = 7                     

                    7.875 + [(100 - 99.709)/7]
Investment yield =  --------------------------   
                         (100 + 99.709)/2

Investment yield = (7.875 + .0415714) / (99.8545)
Investment yield = 7.9165714 / 99.8545
Investment yield = .0792810 = 7.93%

Source: New York Fed 
