Treasury Bill Discount Yield vs Investment Yield

A treasury bill discount yield vs investment yield confusion stems from the fact that T-bills are quoted in two different ways in the market: the bank-discount yield (used by dealers and the Federal Reserve) and the investment yield or bond-equivalent yield (the actual return an investor realizes). The two figures differ because they use different day-count conventions and compound differently, and knowing how to convert between them is essential for comparing money-market investments.

Why Two Yield Figures Exist

Treasury bills are sold at a discount to par value—if you buy a $100 bill maturing in 90 days, you might pay $98.50. The $1.50 difference is your interest income.

The problem arises because there are two natural ways to express that return:

1. Bank-discount yield: Annualizes the discount as a percentage of par value, using a 360-day year.
2. Investment yield (bond-equivalent yield): Annualizes the actual dollar gain as a percentage of the purchase price, using a 365-day year.

The Federal Reserve and T-bill dealers have historically quoted bank-discount yield because it simplifies calculations in the primary market. But investors who buy T-bills in secondary markets or hold them to maturity care about investment yield—the true annual return on their money.

Bank-Discount Yield Formula

The bank-discount yield is calculated as:

$$\text{BDY} = \left( \frac{\text{Discount}}{\text{Par Value}} \right) \times \left( \frac{360}{d} \right) \times 100%$$

Where:

• Discount = Par value minus purchase price
• Par Value = $100 (or any denomination)
• d = days to maturity

Example: A 90-day T-bill with par value of $100 sells for $98.50.

• Discount = $100 − $98.50 = $1.50
• BDY = ($1.50 ÷ $100) × (360 ÷ 90) × 100% = 1.5% × 4 = 6.0%

Investment Yield (Bond-Equivalent Yield) Formula

The investment yield is:

$$\text{IY} = \left( \frac{\text{Discount}}{\text{Purchase Price}} \right) \times \left( \frac{365}{d} \right) \times 100%$$

Where:

• Discount = Par value minus purchase price
• Purchase Price = what you actually paid
• d = days to maturity

Same example: 90-day T-bill, par $100, purchase price $98.50.

• Discount = $1.50
• IY = ($1.50 ÷ $98.50) × (365 ÷ 90) × 100% = 1.523% × 4.056 ≈ 6.18%

Notice the investment yield is higher than the bank-discount yield. This reflects two differences:

1. Day-count: 365 vs 360 days inflates the annualization factor.
2. Base: Dividing by purchase price ($98.50) rather than par ($100) increases the percentage return.

Why These Conventions Matter

When a dealer quotes a T-bill at “6.00%,” they typically mean bank-discount yield. If you want to compare that T-bill to a money-market fund that quotes its yield as investment yield, or to a certificate of deposit that compounds on a 365-day basis, you must convert both to the same basis.

The conversion allows apples-to-apples yield comparison across different money-market instruments—commercial paper, banker’s acceptances, and repurchase agreements all use different quoting conventions.

Conversion Between the Two Yields

To convert bank-discount yield to investment yield:

$$\text{IY} = \text{BDY} \times \frac{365}{360 - (\text{BDY} \times d)}$$

Where d is days to maturity.

Using the 6.00% BDY example (90 days):

• IY = 6.00% × 365 ÷ (360 − (6.00% × 90))
• IY = 6.00% × 365 ÷ (360 − 5.4)
• IY = 6.00% × 365 ÷ 354.6
• IY ≈ 6.18%

This matches our direct calculation above.

To go the other direction (investment yield to bank-discount yield):

$$\text{BDY} = \frac{\text{IY} \times 360}{365 + (\text{IY} \times d)}$$

For the 6.18% IY (90 days):

• BDY = 6.18% × 360 ÷ (365 + (6.18% × 90))
• BDY = 6.18% × 360 ÷ (365 + 5.56)
• BDY = 6.18% × 360 ÷ 370.56
• BDY ≈ 6.00%

Which Yield Do You Use?

For comparing T-bills to each other: Use bank-discount yield. Dealers and the Federal Reserve quote this, and it is the standard in primary-market auctions.

For comparing T-bills to other investments: Convert to investment yield (bond-equivalent yield). If you are evaluating whether a 90-day T-bill or a money-market fund offers a better return, both must be on a 365-day, purchase-price basis.

For actual returns: Investment yield is closer to what you will actually earn, because it uses your actual purchase price as the denominator and the calendar year (365 days) for annualization.

The Fed Funds Rate Connection

The Federal Reserve also manages short-term interest rates through open-market operations, which often involve T-bills and repurchase agreements. The Fed typically quotes its target fed funds rate and treasury bill auction results in bank-discount terms, which is why understanding both conventions is important for interpreting central-bank communication.

Practical Takeaway

Money-market investors should always ask: “Is that yield on a bank-discount or investment basis?” The difference is small for very short maturities (a few days) but can exceed 20 basis points for T-bills maturing in 180+ days. When yields are thinly quoted and competition for returns is fierce, that gap can meaningfully affect the choice between instruments.

See also