MAT-144 · Mathematical Reasoning Topic 07 · Taxes & Stocks
Lesson 06 · Putting it together · topic finale

Diversification and the stock-bond split.

Stocks pay more in the long run but swing wider in the short. Bonds smooth the ride at the cost of some return. A real portfolio is a mix of both, tuned to your time horizon. The closing lesson of the course.

01Risk vs return 02Why diversify 03Mix · time horizon
▸ THE HOOK

Stocks have historically delivered about 10% per year on average over long periods — the highest reliable return of any major asset class. Bonds, by contrast, have returned roughly 5% per year on average. The math seems obvious: if stocks pay twice as much as bonds, why would anyone ever buy bonds?

The answer is in the year-to-year experience. The 10% stock return is an average across many decades. In any individual year, stocks can be up 30% or down 30%. A 100% stock portfolio worth $100,000 at the start of 2007 was worth roughly $63,000 by early 2009 — a 37% drawdown that took until 2013 to fully recover. Bonds, over the same period, drifted gently up. If you needed to withdraw money from your portfolio in 2009 to pay for retirement, tuition, or a medical bill, the 100% stock allocation forced you to sell at exactly the wrong moment.

This is why real investors hold both stocks and bonds. Stocks for long-run growth, bonds for short-run predictability and to fund any near-term spending without selling stocks at a low. The right mix depends on how soon you'll need the money — your time horizon. This closing lesson develops the risk-return trade-off, the math of diversification, and the standard rules of thumb for tuning the mix to your life stage.

Mix them. Tune the mix to your time horizon.

Three properties together summarize any major asset class. Expected return is the long-run average annual return; stocks are higher (about 10%), bonds are lower (about 5%), cash is lowest (about 2-3% in normal rate environments). Volatility (year-to-year variation) follows the same ranking but in the same direction: stocks swing the most, bonds swing some, cash swings almost not at all. The two are tightly linked — higher expected return almost always comes with higher volatility.

Diversification means holding multiple asset classes so that no single one dominates the portfolio's behavior. A 60% stock / 40% bond portfolio has an expected return around 8% (a weighted average of 10% and 5%) but volatility much closer to the bond side than the stock side — because stocks and bonds often move in opposite directions in the short run, so combining them smooths out the swings.

The right time horizon tunes the mix. A 25-year-old saving for retirement at 65 has a 40-year horizon and can afford to ride out the worst stock-market drawdowns; an appropriate allocation might be 90% stocks, 10% bonds. A 65-year-old already drawing on the portfolio cannot afford a 40% drawdown that takes six years to recover; an appropriate allocation might be 40% stocks, 60% bonds. A common rule of thumb: percent in stocks = 110 − age.
RISK vs EXPECTED RETURN historical long-run averages · stock-bond mix tunes the trade-off 10% 8% 6% 4% expected return risk (volatility, year-to-year swings) low medium high 100% bonds ~5% return 60/40 mix ~8% return 100% stocks ~10% return tilt toward stocks for long horizons; tilt toward bonds as retirement approaches

The risk-return trade-off, sketched. Three portfolios on the curve: 100% bonds (low risk, low return), 60/40 mix (medium of both), 100% stocks (high risk, high return). The dashed curve traces all possible blends. Choose a point based on your time horizon.

▸ DEFINITION

Diversification is the practice of spreading investments across multiple asset classes (stocks, bonds, cash, real estate, etc.) to reduce the portfolio's overall volatility. The time horizon is the number of years until you expect to need the money; longer horizons permit greater stock allocation.

Words you'll see in every retirement-planning conversation

  • Asset class A broad category of investment with similar risk-return characteristics. The three classic ones are stocks (equities), bonds (fixed income), and cash (money-market funds, short-term Treasury bills). Real estate, commodities, and others are commonly added in more sophisticated portfolios.
  • Expected return The long-run average annual return of an asset class. Expected in the statistical sense (Lesson 5 of Topic 6: E(X) = Σ x · P(x)) — not a guarantee for any single year.
  • Volatility The standard deviation of annual returns. (Topic 5 standard-deviation math applied to the annual-return distribution.) Higher volatility means wider year-to-year swings around the expected return.
  • Asset allocation The percentages of a portfolio assigned to each asset class. "60/40" is shorthand for 60% stocks, 40% bonds. The biggest single decision in personal investing — far more impactful than picking individual stocks within the stock allocation.
  • Time horizon The number of years until you expect to need the money. Drives the asset allocation: longer = more stocks. A common rule of thumb is percent in stocks = 110 − age, which gradually shifts from stock-heavy in youth to bond-heavy approaching retirement.

Choosing an allocation for two investors.

Apply the time-horizon framework to two investors at different life stages, and compute the expected return of each suggested mix.

"Investor A is 30 years old and saving for retirement at age 65 (35-year horizon). Investor B is 60 years old and plans to retire at 65 (5-year horizon). Using the rule of thumb percent in stocks = 110 − age, suggest an allocation for each, then compute the expected return of each portfolio assuming stocks return 10% and bonds return 5%."

1

Compute Investor A's stock allocation.

stocks % = 110 − 30 = 80%

So Investor A: 80% stocks, 20% bonds. The 35-year horizon gives plenty of time to ride out drawdowns.

→ stock-heavy
2

Compute Investor A's expected return.

Weighted average:

E = 0.80 × 10% + 0.20 × 5%
= 8% + 1% = 9%
→ closer to the stock end
3

Compute Investor B's allocation.

stocks % = 110 − 60 = 50%

So Investor B: 50% stocks, 50% bonds. The 5-year horizon cannot tolerate a deep stock drawdown without delaying retirement.

→ balanced
4

Compute Investor B's expected return.

E = 0.50 × 10% + 0.50 × 5%
= 5% + 2.5% = 7.5%

Lower expected return than Investor A's, but much lower volatility — the right trade-off for the shorter horizon. Investor B sleeps better at night.

→ lower return, lower swings
THE FIGURE The glide path that target-date funds follow

Modern retirement funds (Vanguard 2065, Fidelity Freedom 2055, etc.) automatically shift their stock-bond mix as you age. This chart shows the typical glide path: heavy stocks at 25, equal at about 60, heavier bonds by 75.

THE STOCK / BOND GLIDE PATH how target-date funds shift allocation across a working life (110 − age rule of thumb) 0% 20% 40% 60% 80% 100% ALLOCATION 25 35 45 55 65 75 AGE 85% stocks 35% stocks 15% bonds 65% bonds 50 / 50 around age 60 As retirement nears, the portfolio glides toward bonds — trading expected return for lower volatility.

The portfolio glides from 85% stocks at age 25 to roughly 35% stocks by 75. The 50/50 crossing happens around age 60. Logic: stocks have higher long-run expected returns but more short-term volatility; as retirement approaches, trading some return for stability protects you from a bad market year hitting right when you need to spend the money.

Three portfolios. Compute the expected return.

Same assumptions: stocks return 10%, bonds return 5%. The math is a weighted average.
PROBLEM 01 ☆ ☆   warm-up · 60/40

A portfolio is invested 60% in stocks (10% expected return) and 40% in bonds (5% expected return). What is the portfolio's expected return?

expected return % =
PROBLEM 02 ★ ★ ☆   rule of thumb

Using the rule of thumb stocks % = 110 − age, what stock percentage would you recommend for a 45-year-old?

stocks % =
PROBLEM 03 ★ ★ ★   extreme allocations

A retiree wants very low volatility and chooses a 20% stocks, 80% bonds allocation. With stocks returning 10% and bonds 5%, what is the expected return of this portfolio (to one decimal place)?

expected return % =

Three fast questions before you finish the course.

Tap an answer. Feedback shows up immediately.

Q1. Why do most investors hold both stocks and bonds rather than just the higher-returning stocks alone?

Why B? Stocks have higher expected return but also wider year-to-year swings. Bonds smooth the ride and provide stability when near-term spending needs would otherwise force selling stocks at a low point. A is false (the opposite is true); D is false for most bonds (municipal bonds are tax-exempt for federal but not broadly tax-free).

Q2. Using the rule of thumb stocks % = 110 − age, which allocation fits a 70-year-old retiree?

Why C? 110 − 70 = 40% stocks, leaving 60% bonds. Even at age 70, the rule still allocates a meaningful slice to stocks — because retirement can last 25+ years, and inflation will erode the value of a 100% bond portfolio over that horizon.

Q3. A portfolio with 70% stocks (10% expected return) and 30% bonds (5%) has what expected return?

Why C? Weighted average: 0.70 × 10 + 0.30 × 5 = 7 + 1.5 = 8.5%. B is adding the rates without weighting; A is the 50/50 case; D is roughly the bond-only rate.
▸ WHY THIS MATTERS

The course finale.

You have now finished the content of the course. Across seven topics you have built a complete financial-literacy toolkit:

  • Topic 1 — Linear functions: the algebra that underlies every cost-curve, salary projection, and supply-and-demand diagram in a basic financial model.
  • Topic 2 — Conversions and budgeting: percent of total, percent change, and the math of how money flows between categories of a household budget.
  • Topic 3 — Savings: simple and compound interest, future value, the machinery of how money grows over time.
  • Topic 4 — Loans: amortization, monthly payments, the flip side of savings — how money grows against you when you borrow.
  • Topic 5 — Statistics: descriptive measures, normal distributions, margins of error — the tools to read a data set and a poll headline.
  • Topic 6 — Probability: counting, combinations, expected value, the Law of Large Numbers — the math of uncertainty itself.
  • Topic 7 — Taxes and the stock market: paycheck reading, bracket math, stocks, bonds, and the diversification frame that ties everything together.

The themes recur across topics. Compound interest from Topic 3 is the same exponential growth that drives long-run stock returns in this topic. The progressive-bracket math from Lesson 2 is structurally identical to the percent-change cascades from Topic 2. The expected-value formula from Topic 6 underpins the "expected return" vocabulary of asset allocation in this lesson.

The Final Exam ahead is cumulative across all seven topics; the Final Exam Review (~38 questions) gives you a structured preparation path. Beyond the exam, the math you have built is yours for life. The next time you read a poll headline, calculate a tip, sign a lease, take out a loan, or read an investment quote, you'll be doing it with the full apparatus of seven topics' worth of math behind you.

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