Finance-Theory · Unit 17 · Video 5 · Interactive Practice

Indifference Curves and the Four Conclusions of Portfolio Theory

IKey Formulas

Formula	Name	What it says
$\text{E}[R_p] = r_f + w\,(\text{E}[R_T] - r_f), \quad \sigma_p = w\,\sigma_T$	The tangency line	Every investor's menu: $w < 1$ lends at $r_f$ , $w = 1$ is fully invested, $w > 1$ borrows
$\Delta \text{E}_1 < \Delta \text{E}_2 < \Delta \text{E}_3$ for equal steps $\Delta\sigma$	Convex indifference curve	Each extra point of risk costs more satisfaction than the last (diminishing marginal utility), so it needs a larger rise in expected return
$\sigma_p \le \sum_i w_i\,\sigma_i \quad (w_i \ge 0)$	Conclusion 1: diversification	Strictly less whenever $\rho < 1$ ; half in GM and half in IBM gives $\sqrt{27.82} \approx 5.27\%$ , below their average of about $6.29\%$
$\text{Var}[R_p] \approx \text{average covariance}$	Conclusion 2: covariances matter	In a large, equally weighted portfolio only the shared (systematic) risk is left

Key Insight: Conclusions 3 and 4: hold the efficient frontier, and with a riskless asset hold the tangency portfolio mixed with T-bills. Your indifference curves choose only $w$ , where on the line you sit; the optimum is where your highest attainable curve touches the line. All of it assumes standard deviation is the whole of risk and that expected returns, volatilities and covariances are taken as given.

IIBuilding Your Own Curve

Starting from $(6\%,\ 1.4\%)$ monthly, equal steps in risk that demand ever larger rises in return bend the curve.

E[R] asked at σ = 8% 1.6% E[R] asked at σ = 10% 2%

IIIThe Highest Curve You Can Reach

Only points on the tangency line are attainable, so the best curve is the one that just touches it.

Curve meets σ = 0 at 2%

IVDifferent Seats, One Line

Risk aversion decides how far along the line you sit; the risky portfolio you hold never changes.

Risk aversion above average

💡 The theory is static: if the T-bill rate or the best attainable Sharpe ratio changes, the line moves and every investor's optimum has to be found again.

VQuiz Questions

Problem 1 · Reading a Curve from a Table

Given (illustrative, per year): an investor is equally well off at each of $(\sigma, \text{E}[R]) = (5\%,\ 3\%)$ , $(10\%,\ 4.5\%)$ and $(15\%,\ 7\%)$ . Find the extra expected return the second step in risk, from $10\%$ to $15\%$ , demands, and the shape it implies.

✅ Correct! The first

5

-point step costs

1.5

points of return, the second costs

2.5

: the slope grows with risk, which is the bow of a convex curve.

❌ Not quite.

+1.5

is the first step,

4.5\% - 3\%

; the second step is

7\% - 4.5\% = 2.5

points, so the steps are not equal.

❌ Right rise, wrong shape. A larger rise for the same step in risk means the slope is growing: the curve bends upward like a bowl, which is convex.

❌ That is the total.

7\% - 3\% = 4

points covers both steps, from

5\%

15\%

; the second step alone is

7\% - 4.5\% = 2.5

points.

❌ Not quite. Subtract the expected returns at

\sigma = 10\%

and

\sigma = 15\%

, then compare with the first step.

Show solution

The three points are equally spaced in risk, $5$ points apart. The rises in expected return are

4.5\% - 3\% = 1.5 \text{ points}, \qquad 7\% - 4.5\% = 2.5 \text{ points}

Per point of risk that is $1.5/5 = 0.3$ on the first step and $2.5/5 = 0.5$ on the second. The slope grows as risk grows, so the curve is convex, bowed like a bowl: each extra point of risk needs more compensation than the one before.

Problem 2 · Strictly Better Off

Given: the same investor, indifferent among $(5\%,\ 3\%)$ , $(10\%,\ 4.5\%)$ and $(15\%,\ 7\%)$ . Find the portfolio this investor strictly prefers to $(10\%,\ 4.5\%)$ .

✅ Correct!

(5\%,\ 3.2\%)

has the risk of

(5\%,\ 3\%)

and more return, so it beats a point that is exactly as good as

(10\%,\ 4.5\%)

❌ More return is not enough. At

\sigma = 15\%

this investor needs

7\%

just to be equally well off;

6\%

falls short, so

(15\%,\ 6\%)

is worse.

❌ Less risk, but no better.

(5\%,\ 3\%)

is on the same indifference curve: the investor is equally well off there, not strictly better off.

❌ Not quite. Same risk and less expected return than

(10\%,\ 4.5\%)

leaves the investor worse off.

Show solution

Compare each candidate with the curve point at the same $\sigma$ :

$(15\%,\ 6\%)$ : the curve needs $7\%$ at $15\%$ ; $6\% < 7\%$ , so it lies below the curve, worse.
$(5\%,\ 3\%)$ : on the curve, equally good, not strictly better.
$(10\%,\ 4.4\%)$ : below the curve point $(10\%,\ 4.5\%)$ , worse.
$(5\%,\ 3.2\%)$ : $3.2\% > 3\%$ at the same risk, so it lies above the curve, on a higher one, better.

Being on a higher curve, to the north-west, is what makes a portfolio strictly preferred; a higher expected return alone is not.

Problem 3 · Reading the Optimum

Given (illustrative, per year): $r_f = 1\%$ ; the tangency portfolio has $\text{E}[R_T] = 9\%$ and $\sigma_T = 16\%$ . An investor's highest attainable indifference curve touches the tangency line at $\sigma = 6\%$ . Find the expected return at the optimum and how the investor's wealth is split.

What is $\text{E}[R]$ at the optimum?

How is the investor's wealth split?

✅ Correct!

w = 6/16 = 0.375

, so

\text{E}[R] = 1\% + 0.375 \times 8\% = 4\%

, with

62.5\%

lent at

1\%

in T-bills.

❌ Missing the intercept.

0.5 \times 6\% = 3\%

is the rise along the line; the line starts at

r_f = 1\%

, so add it.

❌ Missing a term.

0.375 \times 9\% = 3.375\%

is only the tangency portfolio's contribution; the

62.5\%

in T-bills earns

0.625 \times 1\% = 0.625\%

more.

❌ Not quite.

1\% + 0.375 \times 9\%

counts the T-bill rate twice; the line rises by the premium

\text{E}[R_T] - r_f = 8\%

, not by

9\%

❌ Weights swapped. The weight in the tangency portfolio is

\sigma / \sigma_T = 6/16 = 0.375

; the remaining

0.625

goes to T-bills.

❌ Not quite.

\sigma = 6\%

is below

\sigma_T = 16\%

, so the optimum sits between T-bills and the tangency portfolio: lending, not borrowing.

❌ Same risky portfolio for everyone. Any risky portfolio other than the tangency portfolio lies below the tangency line, so at

\sigma = 6\%

the mix on the line earns more; preferences change the mix with T-bills, not the risky portfolio.

❌ Check the return. Use

\text{E}[R] = r_f + w\,(\text{E}[R_T] - r_f)

with

w = \sigma / \sigma_T

❌ Check the split. On the tangency line

\sigma = w\,\sigma_T

; solve for

w

and put the rest in T-bills.

Show solution

Every point on the line is a mix of T-bills and the tangency portfolio, with $\sigma = w\,\sigma_T$ :

w = \frac{\sigma}{\sigma_T} = \frac{6}{16} = 0.375, \qquad 1 - w = 0.625

\text{E}[R] = r_f + w\,(\text{E}[R_T] - r_f) = 1\% + 0.375 \times (9\% - 1\%) = 1\% + 3\% = 4\%

Check as a weighted average: $0.625 \times 1\% + 0.375 \times 9\% = 0.625\% + 3.375\% = 4\%$ . The investor lends $62.5\%$ of wealth at $1\%$ and holds $37.5\%$ in the tangency portfolio, a point between T-bills and the tangency portfolio: a fairly risk-averse investor.

Problem 4 · Two Investors, One Risky Portfolio

Given (illustrative, per year): $r_f = 1\%$ ; the tangency portfolio ( $\text{E}[R_T] = 9\%$ , $\sigma_T = 16\%$ ) is $40\%$ stock M and $60\%$ stock N. Ana's optimum is at $\sigma = 3\%$ ; Ben's is at $\sigma = 26\%$ . Find Ben's holding of M as a share of his wealth, and the M : N ratio in Ana's stock holdings.

Ben's holding of M, as a share of his wealth

The M : N ratio in Ana's stock holdings

✅ Correct! Ben holds

w = 26/16 = 1.625

of the tangency portfolio, so

1.625 \times 40\% = 65\%

of his wealth is in M; Ana holds far less, but in the same

2 : 3

proportions.

❌ Not scaled.

40\%

is M's share of the tangency portfolio; Ben holds

1.625

times his wealth in that portfolio, so scale by

w

❌ That is N.

1.625 \times 60\% = 97.5\%

is Ben's holding of N; M is

40\%

of the tangency portfolio.

❌ That is the whole risky position.

162.5\%

w = 26/16

, Ben's total in the tangency portfolio; M is only

40\%

of it.

❌ Swapped. M is

40\%

and N is

60\%

of the tangency portfolio, so M : N is

40 : 60 = 2 : 3

❌ That is the key misconception. Risk aversion only moves Ana toward T-bills along the line; the risky part she holds is the same tangency portfolio, in the same proportions.

❌ Not needed. Her indifference curves set only how much she puts in the tangency portfolio; the M : N split inside it is fixed for everyone.

❌ Check Ben's holding. Find

w = \sigma / \sigma_T

, then multiply by M's share of the tangency portfolio.

❌ Check Ana's ratio. Whatever

w

she chooses, both stocks are scaled by the same

w

Show solution

Each investor holds $w = \sigma / \sigma_T$ in the tangency portfolio and $1 - w$ in T-bills.

w_{\text{Ben}} = \frac{26}{16} = 1.625: \quad \text{M} = 1.625 \times 40\% = 65\%, \quad \text{N} = 1.625 \times 60\% = 97.5\%, \quad \text{T-bills} = 1 - 1.625 = -62.5\%

w_{\text{Ana}} = \frac{3}{16} = 0.1875: \quad \text{M} = 7.5\%, \quad \text{N} = 11.25\%, \quad \text{T-bills} = 81.25\%

Ben borrows $62.5\%$ of his wealth at $1\%$ ; Ana lends $81.25\%$ . Yet both hold M and N in the ratio $65 : 97.5 = 7.5 : 11.25 = 2 : 3$ , the tangency portfolio's own proportions. This is conclusion 4: preferences choose the mix with T-bills, never the risky portfolio.

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