Math on the Series 65 Exam
The Series 65 exam includes approximately 10-15 math-based questions, but they test understanding more than calculation ability.
- Basic calculator provided at the testing center (four functions only)
- Focus on concepts rather than complex calculations
- Know the relationships between yields, risk measures, and returns
Many candidates worry about the math on the Series 65 exam, but here is the truth: you do not need to be a math expert to pass. The exam tests whether you understand financial concepts and can apply them appropriately to client situations.
Most formula questions ask you to interpret results, compare investments, or identify which formula applies to a scenario. When calculations are required, they use simple numbers that work cleanly with a basic calculator.
This guide covers every formula you may encounter on the Series 65, organized by category. For each formula, we explain what it measures, when to use it, and provide exam-focused examples.
With 10-15 math questions representing a meaningful portion of your score, effective formula retention is essential. Our flashcard strategies guide explains how to use FSRS-powered spaced repetition to memorize formulas efficiently. Not through rote repetition, but by understanding when to apply each formula and how to interpret results, which is exactly what the exam tests.
Yield Calculations
Yield measures the income return on an investment. For debt securities, the Series 65 tests three main yield concepts: current yield, yield to maturity, and yield to call.
Current Yield
Example: A $1,000 par bond with a 5% coupon ($50/year) is trading at $900.
Current Yield = $50 ÷ $900 = 5.56%
Current yield ignores time and capital gains/losses. When a bond trades at a discount (below par), current yield is higher than the coupon rate. When it trades at a premium (above par), current yield is lower than the coupon rate.
Yield to Maturity (YTM)
Example: A 10-year, $1,000 par, 4% bond trading at $800.
YTM ≈ [$40 + ($1,000 - $800) ÷ 10] ÷ [($1,000 + $800) ÷ 2]
YTM ≈ [$40 + $20] ÷ $900 = $60 ÷ $900 = 6.67%
YTM is also called “basis” on the exam. It represents the total annualized return assuming the investor holds the bond until maturity and reinvests all coupon payments at the same rate. This is the most complete measure of bond return.
Yield to Call (YTC)
Note: Same formula as YTM, but substitute call price for par value and years to call for years to maturity.
The Bond See-Saw: Yield Relationships
Rather than memorizing formulas, focus on these relationships that the exam frequently tests:
Coupon < Current Yield < YTM < YTC
Yields increase as you move from coupon to YTC.
Coupon = Current Yield = YTM = YTC
All yields equal the coupon rate.
Coupon > Current Yield > YTM > YTC
Yields decrease as you move from coupon to YTC.
These yield relationships are among the most frequently tested concepts on the Series 65. Our flashcard strategies guide provides techniques for memorizing the discount/par/premium patterns using FSRS algorithms, ensuring you can instantly recall which relationship applies when you see a bond trading above or below par on exam day.
Master Yield & Risk Formulas
CertFuel tracks your accuracy on current yield, CAPM, and tax-equivalent yield calculations separately. Our Smart Study algorithm identifies whether you struggle with the formulas themselves or interpreting results, then prioritizes exactly what you need to practice.
Access Free BetaRisk-Adjusted Return Measures
These formulas compare returns relative to the risk taken. They help answer the question: “Is this investment generating enough return for its level of risk?”
CAPM (Capital Asset Pricing Model)
Example: Risk-free rate is 3%, expected market return is 10%, stock beta is 1.5.
Expected Return = 3% + 1.5 × (10% - 3%)
Expected Return = 3% + 1.5 × 7% = 3% + 10.5% = 13.5%
| Component | Meaning |
|---|---|
| Risk-Free Rate (Rf) | Usually the 90-day T-bill rate; return with zero risk |
| Market Return (Rm) | Expected return of the overall market (e.g., S&P 500) |
| Market Risk Premium | Rm - Rf; extra return for taking market risk |
| Beta (β) | Sensitivity to market movements (systematic risk) |
Alpha (Jensen’s Alpha)
Example: CAPM predicts 13.5% return, actual return was 15%.
Alpha = 15% - 13.5% = +1.5% (outperformance)
Investment outperformed expectations for its risk level.
Investment performed exactly as expected for its risk.
Investment underperformed expectations for its risk level.
Sharpe Ratio
Example: Portfolio return 12%, risk-free rate 3%, standard deviation 15%.
Sharpe Ratio = (12% - 3%) ÷ 15% = 9% ÷ 15% = 0.60
Sharpe Ratio uses standard deviation (total risk) in the denominator. It is appropriate for evaluating a portfolio that represents an investor’s entire holdings.
Treynor Ratio
Example: Portfolio return 12%, risk-free rate 3%, beta 1.2.
Treynor Ratio = (12% - 3%) ÷ 1.2 = 9% ÷ 1.2 = 7.5
Treynor Ratio uses beta (systematic risk only). It is appropriate for evaluating a single investment within a diversified portfolio, since unsystematic risk can be diversified away.
Risk Measures
Understanding risk measures is essential for making suitable recommendations and analyzing portfolio performance.
Beta
Beta measures volatility relative to the overall market (systematic risk):
| Beta | Meaning |
|---|---|
| β = 0 | No correlation with market (e.g., T-bills) |
| β = 0.5 | Half as volatile as the market |
| β = 1.0 | Moves exactly with the market |
| β = 1.5 | 50% more volatile than the market |
| β = 2.0 | Twice as volatile as the market |
A portfolio’s beta is the weighted average of the betas of its holdings. If you invest 60% in a stock with beta 1.2 and 40% in a stock with beta 0.8, the portfolio beta is (0.60 × 1.2) + (0.40 × 0.8) = 0.72 + 0.32 = 1.04.
Standard Deviation
Standard deviation measures the dispersion of returns around the mean (total risk/volatility). You will not need to calculate it on the exam, but you must understand what it represents:
| Probability | Range |
|---|---|
| 68% | of returns fall within ±1 standard deviation |
| 95% | of returns fall within ±2 standard deviations |
| 99% | of returns fall within ±3 standard deviations |
A fund has a mean return of 10% and standard deviation of 6%.
- 68% of the time, returns will be between 4% and 16% (10% ± 6%)
- 95% of the time, returns will be between -2% and 22% (10% ± 12%)
Tax Calculations
Tax considerations significantly impact investment returns. These formulas help compare taxable and tax-exempt investments.
Tax-Equivalent Yield
Example: A municipal bond yields 4%. The investor is in the 32% tax bracket.
Tax-Equivalent Yield = 4% ÷ (1 - 0.32) = 4% ÷ 0.68 = 5.88%
This muni bond is equivalent to a taxable bond yielding 5.88%.
The higher the investor’s tax bracket, the more valuable tax-exempt income becomes. A 4% municipal bond is worth more to someone in the 37% bracket (tax-equivalent 6.35%) than to someone in the 22% bracket (tax-equivalent 5.13%).
After-Tax Return
Example: A corporate bond yields 6%. The investor is in the 24% tax bracket.
After-Tax Return = 6% × (1 - 0.24) = 6% × 0.76 = 4.56%
Real Rate of Return
Example: Investment returned 8%, inflation was 3%.
Real Rate of Return = 8% - 3% = 5% purchasing power gain
Mutual Fund Math
Mutual fund calculations focus on pricing, sales charges, and the relationship between NAV and public offering price.
Net Asset Value (NAV)
Example: A fund has $500 million in assets, $20 million in liabilities, and 24 million shares.
NAV = ($500M - $20M) ÷ 24M = $480M ÷ 24M = $20.00 per share
NAV is calculated at the end of each trading day (4:00 PM ET). Orders placed before 4:00 PM receive that day’s NAV; orders placed after receive the next day’s NAV.
Public Offering Price (POP)
Example: NAV is $20.00, sales charge (load) is 5%.
POP = $20.00 ÷ (1 - 0.05) = $20.00 ÷ 0.95 = $21.05
Sales Charge Percentage
Example: POP is $21.05, NAV is $20.00.
Sales Charge % = ($21.05 - $20.00) ÷ $21.05 = 4.99% ≈ 5%
The maximum allowable sales charge for mutual funds is 8.5% of POP. However, most funds charge less, typically 3-5.75% for equity funds.
Expense Ratio
A lower expense ratio means more of your return stays invested. Index funds typically have expense ratios under 0.20%, while actively managed funds may charge 0.50-1.50%.
Time Value of Money
Time value of money concepts help investors understand how money grows over time and make comparisons between present and future values.
Rule of 72
Example: At 8% annual return:
Years to Double = 72 ÷ 8 = 9 years
Rule of 72 Quick Reference
| Annual Return | Years to Double |
|---|---|
| 4% | 18 years |
| 6% | 12 years |
| 8% | 9 years |
| 9% | 8 years |
| 10% | 7.2 years |
| 12% | 6 years |
If you know the time to double, you can find the return: 72 ÷ Years = Return. If an investment quadruples in 20 years (doubles twice in 10 years each), the implied return is 72 ÷ 10 = 7.2%.
Compound Interest
Example: $10,000 invested at 6% for 10 years:
FV = $10,000 × (1.06)¹⁰ = $10,000 × 1.791 = $17,910
While you will not need to calculate (1.06)¹⁰ on the exam, understand that compound interest grows faster than simple interest because you earn returns on your returns.
Complete Formula Sheet
Here is a comprehensive reference of all Series 65 formulas organized by category. Bookmark this page for quick review.
Yield Formulas
Risk-Adjusted Returns
Tax Calculations
Mutual Fund Pricing
Time Value of Money
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Access Free BetaExam Strategy for Math Questions
Here is how to approach math-related questions on the Series 65 exam:
What to Focus On
Relationships over calculations
Know that discount bonds have higher YTM than coupon rate
When to use each formula
Sharpe for total portfolios, Treynor for individual securities
Interpretation
Positive alpha = outperformance, higher Sharpe = better risk-adjusted return
Simple calculations
Current yield, tax-equivalent yield, Rule of 72
NAV timing
Orders before 4 PM get that day’s NAV, after 4 PM get next day’s
Common Exam Traps
- Sharpe vs. Treynor: Sharpe uses standard deviation, Treynor uses beta
- YTM vs. Current Yield: YTM includes capital gain/loss, current yield does not
- Tax-equivalent yield: Use the investor’s marginal tax rate, not effective rate
- Beta of 1.0: Does not mean “no risk,” means same risk as the market
- Negative alpha: Does not mean losses, means underperformance vs. Expectations
These formula-specific traps are just a few of the many calculation and concept errors that trip up Series 65 candidates. Our common mistakes guide identifies all top exam failure patterns, including the specific math-related confusions that cause well-prepared candidates to miss questions they should have gotten correct.
Calculator Strategy
The testing center provides a basic four-function calculator. For each math question:
Math questions take longer than concept questions. If you are struggling with a calculation, flag it and move on. You can return to it after completing the easier questions. Do not let one math problem eat into your time budget.
Formula questions require a different study approach than conceptual questions. You need regular practice to maintain accuracy and speed. Our study schedule guide shows how to incorporate formula drills into your daily study routine while balancing the other three exam sections, ensuring you’re sharp on calculations without over-investing time in just 10-15 questions.
For comprehensive exam preparation including math practice questions, explore our study guides or learn about other exam topics you should master.