Finance-Theory · Unit 17 · Video 3 · Interactive Practice

The Sharpe Ratio: Lending, Borrowing and the Steepest Line

IKey Formulas

Formula	Name	What it says
$\mathsf{E}[R_p] = r_f + \omega_T\,(\mathsf{E}[R_T] - r_f)$	The tangency line	$0 < \omega_T < 1$ : lending at $r_f$ ; $\omega_T = 1$ : fully invested; $\omega_T > 1$ : borrowing at $r_f$
$\sigma_p = \lvert\omega_T\rvert\,\sigma_T$	Risk along the line	The absolute value folds the short side ( $\omega_T < 0$ ) back to the right, below $r_f$
$\text{Sharpe ratio} \equiv \dfrac{\mathsf{E}[R_p] - r_f}{\sigma_p}$	Sharpe ratio	Risk premium per unit of volatility: the slope of the line from $r_f$ through $p$ . Higher is better
$\displaystyle\max_{\omega_1,\dots,\omega_{21}} \text{Sharpe} \;\ge\; \max_{\omega_{21}=0} \text{Sharpe}$	Adding a security	The old portfolio is still available, so the best Sharpe ratio never falls

Key Insight: The tangency portfolio $T$ has the highest Sharpe ratio of any risky portfolio, and every mix of $T$ with T-bills, lending or borrowing at $r_f$ , has exactly the same Sharpe ratio.

IILending, Borrowing, Shorting

One weight $\omega_T$ fixes your place on the line; $\sigma_p = \lvert\omega_T\rvert\,\sigma_T$ folds the short side down and to the right.

Weight in T, ω_T 0.40 Borrowing rate

IIIThe Sharpe Ratio Is a Slope

The line from $r_f = 2\%$ through any feasible portfolio $P$ rises $\mathsf{E}[R_P] - r_f$ over a run of $\sigma_P$ ; none beats $T$ 's.

σ_P 20% E[R_P] 11%

IVA New Stock Can Only Tilt the Line Up

Freeing stock 21's weight from $\omega_{21} = 0$ can only raise the best Sharpe ratio, never lower it.

E[R] of stock 21 9% ρ(stock 21, T) 0.30

💡 The panel mixes $T$ with long positions in stock 21 only. Selling stock 21 short, or re-weighting the twenty old stocks as well, can do still better, so the true new maximum is at least the Sharpe ratio shown.

VQuiz Questions

Problem 1 · A Sharpe Ratio

Given (illustrative, annual): T-bills pay $r_f = 2\%$ . Portfolio $A$ has $\mathsf{E}[R_A] = 15\%$ and $\sigma_A = 26\%$ . Find $A$ 's Sharpe ratio.

✅ Correct!

A

earns

13

points above T-bills on

26

points of volatility:

0.5

of risk premium per unit of risk.

❌ Missing $r_f$ .

15/26 \approx 0.577

divides the whole expected return by

\sigma

; the numerator is the risk premium,

15\% - 2\%

❌ Upside down.

26/13 = 2

is run over rise; the Sharpe ratio is rise over run,

(\mathsf{E}[R_A] - r_f)/\sigma_A

❌ That divides by the variance.

13/26^2 \approx 0.019

; the denominator is the standard deviation

\sigma_A = 26\%

, not

\sigma_A^2

❌ Not quite. Subtract

r_f

from the expected return, then divide by

\sigma_A

Show solution

\text{Sharpe}_A = \frac{\mathsf{E}[R_A] - r_f}{\sigma_A} = \frac{15\% - 2\%}{26\%} = \frac{13}{26} = 0.5

Geometrically, $0.5$ is the slope of the line from the T-bill point $(0, 2\%)$ through $A$ at $(26\%, 15\%)$ : a rise of $13$ over a run of $26$ .

Problem 2 · Below the T-Bill Point

Given (illustrative, annual): $r_f = 2\%$ ; the tangency portfolio has $\mathsf{E}[R_T] = 12\%$ and $\sigma_T = 16\%$ . You short $T$ worth half your wealth, $\omega_T = -0.5$ , and put the proceeds and all your own money into T-bills. Find $\mathsf{E}[R_p]$ and $\sigma_p$ .

What is $\mathsf{E}[R_p]$ ?

What is $\sigma_p$ ?

✅ Correct!

-3\%

8\%

risk: the lending portfolio

\omega_T = 0.5

has the same

8\%

risk and earns

7\%

, so the short position gives up

10

points for nothing.

❌ Sign dropped.

7\%

is the lending portfolio with

\omega_T = +0.5

; with

\omega_T = -0.5

the term

\omega_T(\mathsf{E}[R_T] - r_f)

-5\%

❌ Not the gap.

2\% + (-0.5)(12\%) = -4\%

multiplies

\omega_T

\mathsf{E}[R_T]

; it must multiply the gap

\mathsf{E}[R_T] - r_f = 10\%

❌ Missing the T-bills.

-0.5 \times 12\% = -6\%

counts only the short; the

150\%

in T-bills earns

1.5 \times 2\% = 3\%

❌ Check $\mathsf{E}[R_p]$ . Use

\mathsf{E}[R_p] = r_f + \omega_T(\mathsf{E}[R_T] - r_f)

with

\omega_T = -0.5

❌ A standard deviation is never negative.

\sigma_p^2 = \omega_T^2\sigma_T^2

, so

\sigma_p = \lvert\omega_T\rvert\,\sigma_T

; that absolute value is why the line reflects off the axis.

❌ Not quite.

0.5^2 \times 16 = 4

squares the weight but not

\sigma_T

;

\sigma_p = \sqrt{0.25 \times 256} = 8\%

❌ Wrong weight.

1.5 \times 16 = 24

uses the T-bill weight; T-bills carry no risk, so only

\lvert\omega_T\rvert = 0.5

multiplies

\sigma_T

❌ Check $\sigma_p$ . Only the stake in

T

is risky:

\sigma_p = \lvert\omega_T\rvert\,\sigma_T

Show solution

The weights are $\omega_T = -0.5$ in $T$ and $1 - \omega_T = 1.5$ in T-bills.

\mathsf{E}[R_p] = r_f + \omega_T(\mathsf{E}[R_T] - r_f) = 2\% + (-0.5)(12\% - 2\%) = 2\% - 5\% = -3\%

Check with the weights directly: $1.5 \times 2\% + (-0.5) \times 12\% = 3\% - 6\% = -3\%$ .

\sigma_p^2 = \omega_T^2\,\sigma_T^2 = 0.25 \times 256 = 64 \quad\Rightarrow\quad \sigma_p = \lvert -0.5 \rvert \times 16\% = 8\%

The point $(8\%, -3\%)$ is on the reflected line running down and to the right of the T-bill point: more risk and less return than T-bills alone.

Problem 3 · Borrowing Costs More

Given (illustrative, annual): T-bills pay $2\%$ , but you can borrow only at $6\%$ . The tangency portfolio has $\mathsf{E}[R_T] = 12\%$ and $\sigma_T = 16\%$ . You borrow a quarter of your wealth at $6\%$ and hold $\omega_T = 1.25$ (levering $T$ itself: a simple benchmark, not the best way to take on this much risk at a $6\%$ borrowing rate). Find $\mathsf{E}[R_p]$ and the Sharpe ratio measured against the $2\%$ T-bill rate.

What is $\mathsf{E}[R_p]$ ?

What is the Sharpe ratio $(\mathsf{E}[R_p] - 2\%)/\sigma_p$ ?

✅ Correct! Borrowing at

6\%

costs a point of return relative to borrowing at

2\%

, and the Sharpe ratio drops from

0.625

0.575

: levering

T

at the higher rate falls below the line from

2\%

. It is not the best position at this risk either: with a

6\%

borrowing rate the best set leaves

T

along the frontier to a second tangency point

T_2

, and borrows only beyond

T_2

❌ The loan is not free.

1.25 \times 12\% = 15\%

ignores the interest on the borrowed quarter: subtract

0.25 \times 6\% = 1.5\%

❌ Wrong rate.

14.5\%

charges

2\%

on the loan; here borrowing costs

6\%

, so subtract

0.25 \times 6\%

❌ Not quite.

12\%

T

itself; holding

125\%

of your wealth in

T

changes both the return and the risk.

❌ Check $\mathsf{E}[R_p]$ .

1.25 \times 12\%

minus the interest on the

0.25

you borrowed at

6\%

❌ Only at equal rates. The Sharpe ratio stays at

T

0.625

when you borrow at

r_f

; at

6\%

the levered portfolio sits below that line.

❌ Missing $r_f$ .

13.5/20 = 0.675

uses the whole expected return; the numerator is

\mathsf{E}[R_p] - 2\%

❌ Wrong benchmark.

(13.5 - 6)/20 = 0.375

subtracts the borrowing rate; the question measures the risk premium against the

2\%

T-bill rate.

❌ Check the Sharpe ratio. First

\sigma_p = 1.25 \times 16\%

, then

(\mathsf{E}[R_p] - 2\%)/\sigma_p

Show solution

You hold $1.25$ of your wealth in $T$ and owe $0.25$ at $6\%$ :

\mathsf{E}[R_p] = 1.25 \times 12\% - 0.25 \times 6\% = 15\% - 1.5\% = 13.5\%

Borrowing adds no risk of its own, so $\sigma_p = 1.25 \times 16\% = 20\%$ , and

\text{Sharpe} = \frac{13.5\% - 2\%}{20\%} = \frac{11.5}{20} = 0.575

At a $2\%$ borrowing rate the same position would earn $15\% - 0.5\% = 14.5\%$ , with Sharpe ratio $12.5/20 = 0.625$ , the same as $T$ . So $(20\%, 13.5\%)$ is not on the best attainable set. With unequal rates the best path is no longer one straight line: it bends smoothly from the lending line onto the frontier at $T$ , follows the frontier up to a second tangency point $T_2$ , and only then continues along a flatter line from $6\%$ through $T_2$ . On the illustrative frontier of the first panel, the frontier portfolio with $\sigma = 20\%$ earns about $14.1\%$ with no borrowing at all, and positions riskier than $T_2$ lever $T_2$ , not $T$ .

Problem 4 · The Steepest Line, Then One More Stock

Given (illustrative, annual): $r_f = 2\%$ . Three risky portfolios built from the same twenty stocks: $A$ with $(\sigma, \mathsf{E}[R]) = (26\%, 15\%)$ , $B$ with $(14\%, 9\%)$ and $C$ with $(20\%, 13\%)$ .

Which has the highest Sharpe ratio?

A 21st stock is added to the menu. Its own Sharpe ratio is $0.3$ and its correlation with $C$ is $0.9$ . The best Sharpe ratio now available is…

✅ Correct!

C

's line from

2\%

is the steepest of the three, and adding a security relaxes the constraint

\omega_{21} = 0

, so the best line can only rotate up or stay put.

❌ Return alone is not the ratio.

A

13

points of premium cost

26

points of risk:

(15 - 2)/26 = 0.5

❌ Risk alone is not the ratio.

B

earns only

7

points above T-bills on

14

points of risk:

(9 - 2)/14 = 0.5

❌ They do tie, but not at the top.

A

and

B

both have

0.5

;

C

has

(13 - 2)/20 = 0.55

❌ Check the ranking. Compute

(\mathsf{E}[R] - 2\%)/\sigma

for each portfolio.

❌ You are never forced to buy it. With

\omega_{21} = 0

you still hold

C

and keep

C

's Sharpe ratio, so the maximum cannot fall.

❌ The existing correlations do not change. They are fixed parameters; the new stock only adds a weight to choose, and that weight can be zero.

❌ Negative correlation is not required. Any correlation below

+1

bends the mixes to the left; whether the line rotates up also depends on the stock's expected return.

❌ Not quite. Ask whether the old best portfolio is still on the menu after the stock is added.

Show solution

\text{Sharpe}_A = \frac{15 - 2}{26} = 0.5, \qquad \text{Sharpe}_B = \frac{9 - 2}{14} = 0.5, \qquad \text{Sharpe}_C = \frac{13 - 2}{20} = 0.55

$C$ has the steepest line from the T-bill point, so of these three it is the candidate for the tangency portfolio. $A$ and $B$ lie on the same flatter line from $2\%$ .

Adding the 21st stock: before, you chose $\omega_1, \dots, \omega_{21}$ subject to $\omega_{21} = 0$ ; now $\omega_{21}$ is free. Every portfolio you could hold before, $C$ included, is still available, so

\max_{\omega_1,\dots,\omega_{21}} \text{Sharpe} \;\ge\; \max_{\omega_{21}=0} \text{Sharpe} \;\ge\; 0.55

The new stock's low Sharpe ratio and its high correlation decide only whether, and by how much, the best Sharpe ratio rises; neither can push it below $0.55$ .

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