6.1Lesson 6.1 · Module 6 — Portfolio Theory

Portfolios, Weights, and Returns

How a portfolio is represented by weights, how realized and expected returns are calculated, and why risk is different.

Weights sum to 100% for a fully invested portfolio
Realized return = weighted average
Expected return = weighted average
Risk is NOT a simple weighted average

6.1.1Scene 1 · Allocation

A portfolio is its weights

A portfolio is a collection of assets. Rather than track every share count, we describe it with portfolio weights — the fraction of total wealth held in each asset.

w_{i} = \frac{market value of asset i}{total portfolio value}

Each weight is the dollars invested in an asset divided by the total dollars in the portfolio. Pair the dollar holdings directly with the weights they produce:

Asset	Market value	Weight
Stock A	$10,000	10%
Stock B	$60,000	60%
Stock C	$30,000	30%
Total	$100,000	100%

When the portfolio is fully invested, the weights sum to 1 (100%). This is the one place we state the fully-invested condition in this lesson:

w_{A} + w_{B} + w_{C} = 0.10 + 0.60 + 0.30 = 1

Definition · Fully invested

A portfolio where the weights of all assets sum to 1. No money is set aside and none is borrowed.

Weights need not stay between 0 and 100%. A negative weight is a short sale or borrowed funds; a weight above 100% means leverage.

Position	Weight
Risky assets	150%
Borrowing	−50%
Net	100%

The investor puts up equity, borrows an amount equal to 50% of equity, and invests 150% into risky assets. Net weights still sum to 100%. Leverage amplifies both gains and losses.

6.1.2Scene 2 · Return contributions

A portfolio's return is a weighted average

Whether realized or expected, portfolio return is linear in the weights. Each asset's contribution is its weight times its return.

Realized portfolio return

R_{P} = w_{1} R_{1} + w_{2} R_{2} + \dots + w_{n} R_{n} = i = 1 \sum n w_{i} R_{i}

The portfolio's realized return is the weighted average of the individual assets' realized returns over the same period.

$w_{i}$: Weight of asset i.
$R_{i}$: Realized return of asset i.
$R_{P}$: Realized portfolio return.

Stock A is 60% of the portfolio and returned 10%; Stock B is 40% and returned −5%. Distinguish carefully: Stock A returned 10%, but its contribution to the portfolio is only 6 percentage points, because it is only 60% of the money.

Asset A

own return 10%

weight 0.60

Contribution to R_P

6 percentage points

the part of portfolio return A actually delivers

R_{P} = 0.60 \times 10% + 0.40 \times (- 5%) = 6% - 2% = 4%

Expected portfolio return

E [R_{P}] = w_{1} E [R_{1}] + w_{2} E [R_{2}] + \dots + w_{n} E [R_{n}]

Expected return uses the same weighting, but with expected returns. With E[ $R_{A}$ ] = 8% and E[ $R_{B}$ ] = 5%:

$E [R_{i}]$: Expected return of asset i.
$E [R_{P}]$: Expected portfolio return.

E [R_{P}] = 0.60 \times 8% + 0.40 \times 5% = 4.8% + 2.0%

E[ $R_{P}$ ] = 6.8%

Expected return is linear in the weights — easy to compute. Risk, however, is not. That asymmetry is the gateway to the rest of portfolio theory.

6.1.3Scene 3 · Graded worksheet

Portfolio weights and returns worksheet

You hold a $50,000 stock fund, a $30,000 bond fund, and $20,000 in T-bills — $100,000 total. The stock fund realized 12% (expected 8%); the bond fund realized −4% (expected 5%); T-bills realized 1% (expected 2%). Work through weights, contributions, and totals.

Part 1 — Weights

Weight = market value ÷ total.

Stock fund weight

Bond fund weight

T-bill weight

Part 2 — Realized contributions (weight × realized return)

Contribution = weight × realized return, in percentage points.

Stock realized contribution

Bond realized contribution

T-bill realized contribution

Realized portfolio return (sum of contributions)

Part 3 — Expected contributions (weight × expected return)

Same structure, using expected returns.

Stock expected contribution

Bond expected contribution

T-bill expected contribution

Expected portfolio return (sum of contributions)

6.1.4Scene 4 · Transition to risk

Why risk is different

Return is linear. Risk is not. Two assets with identical volatilities can produce very different portfolio risk depending on how they move together.

Expected return

A clean weighted average. Double an asset's weight and you double its contribution. Linear — easy.

E [R_{P}] = w_{A} E [R_{A}] + w_{B} E [R_{B}]

Risk — what people assume

A tempting but wrong shortcut: weight the volatilities the same way. It ignores how assets move together.

σ_{P} \neq = w_{A} σ_{A} + w_{B} σ_{B}

If $σ_{P}$ is not that, what is missing?

The missing piece is co-movement — how the assets move relative to each other. The next lesson introduces covariance and correlation, and shows exactly why portfolio volatility refuses to behave like a weighted average.

Try itMastery check

Pass with 3 of 5 correct

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

01
$60K in A, $40K in B, total $100K. What is $w_{A}$ ?
02
$w_{A}$ = 60%, $R_{A}$ = 10%, $w_{B}$ = 40%, $R_{B}$ = −5%. What is $R_{P}$ ?
03
Stock A returns 10% with a 60% weight. Its contribution is:
04
Which is generally true?
05
A weight of −50% could represent:

Lesson summary

1Portfolio weights describe the fraction of wealth in each asset.
2Fully invested weights sum to 100%.
3Realized portfolio return is a weighted average of asset returns.
4Expected portfolio return uses the same weighting with expected returns.
5An asset's contribution is its weight times its return, in percentage points.
6Portfolio risk is not generally a weighted average — co-movement matters.

Replay this lesson Continue to Portfolio Risk, Covariance, and Correlation