3.2 Money Market Yield Calculations, Compounding & Valuation

3.2 Money Market Yield Calculations, Compounding & Valuation

Short-term money market instruments are quoted across different day-count conventions and pricing mechanisms. A Treasury Bill quoted at a 4.80% discount rate does not yield 4.80% in annual investment return. To make sound capital allocation decisions, corporate treasurers must master the mathematical relationships between Discount Rate (DR), Money Market Yield (MMY), Bond Equivalent Yield (BEY), and Effective Annual Yield (EAY).

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1. Day Count Conventions in Fixed Income

Before computing yields, treasurers must identify the underlying day count convention governing the instrument:

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2. Core Money Market Yield Formulas

+-----------------------------------------------------------------------------+
|                   MONEY MARKET FORMULA CHEAT SHEET                          |
|                                                                             |
|   1. Dollar Discount ($D):                                                  |
|      D = Face Value - Purchase Price = Face Value * DR * (Days / 360)       |
|                                                                             |
|   2. Bank Discount Yield (Discount Rate - DR):                              |
|      DR = (D / Face Value) * (360 / Days to Maturity)                       |
|                                                                             |
|   3. Money Market Yield (CD Equivalent Yield - MMY):                        |
|      MMY = (D / Purchase Price) * (360 / Days to Maturity)                  |
|          = (360 * DR) / [ 360 - (Days to Maturity * DR) ]                   |
|                                                                             |
|   4. Bond Equivalent Yield (Investment Yield - BEY):                        |
|      BEY = (D / Purchase Price) * (365 / Days to Maturity)                  |
|          = MMY * (365 / 360)                                                |
|          = (365 * DR) / [ 360 - (Days to Maturity * DR) ]                   |
|                                                                             |
|   5. Effective Annual Yield (Compounded - EAY):                             |
|      EAY = [ 1 + (Face Value - Price) / Price ] ^ (365 / Days) - 1          |
|          = (1 + Holding Period Return) ^ (365 / Days) - 1                   |
|                                                                             |
|   6. Tax-Equivalent Yield (TEY):                                            |
|      TEY = Tax-Exempt Municipal Yield / (1 - Marginal Tax Rate)             |
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Detailed Formula Analysis:

1. Bank Discount Yield (Discount Rate / BDY)

Discount securities (T-Bills, CP, BAs) are issued at a price below their par/face value and mature at par. The dealer quotes the Discount Rate ($DR$):

$DR = \frac{\text{Face Value} - \text{Purchase Price}}{\text{Face Value}} \times \frac{360}{\text{Days to Maturity}}$

Given a quoted $DR$, the Purchase Price ($P$) is determined by:

$P = \text{Face Value} \times \left[ 1 - \left( \frac{\text{Days to Maturity} \times DR}{360} \right) \right]$

[!WARNING] Flaws of the Discount Rate:

1. Denominator Error: It divides dollar earnings by Face Value rather than the cash Purchase Price actually paid by the enterprise.
2. Artificial Year: It uses 360 days instead of 365 days. Result: The quoted Discount Rate systematically understates the true annualized return earned on invested capital.

2. Money Market Yield (CD Equivalent Yield / MMY)

The Money Market Yield ($MMY$) corrects the denominator flaw by dividing the dollar earnings by the cash Purchase Price, while keeping the money market 360-day baseline:

$MMY = \frac{\text{Face Value} - \text{Purchase Price}}{\text{Purchase Price}} \times \frac{360}{\text{Days to Maturity}}$

To convert a quoted Discount Rate ($DR$) directly to $MMY$ without computing dollar prices:

$MMY = \frac{360 \times DR}{360 - (\text{Days to Maturity} \times DR)}$

3. Bond Equivalent Yield (Investment Yield / BEY)

To compare a short-term discount instrument with coupon-bearing Treasury notes or corporate bonds that trade on a 365-day year, treasurers compute the Bond Equivalent Yield ($BEY$):

$BEY = \frac{\text{Face Value} - \text{Purchase Price}}{\text{Purchase Price}} \times \frac{365}{\text{Days to Maturity}}$

Direct conversions:

$BEY = MMY \times \frac{365}{360} = \frac{365 \times DR}{360 - (\text{Days to Maturity} \times DR)}$

4. Effective Annual Yield (EAY)

Neither $MMY$ nor $BEY$ accounts for the compounding of interest during the year. The Effective Annual Yield ($EAY$) reflects the true annualized compound return assuming that proceeds are reinvested at the same rate across all holding periods in a 365-day year:

$\text{Holding Period Return (HPR)} = \frac{\text{Face Value} - \text{Purchase Price}}{\text{Purchase Price}}$

$EAY = (1 + \text{HPR})^{\frac{365}{\text{Days to Maturity}}} - 1$

The Immutable Yield Hierarchy:

For any short-term instrument purchased at a discount with maturity under one year:

$\mathbf{EAY > BEY > MMY > DR}$

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3. Comprehensive Step-by-Step Worked Scenarios

Worked Example 1: Full Multi-Yield Conversion on a 90-Day T-Bill

Scenario: A corporate treasury department purchases a $10,000,000 face value, 90-day Treasury Bill quoted at a Bank Discount Rate ($DR$) of 4.80%.

Step 1: Calculate Total Dollar Discount ($D$)

$D = \text{Face Value} \times DR \times \frac{\text{Days}}{360} = \$10,000,000 \times 0.0480 \times \frac{90}{360} = \$10,000,000 \times 0.0120 = \$120,000$

Step 2: Calculate Cash Purchase Price ($P$)

$P = \text{Face Value} - D = \$10,000,000 - \$120,000 = \$9,880,000$

Step 3: Calculate Money Market Yield ($MMY$)

$MMY = \frac{\$120,000}{\$9,880,000} \times \frac{360}{90} = 0.01214575 \times 4 = \mathbf{4.8583\%}$ Direct Verification Formula Check: $MMY = \frac{360 \times 0.0480}{360 - (90 \times 0.0480)} = \frac{17.28}{360 - 4.32} = \frac{17.28}{355.68} = \mathbf{4.8583\%}$

Step 4: Calculate Bond Equivalent Yield ($BEY$)

$BEY = \frac{\$120,000}{\$9,880,000} \times \frac{365}{90} = 0.01214575 \times 4.055556 = \mathbf{4.9258\%}$ Direct Conversion from MMY: $BEY = 4.8583\% \times \frac{365}{360} = \mathbf{4.9258\%}$

Step 5: Calculate Effective Annual Yield ($EAY$)

$\text{HPR} = \frac{\$120,000}{\$9,880,000} = 0.012145749$ $EAY = (1 + 0.012145749)^{\frac{365}{90}} - 1 = (1.012145749)^{4.0555556} - 1 = 1.050183 - 1 = \mathbf{5.0183\%}$

Comparative Yield Summary Table:

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Worked Example 2: Tax-Equivalent Yield (TEY) Analysis

Scenario: A corporate treasurer in a company subject to a 21% Federal corporate income tax rate and a 4% State corporate tax rate (effective combined marginal tax rate = 25%) is evaluating two alternative 180-day investments:

• Option A: A tax-exempt Municipal Variable Rate Demand Note (VRDN) yielding 3.45%.
• Option B: A taxable Tier-1 Commercial Paper yielding a BEY of 4.50%.

Step 1: Calculate Tax-Equivalent Yield ($TEY$) for Option A

$TEY = \frac{\text{Tax-Exempt Yield}}{1 - \text{Marginal Tax Rate}} = \frac{3.45\%}{1 - 0.25} = \frac{3.45\%}{0.75} = \mathbf{4.6000\%}$

Step 2: Calculate After-Tax Yield of Option B (Taxable CP)

$\text{After-Tax Yield} = \text{Taxable Yield} \times (1 - t) = 4.50\% \times (1 - 0.25) = 4.50\% \times 0.75 = \mathbf{3.3750\%}$

Decision Analysis:

The municipal note offers a Tax-Equivalent Yield of 4.60%, which exceeds the taxable Commercial Paper yield of 4.50%. Correspondingly, the municipal note delivers an after-tax net yield of 3.45% versus only 3.38% for the commercial paper. The treasurer should select Option A (Municipal VRDN).

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Worked Example 3: Overnight Repurchase Agreement Interest

Scenario: Treasury invests $50,000,000 in an overnight reverse repo with a primary dealer at a Repo Rate of 5.30% (Actual/360) against U.S. Treasury collateral requiring a 2% haircut.

Step 1: Calculate Overnight Interest Earned

$\text{Interest} = \text{Principal} \times \text{Repo Rate} \times \frac{1}{360} = \$50,000,000 \times 0.0530 \times \frac{1}{360} = \$7,361.11$ $\text{Settlement Repurchase Amount} = \$50,000,000 + \$7,361.11 = \mathbf{\$50,007,361.11}$

Step 2: Determine Required Collateral Market Value

$\text{Required Collateral} = \frac{\text{Cash Loan Amount}}{1 - \text{Haircut}} = \frac{\$50,000,000}{1 - 0.02} = \frac{\$50,000,000}{0.98} = \mathbf{\$51,020,408.16}$