Finance-Theory · Unit 16 · Video 5 · Interactive Practice

Mixing In T-Bills: A Straight Line and Leverage

IKey Formulas

Formula	Name	What it says
$\text{E}[R_p] = \omega_{\text{TBill}}\,R_{\text{TBill}} + \omega_{\text{STK}}\,\mu_{\text{STK}}$	Expected return	Still the weighted average of the two means
$\text{Var}[R_p] = \omega_{\text{STK}}^2\,\sigma_{\text{STK}}^2$	Variance with T-bills	The T-bill's variance term and the covariance term are both zero
$\sigma_p = \omega_{\text{STK}}\,\sigma_{\text{STK}}$ for $\omega_{\text{STK}} \ge 0$	Standard deviation	Linear in the stock weight, like the mean, so the mixes lie on a straight line
$\omega_{\text{TBill}} = 1 - \omega_{\text{STK}} < 0$	Leverage	Shorting T-bills is borrowing at the T-bill rate; it pushes the point past the stock along the same line

Key Insight: Two risky assets trace a bullet; a riskless asset and any risky one trace a straight line. On a bullet, only the top half, from the minimum-variance point up, is worth holding: that is the efficient frontier.

IIT-Bills Turn the Bullet into a Line

With $\sigma_{\text{TBill}} = 0$ , both $\text{E}[R_p]$ and $\sigma_p$ are linear in $\omega_{\text{STK}}$ , so every mix lands on one straight line.

Stock weight ω_STK 0.60

IIILeverage Multiplies a Bad Month

A levered return is $\omega_{\text{STK}}$ times the fund's return minus the borrowing cost, so a modest loss can erase your own capital.

Stock weight ω_STK 1.75 Kestrel's return this month −8%

💡 In 2008 the levered positions were far less smooth than a stock index, and estimated correlations jumped toward one, so the true risk was far larger than the $\sigma$ the leverage had been chosen for.

IVOnly the Top Half of the Bullet

Every portfolio on the bottom half has a partner straight above it with the same $\sigma_p$ and a higher mean.

Motorola weight ω_MOT 0.00

💡 The theory builds the frontier but does not say where on it to sit: a cautious saver near the nose and a hedge fund chasing 2% a month both choose by their own preferences over risk and reward.

VQuiz Questions

Problem 1 · A Lending Mix

Given (illustrative): T-bills pay $0.25\%$ per month with no risk; the Kestrel Index has $\mu = 0.9\%$ and $\sigma = 5\%$ per month. You put $30\%$ in T-bills and $70\%$ in Kestrel. Find the portfolio's expected return and standard deviation per month.

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

What is $\sigma_p$ ?

✅ Correct! The mean is the weighted average,

0.3 \times 0.25 + 0.7 \times 0.9 = 0.705\%

, and only the stock term survives in the variance, so

\sigma_p = 0.7 \times 5 = 3.5\%

❌ Not quite.

0.575\%

is the plain average of

0.25\%

and

0.9\%

, which is the

50/50

mix; weight each mean by its own portfolio weight.

❌ Weights swapped.

0.445\%

puts

0.7

on the T-bill rate and

0.3

on Kestrel; the

70\%

belongs to Kestrel.

❌ Missing a term.

0.7 \times 0.9 = 0.63\%

is only Kestrel's contribution; the

30\%

in T-bills still earns

0.3 \times 0.25 = 0.075\%

❌ Not quite.

\sqrt{0.7} \times 5 \approx 4.18

takes the square root of the weight; the variance is

0.7^2 \times 5^2

, and its square root is

0.7 \times 5

❌ Not quite.

0.7^2 \times 5 = 2.45

squares the weight but not

\sigma

;

\text{Var}[R_p] = 0.7^2 \times 5^2 = 12.25

, so

\sigma_p = \sqrt{12.25} = 3.5\%

❌ That is the variance.

0.7^2 \times 5^2 = 12.25

\text{Var}[R_p]

\%^2

; take the square root for the standard deviation.

❌ Check the mean. Weight each mean by its portfolio weight and add.

❌ Check the risk. With

\sigma_{\text{TBill}} = 0

, only the stock term of the variance survives.

Show solution

The weights are $\omega_{\text{TBill}} = 0.3$ and $\omega_{\text{STK}} = 0.7$ , summing to one.

\text{E}[R_p] = 0.3 \times 0.25 + 0.7 \times 0.9 = 0.075 + 0.63 = 0.705\%

The T-bill's variance is zero, and so is its covariance with Kestrel, because its return is not random:

\text{Var}[R_p] = 0.3^2 \times 0^2 + 0.7^2 \times 5^2 + 2 \times 0.3 \times 0.7 \times 0 = 12.25\ (\%^2)

\sigma_p = \sqrt{12.25} = 0.7 \times 5 = 3.5\%

The portfolio earns $0.705\%$ per month at a standard deviation of $3.5\%$ per month, a point on the segment between T-bills and Kestrel.

Problem 2 · The Cost of Borrowing

Given (illustrative): the same T-bills ( $0.25\%$ per month) and Kestrel Index ( $\mu = 0.9\%$ , $\sigma = 5\%$ per month). You borrow $75\%$ of your wealth at the T-bill rate and put it, with all your own money, into Kestrel, so $\omega_{\text{STK}} = 1.75$ . Find the expected return per month.

✅ Correct!

1.75 \times 0.9 - 0.75 \times 0.25 = 1.575 - 0.1875 = 1.3875\%

: the borrowed money earns Kestrel's return but costs the T-bill rate.

❌ The loan is not free.

1.75 \times 0.9 = 1.575\%

ignores the interest; the T-bill weight is

-0.75

, so you pay

0.75 \times 0.25 = 0.1875\%

❌ Sign error.

1.7625\%

adds the interest; with

\omega_{\text{TBill}} = -0.75

the T-bill term is

-0.75 \times 0.25

, a cost.

❌ Not quite. Leverage changes the weights, so it changes the mean: the expected return moves up the line along with the risk.

Show solution

Borrowing $75\%$ of your wealth is a short position in T-bills: $\omega_{\text{TBill}} = -0.75$ and $\omega_{\text{STK}} = 1.75$ , which still sum to one.

\text{E}[R_p] = -0.75 \times 0.25 + 1.75 \times 0.9 = -0.1875 + 1.575 = 1.3875\%

The risk rises in fixed proportion, $\sigma_p = 1.75 \times 5 = 8.75\%$ per month, so the levered point sits on the dashed extension of the T-bill–Kestrel line, past Kestrel.

Problem 3 · Hitting a Target Return

Given (illustrative): T-bills at $0.25\%$ per month and the Kestrel Index with $\mu = 0.9\%$ , $\sigma = 5\%$ per month. You want an expected return of $1.16\%$ per month from these two alone. Find the stock weight you need and the resulting standard deviation.

What is $\omega_{\text{STK}}$ ?

What is $\sigma_p$ ?

✅ Correct! Each unit of stock weight adds

0.9 - 0.25 = 0.65

points to the T-bill rate, so

\omega_{\text{STK}} = 0.91 / 0.65 = 1.4

and

\sigma_p = 1.4 \times 5 = 7\%

: you borrow

40\%

of your wealth.

❌ Not quite.

1.16 / 0.9 \approx 1.29

forgets the T-bill term; with

\omega_{\text{TBill}} = 1 - \omega_{\text{STK}}

, the mean is

0.25 + 0.65\,\omega_{\text{STK}}

❌ Not quite.

1.16 / 0.65 \approx 1.78

divides the whole target by

0.65

; first subtract the

0.25\%

every mix starts from:

(1.16 - 0.25) / 0.65

❌ One step short.

1.16 - 0.25 = 0.91

is the extra return needed; each unit of stock weight supplies only

0.65

of it, so divide by

0.65

❌ Not quite.

\approx 6.44\%

5\%

times the weight

1.16 / 0.9 \approx 1.29

, which leaves out the T-bill term and earns less than

1.16\%

❌ Not quite.

\approx 8.92\%

5\%

times the weight

1.16 / 0.65 \approx 1.78

, which overshoots the target return.

❌ Not quite.

1.4^2 \times 5 = 9.8

squares the weight without squaring

\sigma

;

\sigma_p = \sqrt{1.4^2 \times 5^2} = 1.4 \times 5

❌ Check the weight. Write the mean as

0.25 + (0.9 - 0.25)\,\omega_{\text{STK}}

and solve for

\omega_{\text{STK}}

❌ Check the risk.

\sigma_p = \omega_{\text{STK}} \times 5\%

with the weight that actually hits

1.16\%

Show solution

Substitute $\omega_{\text{TBill}} = 1 - \omega_{\text{STK}}$ into the mean:

\text{E}[R_p] = (1 - \omega_{\text{STK}}) \times 0.25 + \omega_{\text{STK}} \times 0.9 = 0.25 + 0.65\,\omega_{\text{STK}}

0.25 + 0.65\,\omega_{\text{STK}} = 1.16 \quad\Rightarrow\quad \omega_{\text{STK}} = \frac{0.91}{0.65} = 1.4

So $\omega_{\text{TBill}} = -0.4$ : for every $100 of your own money you borrow $40 at the T-bill rate and hold $140 of Kestrel.

\sigma_p = 1.4 \times 5 = 7\%

Check: $-0.4 \times 0.25 + 1.4 \times 0.9 = -0.1 + 1.26 = 1.16\%$ per month.

Problem 4 · Straight Up from a Risky Portfolio

Given (illustrative): a portfolio Q of risky stocks has $\sigma = 6\%$ and $\text{E}[R] = 0.95\%$ per month. T-bills pay $0.25\%$ per month, and the Kestrel Index has $\mu = 0.9\%$ , $\sigma = 5\%$ per month, with borrowing at the T-bill rate. Find the expected return of the T-bill–Kestrel mix with the same $6\%$ standard deviation as Q, and decide whether that mix dominates Q.

Expected return of the mix with $\sigma_p = 6\%$ ?

Does the mix dominate Q?

✅ Correct!

\omega_{\text{STK}} = 6/5 = 1.2

gives

\text{E}[R_p] = -0.2 \times 0.25 + 1.2 \times 0.9 = 1.03\%

at exactly

6\%

, so moving straight up from Q to the line gains

0.08

points at no extra risk.

❌ The loan is not free.

1.2 \times 0.9 = 1.08\%

leaves out the T-bill weight

-0.2

, which costs

0.2 \times 0.25 = 0.05\%

❌ Sign error.

1.13\%

adds the interest on the

20\%

borrowed; with

\omega_{\text{TBill}} = -0.2

it is a cost,

-0.05\%

❌ Missing the starting point.

1.2 \times (0.9 - 0.25) = 0.78

is only the extra return over T-bills; add the

0.25\%

back.

❌ Not quite. With

\sigma_{\text{TBill}} = 0

, borrowing adds no hidden variance:

\sigma_p = 1.2 \times 5 = 6\%

exactly.

❌ Not quite. Comparing Q with Kestrel alone is a trade-off (more return, more risk); the relevant comparison is with the mix at Q's own

6\%

risk, which earns more.

❌ Check the mean. Find

\omega_{\text{STK}}

from

\sigma_p = 5\,\omega_{\text{STK}} = 6

, then use both weights.

❌ Check the comparison. Dominance means a weakly higher mean and weakly lower risk, with at least one strict.

Show solution

Step 1: the weight. On the T-bill–Kestrel line $\sigma_p = 5\,\omega_{\text{STK}}$ , so $\sigma_p = 6\%$ needs $\omega_{\text{STK}} = 1.2$ and $\omega_{\text{TBill}} = -0.2$ (borrow $20\%$ of your wealth).

Step 2: the mean.

\text{E}[R_p] = -0.2 \times 0.25 + 1.2 \times 0.9 = -0.05 + 1.08 = 1.03\%

Step 3: compare. The mix and Q have the same risk, $6\%$ , and the mix has the higher mean, $1.03\% > 0.95\%$ , so the mix dominates Q. This is the same move as from the bottom half of a bullet: straight up, to a higher return at exactly the same risk.

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