6.5Lesson 6.5 · Module 6 — Portfolio Theory

The Risk-Free Asset, Tangency Portfolio, and Sharpe Ratio

Why risk-free plus risky creates a straight line, what the Sharpe ratio measures, and why the tangency portfolio is the maximum-Sharpe portfolio.

Risk-free asset has σ = 0
Risky + risk-free = straight line
Slope = Sharpe ratio
Tangency = maximum Sharpe
Two-fund separation

Start

Learning objectives

By the end of this lesson, you should be able to:

1Explain why a risk-free asset appears at (0, r_f) on the risk-return graph.
2Derive why combining a fixed risky portfolio with the risk-free asset produces a straight line.
3Define the Sharpe ratio as the slope of the allocation line.
4Identify the tangency portfolio as the maximum-Sharpe risky portfolio.
5Distinguish lending, full investment, and leverage positions along the same line.
6State the two-fund separation principle and its assumptions.
7Recognize the limitations of the mean-variance framework with a risk-free asset.

6.5.1Scene 1 · The risk-free point

The risk-free asset

A risk-free asset has a known return with zero volatility. On the risk-return graph it appears at (0, r_f) — directly on the vertical axis. A short-term Treasury bill is the standard example: the government promises a specific nominal return, so over the bill's maturity σ_f = 0.

Important caveat

Risk-free in this model does not mean free of every economic risk. Treasury bills still carry:

• Inflation risk — nominal return is known, real purchasing power is not.
• Reinvestment risk — future rates are unknown when the bill matures.
• Price risk if sold early — selling before maturity exposes you to market prices.

Definition · Risk-free rate (

r_{f}

)

The return on an asset with zero volatility in this model. It anchors the allocation line and is the baseline for measuring excess returns.

6.5.2Scene 2 · Build the line manually

Risky + risk-free = a straight line

Combine a fixed risky portfolio with the risk-free asset, changing only y — the fraction in the risky portfolio. Build the line point by point before any algebra.

Complete portfolio return

E [R_{C}] = r_{f} + y (E [R_{P}] - r_{f})

Complete portfolio volatility

σ_{C} = y σ_{P}

Because $σ_{f} = 0$ and $Cov (R_{f}, R_{P}) = 0$ , the risk-free asset contributes no variance — volatility scales linearly with y.

Fill in the expected return and volatility for each value of y. Once a row is correct, its point appears on the chart. Use $r_{f} = 2%$ , $E [R_{P}] = 8%$ , $σ_{P} = 12%$ .

y = 0.00

E[R_C] (%)

σ_C (%)

y = 0.25

E[R_C] (%)

σ_C (%)

y = 0.50

E[R_C] (%)

σ_C (%)

y = 0.75

E[R_C] (%)

σ_C (%)

y = 1.00

E[R_C] (%)

σ_C (%)

Each additional 25% in y adds exactly 3pp volatility and 1.5pp expected return.

Why it is a straight line

Step 1 · Both quantities scale with y

E [R_{C}] - r_{f} = y (E [R_{P}] - r_{f}), σ_{C} = y σ_{P}

The expected excess return is y × (E[R_P] − r_f), and the volatility is y × σ_P. Both are proportional to y.

The risk-free asset contributes no variance, so σ_C is perfectly linear in y.

Allocation line equation

E [R_{C}] = r_{f} + \frac{E [ R _{P} ] - r _{f}}{σ _{P}} σ_{C}

The complete portfolio's expected return is a linear function of its volatility. The slope is the Sharpe ratio of the risky portfolio.

$r_{f}$: Risk-free rate (the y-intercept).
$σ_{C}$: Volatility of the complete portfolio.

Every combination of rf and portfolio P lies on this line — starting at (0, $r_{f}$ ) with slope equal to P's Sharpe ratio.

6.5.3Scene 3 · Compare allocation lines

Not every allocation line is equally good

Different risky portfolios produce different lines from (0, r_f). A steeper line delivers more excess return per unit of risk. Compute the Sharpe ratios first, then see the lines reveal the winner.

Three portfolios are available. With $r_{f} = 2%$ , compute the excess return and Sharpe ratio for each, then identify the maximum-Sharpe portfolio.

Portfolio A

E[R] = 7%, σ = 10%

Excess return (%)

Sharpe ratio

Portfolio B

E[R] = 8.5%, σ = 12%

Excess return (%)

Sharpe ratio

Portfolio C

E[R] = 10%, σ = 16%

Excess return (%)

Sharpe ratio

Which portfolio has the highest Sharpe ratio?

Why isn't Portfolio C selected, despite the highest expected return?

6.5.4Scene 4 · Tangency

The steepest line touches the frontier

Rotate the line upward from (0, r_f). The steepest line that still touches the risky efficient frontier is tangent to it — and that contact point is the tangency portfolio, the maximum-Sharpe risky portfolio.

Sharpe ratio

S_{P} = \frac{E [ R _{P} ] - r _{f}}{σ _{P}}

Expected excess return per unit of volatility — the slope of the allocation line from (0, $r_{f}$ ).

$E [R_{P}]$: Expected return of the risky portfolio.
$r_{f}$: Risk-free rate.
$σ_{P}$: Volatility of the risky portfolio.

A higher Sharpe ratio means a steeper line. It is not highest return, not lowest volatility, not a probability of profit, and not alpha.

Definition · Tangency portfolio

The risky portfolio where the steepest allocation line from (0, r_f) touches the risky efficient frontier — the maximum-Sharpe risky portfolio.

Asset	Tangency weight
GM	27.27%
IBM	48.85%
Motorola	23.89%
E[R] / σ / Sharpe	1.36% / 5.54% / 0.223

Historical monthly instructional estimates · MIT 15.401 · 1946–2001 — not current estimates. With r_f = 0.12% (monthly), the tangency portfolio has the highest Sharpe ratio achievable with these three assets.

6.5.5Scene 5 · Scale exposure

Lending, full investment, and leverage

One tangency portfolio, three positions along the same line. Only the split between risky and risk-free changes — the internal risky weights stay the same.

Position	y	rf weight	E[R_C]	σ_C
Lending	0.40	+60%	4.4%	4.8%
Full investment	1.00	0%	8.0%	12.0%
Leverage	1.50	-50%	11.0%	18.0%

At y = 1.5 the investor borrows 50% at $r_{f}$ and invests 150% in the tangency portfolio. Expected return rises to 11%, volatility to 18%. Leverage increases both in the same proportion — it does not improve the Sharpe ratio.

Three investors choose different y with $r_{f} = 2%$ , $E [R_{T}] = 8%$ , $σ_{T} = 12%$ . Compute the risk-free weight, expected return, and volatility for each.

Investor A — Conservative · y = 0.40

Risk-free weight (%)

E[R_C] (%)

σ_C (%)

Investor B — Moderate · y = 1.00

Risk-free weight (%)

E[R_C] (%)

σ_C (%)

Investor C — Aggressive · y = 1.50

Risk-free weight (%)

E[R_C] (%)

σ_C (%)

Note

All three investors sit on the same straight line from $(0, r_{f})$ through the tangency portfolio. They differ only in their position along it — determined by y.

6.5.6Scene 6 · Two-fund separation

Separate the risky choice from the risk level

Every investor makes two independent decisions: which risky portfolio (the tangency — the same for everyone) and how much total risk (the split y — different for each).

Decision 1 · Which risky portfolio?

The tangency portfolio — maximum Sharpe ratio. This decision is the same for every investor.

Decision 2 · How much total risk?

Choose y — the split between tangency and rf. This decision is different for each investor.

Investor type	Tangency	Risk-free	Position
Conservative	40%	+60%	Lending
Moderate	100%	0%	Full
Aggressive	130%	-30%	Leverage

All three hold the same risky portfolio. They differ only in the fraction invested in it. The risky-mix decision is separated from the risk-level decision.

Why the line replaces the curve

Under equal borrowing/lending rates, the tangency line dominates the curved frontier at every risk level (except at the tangency point itself). The curved frontier is still needed to find the tangency point — once found, the line becomes the new efficient set.

Borrowing assumption

This result assumes investors can borrow and lend at the same rate r_f. In reality borrowing rates exceed lending rates — which would create a kink at the tangency point. We state this assumption explicitly.

Assumptions and limitations

1Equal borrowing/lending rate. In reality borrowing exceeds lending, producing a kinked efficient set above the tangency point.
2Stable estimates. Small changes in expected returns or covariances can shift the tangency weights significantly.
3Mean-variance framework. Standard deviation treats upside and downside symmetrically, ignoring skewness, drawdowns, tail risk, liquidity, costs, and taxes.
4Leverage is not frictionless. Borrowing introduces margin calls, collateral, and the risk of forced selling at unfavorable prices.
5Tangency ≠ market portfolio. Identifying it with the market portfolio requires equilibrium assumptions beyond this lesson.

6.5.7Final check · Sharpe and the allocation line

rf = 3%, X(9%, 10%), Y(11%, 16%)

Tie it together: compute Sharpe ratios, build a combination with rf, and reason about which risky portfolio to combine with the risk-free asset.

Use $r_{f} = 3%$ . Two risky portfolios: X at $(σ = 10%, E [R] = 9%)$ and Y at $(σ = 16%, E [R] = 11%)$ .

Sharpe ratios

Sharpe ratio of X (S_X)

Sharpe ratio of Y (S_Y)

Combine X with rf using y = 0.50

E[R_C] (%)

σ_C (%)

Why is X a better risky portfolio than Y for combining with rf?

The 50/50 combination (y = 0.50) and full investment (y = 1.0) in X both…

6.5.8Optional · Common questions

Questions you may still have

Try itMastery check

Pass with 4 of 6 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
rf = 2%, E[ $R_{P}$ ] = 8%, σ_P = 12%, y = 0.5. What is E[ $R_{C}$ ]?
02
Same setup. What is σ_C?
03
A portfolio has Sharpe = 0.542, rf = 2%, σ = 12%. What is E[R]?
04
If y = 1.5, what is the risk-free weight?
05
Two-fund separation means:
06
Is the tangency portfolio the market portfolio?

Lesson summary

1A risk-free asset appears at (0, $r_{f}$ ) on the risk-return graph.
2Combining a fixed risky portfolio with rf creates a straight allocation line.
3The line's slope equals the Sharpe ratio: (E[ $R_{P}$ ] - $r_{f}$ )/σ_P.
4The tangency portfolio is the maximum-Sharpe risky portfolio.
5Investors can scale total risk using the same risky portfolio (two-fund separation).
6Lending (y<1) and leverage (y>1) are positions along the same line.
7Leverage does not improve the Sharpe ratio along the same line.
8These results depend on equal borrowing/lending rates, stable estimates, and mean-variance assumptions.
9The tangency portfolio is NOT automatically the market portfolio.

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