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MAT-144 · Topic 07 · Cheat Sheet

Taxes & Stocks

The formulas, the moves, and the traps for Topic 7.

Key formulas

Two halves: progressive tax math on top, stock-and-bond return math on the bottom. Both halves are simple ratios once the vocabulary clicks.

Taxable income
taxable income = gross income − deductions
Start with everything you earned (W-2 wages, 1099 income, interest). Subtract the standard deduction (or your itemized total, whichever is larger). What's left is what the brackets apply to.
Federal income tax (progressive brackets)
tax = Σ (dollars in each bracket) × (bracket rate)
The whole income is not taxed at one rate. Only the dollars that fall inside each bracket pay that bracket's rate. Walk the brackets bottom-up, multiply the slice in each bracket by its rate, then sum.
Marginal vs effective tax rate
Marginal rate  =  rate on the next dollar earned
Effective rate  =  total tax paidtaxable income × 100
The marginal rate is the bracket your last dollar sits in (typically 22% or 24% for a middle-income filer). The effective rate is what you actually pay on average — always lower than the marginal because the lower brackets are taxed at lower rates.
Refund or amount due
refund = withholding − tax owed
Withholding > tax owed → refund (the IRS holds your money interest-free until April). Withholding < tax owed → you owe the difference. A “big refund” is not free money; it's an interest-free loan to the government.
Stock dividend yield
dividend yield = annual dividend per sharecurrent share price × 100
If a stock pays $2.40 in dividends per year and trades at $60, the yield is 4%. Mature companies (utilities, banks) yield 3-5%; growth companies (tech) often pay 0%.
Total return on a stock
total return = capital gain + dividends received
% return = total returnamount invested × 100
Buy 100 shares at $50 ($5,000 invested). Sell at $58 ($800 gain) after collecting $200 in dividends. Total return = $1,000 = 20%. Don't forget the dividends.
Bond annual coupon & current yield
annual coupon = face value × coupon rate
current yield = annual couponmarket price × 100
A $1,000 face-value bond with a 5% coupon pays $50/year regardless of market price. If the bond now trades at $950, the current yield is $50/$950 = 5.26%. Bond prices and yields move inversely.
Stock-bond split (rule of thumb)
stock %  ≈  110 − age
bond %  ≈  age − 10
At 25: about 85% stocks / 15% bonds. At 65: about 45% stocks / 55% bonds. As retirement nears, the allocation glides toward the less-volatile bond side. Modern target-date funds do this automatically.

Common mistakes

The traps that show up on both the ALEKS review and the DQs.

  1. Applied the marginal rate to the whole income. Sitting in the 22% bracket does not mean 22% of your whole income goes to taxes. The lower-bracket dollars are taxed at lower rates. Walk the brackets one slice at a time.
  2. Forgot to subtract the deduction first. Brackets apply to taxable income (gross minus deductions), not gross income. A $60K gross with a $14,600 standard deduction has only $45,400 of taxable income.
  3. Confused gross, net, and taxable. Gross = what you earned. Taxable = gross minus deductions, what brackets are applied to. Net (or take-home) = what's left after federal + state + FICA + benefits.
  4. Reported the marginal rate when asked for the effective rate. Effective rate = total tax / taxable income. Always lower than the marginal rate (unless your whole income is in the bottom bracket).
  5. Computed yield using price-per-share but dividend per year on the whole position. Both numerator and denominator have to be on the same per-share basis (or both on the same total-position basis). Mix them and the answer is off by a factor of (shares).
  6. Forgot dividends in total return. Total return = price change plus dividends received. Capital gain alone is the price change. ALEKS asks for both forms; read carefully.
  7. Used coupon rate instead of current yield. The coupon rate is fixed at the bond's issue. Current yield depends on the market price and changes daily. ALEKS usually wants current yield.
  8. Reversed bond price vs yield. When prices go up, yields go down (and vice versa). The coupon payment is fixed, so a higher purchase price = smaller percentage return.
  9. Treated stocks and bonds as the same kind of risk. Stocks have higher expected returns and higher short-term volatility. Bonds have lower returns and act as a stabilizer. Diversification works because they don't move in lockstep.
  10. Quoted the rule-of-thumb allocation as a hard rule. “110 minus age” is a starting point, not a prescription. Real allocation depends on goals, time horizon, and risk tolerance; the formula is the conversation opener.

Quick reference

Vocabulary, the 2024 federal brackets, and Excel patterns for paycheck and portfolio math.

Paycheck vocabulary
Gross pay — total earnings before any deductions.
FICA — 7.65% off the top (6.2% Social Security + 1.45% Medicare).
Federal withholding — pre-paid federal income tax, calibrated by your W-4.
Pre-tax benefits — 401(k), HSA, health-insurance premiums (lower taxable income).
Net pay — what hits your checking account. Take-home.
2024 federal brackets — single filer
10%  $0 – $11,600
12%  $11,600 – $47,150
22%  $47,150 – $100,525
24%  $100,525 – $191,950
32%  $191,950 – $243,725
35%  $243,725 – $609,350
37%  $609,350 +
Standard deduction (single): $14,600. Brackets shift each year for inflation — check the current year's IRS table when filing for real.
Investment vocabulary
Share — one unit of ownership in a company.
Dividend — periodic cash payment to shareholders.
Capital gain — profit from selling a share for more than you paid.
Face value (par) — what a bond pays at maturity (often $1,000).
Coupon rate — fixed % of face value paid as annual interest.
Current yield — annual coupon ÷ market price (today's effective return).
Maturity — the date the bond's face value is returned.
Excel for paychecks & portfolios
=A2*0.0765  FICA off gross
=SUMPRODUCT(slices, rates)  bracket-by-bracket tax
=(B2-A2)/A2  percent change (price gain)
=(price_now*shares + dividends - cost)/cost  total return %
=face*coupon_rate/price  bond current yield
=stock_pct*total + bond_pct*total  allocation
Why diversification
Stocks and bonds tend to move differently in response to the same news. A diversified portfolio gives up a bit of upside in exchange for smaller drawdowns — lower variance, similar long-run mean. The math is essentially the Topic 5 standard-deviation argument applied to portfolio returns.
MAT-144 · Topic 07 · Cheat Sheet · mat144.com