Discount yield

A Treasury bill pays no coupon. You buy it below par and collect Rs 100 at maturity, and the gap is your return. The only question is how to annualise that gap, and there are two answers.

Bond equivalent yield divides the gain by what you actually paid, and annualises on a 365-day year. It is the honest return.

Discount yield divides the same gain by the face value instead, and annualises on a 360-day year. It is not the honest return — it is smaller, and deliberately so, because it exists to make quoting and settling simple rather than to measure performance.

The reason it matters is that India's 91-day T-Bill futures contract is built on it. The contract is quoted as 100 minus the discount yield, and its final settlement value is computed from the discount yield at the RBI auction on expiry day. Get the convention wrong and every number in the contract is wrong.

The difference between the two measures is entirely in the denominator and the year.

Bond equivalent yield : gain ÷ PRICE,      annualised on 365
Discount yield        : gain ÷ FACE VALUE, annualised on 360

Since price is always below face value on a discount instrument, dividing by the larger number gives the smaller yield. Discount yield therefore always sits below the bond equivalent yield on the same bill, and the gap widens as the discount deepens.

The futures contract wraps that into a price quote. If the discount yield is 5%, the futures quote is 100 − 5 = 95.0000. Because a 91-day bill is roughly a quarter of a year, the contract value uses one quarter of the yield:

Contract value = 2,000 × (100 − 0.25 × y)

which is why a tick is worth so little. The minimum price move is Rs 0.0025, but the 0.25 factor scales it: 2,000 × 0.0025 × 0.25 = Rs 1.25 per lot, against Rs 5 on a bond futures lot.

                  (Face value − Price)        360
Discount yield = ───────────────────── × ────────────────────
                     Face value           Days to maturity


                     (Face value − Price)        365
Compare, BEY   =  ───────────────────────── × ────────────────────
                          Price                Days to maturity

And the contract built on it:

91-day T-Bill futures price = 100 − y
Contract value             = 2,000 × (100 − 0.25 × y)
Final settlement value     = Rs 2,000 × (100 − 0.25 × y)
     y = weighted average discount yield at the RBI 91-day auction on expiry day

A 91-day Treasury bill is trading at Rs 98.75 with the full 91 days to run.

Discount yield = (100 − 98.75) ÷ 100 × 360/91
               = 0.0125 × 3.956
               = 4.9451%

Bond equivalent = (100 − 98.75) ÷ 98.75 × 365/91
                = 0.012658 × 4.011
                = 5.0772%

The same bill, the same 91 days, 13 basis points apart — purely because one measure divides by 100 and the other by 98.75, and one year has 360 days and the other 365.

Now trade it. A treasurer expecting short rates to rise sells 20 lots of 91-day T-Bill futures at the workbook's quotation of Rs 95.00, which is a discount yield of 5.00%:

Contract value = 2,000 × (100 − 0.25 × 5) × 20 lots
               = 2,000 × 98.75 × 20
               = Rs 39,50,000

The RBI auction on expiry day prints a weighted average discount yield of 5.60%, so the contract settles at 100 − 5.60 = Rs 94.40.

Settlement value = 2,000 × (100 − 0.25 × 5.60) × 20 = Rs 39,44,000
Gain to the short                                   = Rs     6,000

Sixty basis points of yield on Rs 40 lakh of notional produced Rs 6,000 — which is exactly 60 ticks × Rs 1.25 × 20 lots. The quarter-year factor is doing all the work, and a candidate who forgets it will report a figure four times too large.

Chapter 1, section 1.10.2.3 (Yield for Money Market), introduces discount yield immediately after the bond equivalent yield and says plainly that it "is important for trading in 91-day T-Bills Futures". Chapter 3 then uses it throughout the 91-day T-Bill futures contract specification — quotation, contract value, daily settlement value and final settlement value — and section 3.4.2 works the lot-size arithmetic.

Questions ask you to compute a discount yield from price and days, to distinguish it from bond equivalent yield, to convert a discount yield into a futures quote, and to compute the contract value or tick value of a T-Bill futures lot.

• Divide by face value, not by price. That single substitution is the whole difference from bond equivalent yield, and it is the most common error on this topic.
• Use 360 days, not 365. Discount yield deliberately uses a 30-day month and 360-day year; the Indian money market's bond equivalent yield uses 365.
• Discount yield always understates the true return. It is never the higher of the two measures on the same instrument.
• The futures quote is 100 minus the yield, so a rising rate means a falling quote — a short position profits when short rates rise.
• Do not drop the 0.25 in the contract value. 2,000 × (100 − 0.25y) reflects the 91-day tenor; using 2,000 × (100 − y) inflates every answer.
• Tick value differs by contract. Rs 1.25 on a T-Bill futures lot, Rs 5 on a bond futures lot, Rs 102.75 on an Overnight MIBOR contract — same Rs 0.0025 tick size, three different rupee values.