Finance-Theory · Unit 17 · Video 2 · Interactive Practice

The Tangency Portfolio: Why Every Investor Holds the Same Stocks

IKey Formulas

Formula	Name	What it says
$q$ dominates $p$ when $\mathsf{E}[R_q] \ge \mathsf{E}[R_p]$ and $\sigma_q \le \sigma_p$ , at least one strictly	Dominance test	$q$ sits in $p$ 's north-west quadrant: straight up, straight west, or between
$\mathsf{E}[R] = r_f + \omega\,(\mathsf{E}[R_p] - r_f)$	Expected return of a T-bill mix	Weight $\omega$ on portfolio $p$ , the rest $1 - \omega$ in T-bills
$\sigma = \omega\,\sigma_p$	Risk of a T-bill mix	T-bills carry no risk, so only the share in $p$ contributes ( $0 \le \omega \le 1$ here)
$\dfrac{\mathsf{E}[R_p] - r_f}{\sigma_p}$	Steepness of the line through $p$	Extra expected return per unit of risk; at any positive risk, the steeper line pays more

Key Insight: Both coordinates are linear in $\omega$ , so every T-bill mix lies on the straight line from $r_f$ to $p$ . The steepest such line that still reaches a feasible portfolio is tangent to the efficient frontier, and every investor who likes expected return and dislikes risk holds its touching point, the tangency portfolio, choosing only how much to keep in T-bills.

IIThe North-West Quadrant

A portfolio beats another only if it lies in that portfolio's north-west quadrant: no more risk, no less expected return.

Test point σ (%) 24 Test point E[R] (%) 9

IIIT-Bills Plus Any Portfolio

Mixing T-bills with any portfolio $p$ moves you along the straight segment from $r_f$ to $p$ .

Risky portfolio p Weight ω on p 3/4

IVRotating Up to the Tangent Line

Pivoting the line about $r_f$ , the steepest line that still reaches a feasible portfolio touches the frontier exactly once.

Slope of the line 0.31

💡 A cautious retiree and a hedge fund manager both hold $T$ ; they differ only in $\omega$ , how much of their money sits in $T$ rather than in T-bills.

VQuiz Questions

Problem 1 · Mixing T-Bills with a Portfolio

Given: portfolio $C$ has $\mathsf{E}[R_C] = 14\%$ and $\sigma_C = 30\%$ per year, and T-bills pay $r_f = 3\%$ . You put $\tfrac{1}{5}$ of your money in T-bills and the rest in $C$ . Find the expected return and standard deviation of your mix.

✅ Correct! With

\omega = \tfrac{4}{5}

the mix sits four-fifths of the way along the segment from

r_f

C

❌ Not quite. That uses

\omega = \tfrac{1}{5}

, but

\tfrac{1}{5}

is the share in T-bills; the weight on

C

\omega = 1 - \tfrac{1}{5} = \tfrac{4}{5}

❌ Not quite.

\tfrac{4}{5} \times 14\% = 11.2\%

leaves out the T-bills, which still earn

3\%

on their fifth:

11.2\% + \tfrac{1}{5} \times 3\% = 11.8\%

❌ Close, but… the expected return is right, yet

\sqrt{\tfrac{4}{5}} \times 30\%

scales the variance by

\omega

. T-bills are riskless, so the standard deviation itself scales:

\sigma = \omega\,\sigma_C

Show solution

The weight on $C$ is $\omega = 1 - \tfrac{1}{5} = \tfrac{4}{5}$ .

\mathsf{E}[R] = r_f + \omega\,(\mathsf{E}[R_C] - r_f) = 3\% + \tfrac{4}{5}\,(14\% - 3\%) = 3\% + 8.8\% = 11.8\%

\sigma = \omega\,\sigma_C = \tfrac{4}{5} \times 30\% = 24\%

The mix sits on the straight line from $(0\%, 3\%)$ to $C = (30\%, 14\%)$ in the (risk, return) plane, four-fifths of the way toward $C$ .

Problem 2 · Which One Dominates?

Given: portfolio $B$ has $\mathsf{E}[R_B] = 11\%$ and $\sigma_B = 20\%$ per year; T-bills pay $r_f = 3\%$ . Which of these is strictly preferred to $B$ by every investor who likes expected return and dislikes risk?

✅ Correct! Each candidate trades more return for more risk or less risk for less return, so none lies in

B

's north-west quadrant.

❌ Not quite.

C

has the higher expected return but also the higher risk: it lies north-east of

B

, outside the preferred quadrant.

❌ Not quite.

A

is less risky but also earns less: it lies south-west of

B

, outside the preferred quadrant.

❌ Not quite. Work out the mix:

\mathsf{E}[R] = 3\% + \tfrac{3}{5}(14\% - 3\%) = 9.6\%

and

\sigma = \tfrac{3}{5}(30\%) = 18\%

. Less risk than

B

, but less expected return too.

Show solution

$q$ is strictly preferred to $B$ only if $\mathsf{E}[R_q] \ge 11\%$ and $\sigma_q \le 20\%$ , with at least one inequality strict. Check each candidate:

$C$ : $14\% \ge 11\%$ but $30\% > 20\%$ . Fails the risk condition (north-east).
$A$ : $12\% \le 20\%$ but $8\% < 11\%$ . Fails the return condition (south-west).
The mix, $\omega = \tfrac{3}{5}$ : $\mathsf{E}[R] = 3\% + \tfrac{3}{5}(14\% - 3\%) = 9.6\%$ , $\sigma = \tfrac{3}{5}(30\%) = 18\%$ . Again south-west.

None of them dominates $B$ , and $B$ dominates none of them either: a higher return alone, or a lower risk alone, is not enough.

Problem 3 · Three Ways to the Same Risk

Given: T-bills pay $r_f = 3\%$ , and three portfolios have $A$ : $\mathsf{E}[R_A] = 8\%$ , $\sigma_A = 12\%$ ; $B$ : $\mathsf{E}[R_B] = 11\%$ , $\sigma_B = 20\%$ ; $C$ : $\mathsf{E}[R_C] = 14\%$ , $\sigma_C = 30\%$ (all per year). Three investments each have $\sigma = 12\%$ : $A$ alone; $\tfrac{3}{5}$ in $B$ with $\tfrac{2}{5}$ in T-bills; $\tfrac{2}{5}$ in $C$ with $\tfrac{3}{5}$ in T-bills.

Expected return of 3/5 in B, 2/5 in T-bills?

Which of the three earns the most at σ = 12%?

✅ Correct! At

\sigma = 12\%

the three earn

8\%

7.8\%

and

7.4\%

: the line through

A

is the steepest, so it pays the most at this risk and at every other positive risk.

❌ Not quite.

6.2\%

puts

\tfrac{2}{5}

B

; here

\omega = \tfrac{3}{5}

is the weight on

B

❌ Not quite.

\tfrac{3}{5} \times 11\% = 6.6\%

forgets that the T-bill share still earns

3\%

❌ Not quite.

3\% + \tfrac{3}{5} \times 11\%

counts the riskless rate twice: the weight multiplies the gap

\mathsf{E}[R_B] - r_f = 8\%

, not

11\%

❌ Check the expected return. Use

\mathsf{E}[R] = r_f + \omega\,(\mathsf{E}[R_B] - r_f)

with

\omega = \tfrac{3}{5}

❌ Not quite. The

B

mix earns

7.8\%

, but

A

alone earns

8\%

at the same

12\%

risk: the line through

A

is steeper,

\tfrac{5}{12} > \tfrac{8}{20}

❌ Not quite.

C

has the highest expected return on its own, but diluted to

\sigma = 12\%

it earns only

3\% + \tfrac{2}{5}(11\%) = 7.4\%

: its line is the flattest.

❌ Not quite. Equal risk does not mean equal return. Each mix sits on a different line out of

r_f

, and at

\sigma = 12\%

the lines are at different heights.

❌ Not quite. Compute all three expected returns at

\sigma = 12\%

and compare.

Show solution

Each investment reaches $\sigma = 12\%$ , since $\sigma = \omega\,\sigma_p$ : $1 \times 12\%$ , $\tfrac{3}{5} \times 20\%$ , $\tfrac{2}{5} \times 30\%$ . Their expected returns:

A:\ 8\%, \qquad B \text{ mix}:\ 3\% + \tfrac{3}{5}(11\% - 3\%) = 7.8\%, \qquad C \text{ mix}:\ 3\% + \tfrac{2}{5}(14\% - 3\%) = 7.4\%

Same risk, highest return: $A$ alone dominates both mixes (it lies straight above them). The reason is the steepness of each line out of $r_f$ :

\frac{8 - 3}{12} = \frac{5}{12} \approx 0.417, \qquad \frac{11 - 3}{20} = 0.4, \qquad \frac{14 - 3}{30} = \frac{11}{30} \approx 0.367

The line through $A$ is the steepest of the three, so at any positive level of risk it offers the highest expected return.

Problem 4 · The Retiree's Portfolio

Given: suppose the tangency portfolio is $A$ , with $\mathsf{E}[R_A] = 8\%$ and $\sigma_A = 12\%$ per year, holding three stocks, Osprey, Plover and Robin, in weights $\tfrac{1}{2}$ , $\tfrac{3}{10}$ and $\tfrac{1}{5}$ . T-bills pay $r_f = 3\%$ . A cautious retiree with $150,000 keeps $\tfrac{4}{5}$ of it in T-bills and puts the rest in $A$ .

How much does she hold in Plover?

Her portfolio's expected return and risk?

✅ Correct! She holds Plover and Robin in the ratio

\tfrac{3}{10} : \tfrac{1}{5} = 3 : 2

, exactly as a hedge fund manager holding

A

does; only her T-bill share is different, and it puts her low on the line, close to

r_f

❌ Not quite.

\tfrac{3}{10} \times \text{\textdollar}150{,}000

treats all her money as invested in

A

; only

\tfrac{1}{5}

of it is.

❌ Not quite. That applies Plover's weight to the

\tfrac{4}{5}

she keeps in T-bills; T-bills hold no stocks.

❌ Not quite. $30,000 is her whole holding of

A

; Plover is only

\tfrac{3}{10}

of it.

❌ Check the Plover holding. Multiply her wealth by the share in

A

, then by Plover's weight inside

A

❌ Not quite. That uses

\omega = \tfrac{4}{5}

, the T-bill share; her weight on

A

\omega = \tfrac{1}{5}

❌ Close, but… the expected return is right, but T-bills are riskless, so her risk shrinks with her weight in

A

\sigma = \tfrac{1}{5} \times 12\%

❌ Not quite.

\tfrac{1}{5} \times 8\% = 1.6\%

forgets the

3\%

her T-bills earn on the other

\tfrac{4}{5}

❌ Check the mix. Use

\omega = \tfrac{1}{5}

in both formulas.

Show solution

Her weight on $A$ is $\omega = 1 - \tfrac{4}{5} = \tfrac{1}{5}$ , so she has $\tfrac{1}{5} \times \text{\textdollar}150{,}000 = \text{\textdollar}30{,}000$ in $A$ , and Plover is $\tfrac{3}{10}$ of that:

\tfrac{3}{10} \times \text{\textdollar}30{,}000 = \text{\textdollar}9{,}000

Her expected return and risk:

\mathsf{E}[R] = 3\% + \tfrac{1}{5}\,(8\% - 3\%) = 4\%, \qquad \sigma = \tfrac{1}{5} \times 12\% = 2.4\%

Her stock holdings, $15,000 in Osprey, $9,000 in Plover and $6,000 in Robin, are in the proportions $\tfrac{1}{2} : \tfrac{3}{10} : \tfrac{1}{5}$ of the tangency portfolio. Every investor holds the same stocks in the same proportions; risk tolerance only sets $\omega$ .

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