Lesson 6 of 6 11–14% of the exam

By Daniil Tarakanov Updated Oct 22, 20262027 curriculum

From the capital allocation line to the capital market line and the CAPM

The capital asset pricing model (CAPM) ends a chain. A risk-free asset plus the best risky portfolio gives a straight line, the capital allocation line, whose slope is the Sharpe ratio. For the market it is the capital market line, and expected returns follow r_f + β(E[r_m] − r_f).

8 min cram, 44 min full read. Based on the 2027 CFA® Program Level I curriculum, Quantitative Methods, Learning Module 8.

Tailor this to you: take the self-assessment

Before this: Risk preferences, mean–variance utility and indifference curves, The minimum-variance portfolio and the efficient frontier

On the exam

Explain the selection of an optimal portfolio, given an investor’s risk aversion and the capital allocation line, and how this extends to the market portfolio and the capital market line

Explain how an investor's risk aversion and the capital allocation line select the optimal portfolio, and how this extends to the market portfolio and the capital market line. Expect the risky weight and risk for a target return (above 100% means borrowing), the Sharpe ratio along a line, a capital market line return, beta from a correlation or a covariance, a CAPM return or the market return backed out from it, and telling the three lines apart.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What happens when you add a risk-free asset?

The mix of a risk-free asset and a risky portfolio rests on four extra assumptions:

The risk-free asset has a standard deviation of zero.
The risky portfolio’s expected return is above the risk-free rate.
Only these two assets are held.
The investor can lend and borrow freely at the risk-free rate.

Call the risky portfolio P, the combined position p, and the weight in the risk-free asset $w_f$ . The combined return and risk follow directly. The risk-free asset adds no risk of its own, so the risk is just the risky weight times the risky portfolio’s standard deviation.

Return of a risk-free and risky mix Memorize

E[r_p] = w_f\, r_f + (1 - w_f)\, E[r_P]

Risk of a risk-free and risky mix Memorize

\sigma_p = (1 - w_f)\, \sigma_P

Eliminating the weight gives a straight line from the risk-free rate through the risky portfolio, the capital allocation line (CAL).

Capital allocation line Memorize

E[r_p] = r_f + \dfrac{E[r_P] - r_f}{\sigma_P}\, \sigma_p

Its slope, $(E[r_P] - r_f) \div \sigma_P$ , is the Sharpe ratio (lesson 4), and it is the same at every point on the line. The best CAL touches the efficient frontier at the tangency portfolio. A CAL through any other risky portfolio is flatter and lies below it.

Points between the risk-free rate and P are lending portfolios, with part of the money in the risk-free asset. Points beyond P are borrowing portfolios, with more than 100% of the investor’s own money in P. Together these are the lending and borrowing portfolios on the line.

Risky weight for a target return Memorize

w_P = \dfrac{r_p - r_f}{E[r_P] - r_f}, \quad \sigma_p = w_P\, \sigma_P

Worked example A 3.5% bill and one risky portfolio

The risk-free rate is 3.5%. The risky portfolio P has E[r_P] = 9.5% and σ_P = 15%. The slope is (9.5 − 3.5) ÷ 15 = 0.40.

Target 7.5%. Weight in P = (7.5 − 3.5) ÷ (9.5 − 3.5) = 4 ÷ 6 = 66.67%, so $w_f$ = 33.33% (lending). σ_p = 0.6667 × 15% = 10.00%. Sharpe ratio (7.5 − 3.5) ÷ 10 = 0.40.
Target 12.5%. Weight in P = 9 ÷ 6 = 150%, so $w_f$ = −50%: the investor borrows half of their own money at 3.5%. σ_p = 1.5 × 15% = 22.50%. Sharpe ratio (12.5 − 3.5) ÷ 22.5 = 0.40.
A less efficient portfolio. A CAL through a portfolio N below the frontier (E[r] 8%, σ 18%) has slope (8 − 3.5) ÷ 18 = 0.25. Every point on it is beaten by the 0.40 line.

The 12.5% target on your calculator

(12.5−3.5)÷(9.5−3.5)= 1.5, the weight in the risky portfolio
×15= 22.5, the standard deviation
(12.5−3.5)÷22.5= 0.4, the Sharpe ratio

12.5ENTER3.5−9.5ENTER3.5−÷ 1.5, the weight in the risky portfolio
15× 22.5, the standard deviation
12.5ENTER3.5−22.5÷ 0.4, the Sharpe ratio

ENTER enter (on the HP, puts the number on the stack)

Rearrange itRisk-free asset plus a risky portfolio: solve for the weight, the risk, the return or the Sharpe ratio

The question asks for

The risk-free rate is 3.50%. A risky portfolio has an expected return of 9.50%. An investor wants an expected return of 12.50% by combining the two, borrowing at the risk-free rate if needed. The weight in the risky portfolio should be closest to:

Risk-free rate, r_f%Risky portfolio's expected return, E[r_P]%Combined portfolio's expected return, E[r_p]%

w_{P} = \frac{E [ r _{p} ] - r _{f}}{E [ r _{P} ] - r _{f}}

(12.5 − 3.5) ÷ (9.5 − 3.5) = 1.5, so 150.00% (borrowing)

Weight in the risky portfolio150.00%

On your calculator

(12.5−3.5)÷(9.5−3.5)=as a decimal

12.5ENTER3.5−9.5ENTER3.5−÷as a decimal

ENTER enter (on the HP, puts the number on the stack)

Where on the line does each investor sit?

At the point where the investor’s highest attainable indifference curve (lesson 5) touches the CAL. The optimal investor portfolio sits further left for a more risk-averse investor, with more in the risk-free asset (lending). A more risk-tolerant investor sits further right, past the risky portfolio, by borrowing (leverage).

All investors use the same risky portfolio. Only the split between it and the risk-free asset differs.

The market portfolio and the capital market line

If every investor sees the same risky assets the same way, they all hold the same tangency portfolio. That portfolio must then be the market portfolio : all risky assets at their market-value weights. Its CAL is the capital market line (CML), and the market portfolio has the highest Sharpe ratio available.

Capital market line Memorize

E[r_p] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m}\, \sigma_p

The mutual fund theorem says every investor can reach an efficient portfolio with just two funds: the market portfolio and the risk-free asset, lending or borrowing to set the risk. (Portfolio Management calls this two-fund separation.) In practice the market is proxied by a broad index, which never holds every investable asset.

Worked example An optimal portfolio on the capital market line

Take portfolio P above as the market: r_f 3.5%, E[r_m] 9.5%, σ_m 15%. An investor holds an optimal portfolio with σ_p = 9%.

Capital market line. E[r_p] = 3.5 + 0.40 × 9 = 7.10%.
Beta. On the CML the portfolio is perfectly correlated with the market, so its beta is σ_p ÷ σ_m = 9 ÷ 15 = 0.60.
Check with the capital asset pricing model (CAPM) below. 3.5 + 0.6 × 6 = 7.10%, the same.

CML, beta and the CAPM check on your calculator

(9.5−3.5)÷15×9+3.5= 7.1, the CML return
9÷15= 0.6, the beta
×(9.5−3.5)+3.5= 7.1, the same through the CAPM

9.5ENTER3.5−15÷9×3.5+ 7.1, the CML return
9ENTER15÷ 0.6, the beta
9.5ENTER3.5−×3.5+ 7.1, the same through the CAPM

ENTER enter (on the HP, puts the number on the stack)

InteractiveOne line for everyone: the capital market line and beta

Try this 0 of 3

Hold 150% in the market portfolio
Switch on the non-efficient portfolio and compare the slopes
Beta tab: set ρ to 1 and compare beta with σ_i ÷ σ_m

Share of wealth in the market portfolio100%

The capital market line (CML) is the capital allocation line (CAL) through the market portfolio

Efficient frontier
Dominated half
CML

CML slope (Sharpe ratio)0.400

In the risky portfolio100.0%

E(Rp)9.50%

σp15.00%

PositionAll in the risky portfolio

Weight in the risk-free asset, w_f0.0%

Show working

CML slope = (9.50% − 3.50%) ÷ 15.00% = 0.400

E(Rp) = Rf + w × (E(R) − Rf) = 3.50% + 1 × (9.50% − 3.50%) = 9.50%

σp = w × σ = 1 × 15.00% = 15.00% (the risk-free asset adds no risk)

Weight for a target return: w_P = (r_p − r_f) ÷ (E[r_P] − r_f). For 9.5%: (9.5 − 3.5) ÷ (9.5 − 3.5) = 100% in the market portfolio, so w_f = 0%; σ_p = w_P × σ_P = 1 × 15% = 15%.

Sharpe ratio of this position: (9.5 − 3.5) ÷ 15 = 0.40, the slope of the line.

Calculator. TI BA II Plus: 9.5 [−] 3.5 [=] [×] 1 [+] 3.5 [=] // 9.5, E[r_p]; 15 [×] 1 [=] // 15, σ_p. HP 12C: 9.5 [ENTER] 3.5 [−] 1 [×] 3.5 [+] // 9.5; 15 [ENTER] 1 [×] // 15.

Everyone holds the market and a risk-free asset; beta, not total risk, sets an asset's expected return.

Beta and the capital asset pricing model

The CAPM assumptions add four to the ones above:

Investors share the same views of risk and return.
They all hold mean–variance efficient portfolios, each at their own risk aversion.
The market is in equilibrium: every security is owned.
Investors are price takers.

A single asset is not perfectly correlated with the market, so its total σ is the wrong measure of its risk. Diversification removes non-systematic risk (security-specific risk) at no cost, so only systematic risk (market risk) earns a premium. Beta measures it.

Beta Memorize

\beta_i = \dfrac{\sigma_{i,m}}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i\,\sigma_m}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i}{\sigma_m}

A beta above 1 means an expected return above the market’s, and a beta below 1 means below. The market’s own beta is $\sigma_{m,m} \div \sigma_m^2$ = 1.

The expected return that follows is the security market line (SML), shown here in its covariance form, and then in the beta form called the capital asset pricing model (CAPM).

Security market line Memorize

E[r_i] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m^2}\, \sigma_{i,m}

Capital asset pricing model Memorize

E[r_i] = r_f + \beta_i \left(E[r_m] - r_f\right)

The difference $E[r_m] - r_f$ is the market risk premium . It is expected, not guaranteed.

Worked example Stock K

Stock K has σ_K = 30% and a correlation of 0.55 with the market. The market has σ_m = 15% and E[r_m] = 9.5%. The risk-free rate is 3.5%.

Beta. β_K = 0.55 × 30 ÷ 15 = 1.10. Check with the covariance: σ_K,m = 0.55 × 0.30 × 0.15 = 0.02475, and 0.02475 ÷ 0.0225 = 1.10.
Expected return. E[r_K] = 3.5 + 1.10 × (9.5 − 3.5) = 3.5 + 6.6 = 10.10%. The covariance form of the SML gives the same.
The trap. σ_K ÷ σ_m = 2.0 would give 3.5 + 2.0 × 6 = 15.5%, the same as reading the CML at σ = 30%. Stock K sits below the CML because most of its risk is diversifiable and not paid for.

Beta, then the CAPM return

0.55×30÷15= 1.1, the beta
×(9.5−3.5)+3.5= 10.1

0.55ENTER30×15÷ 1.1, the beta
9.5ENTER3.5−×3.5+ 10.1

ENTER enter (on the HP, puts the number on the stack)

Rearrange itBeta: solve from the correlation, from the covariance, or for the correlation

The question asks for

An asset has a standard deviation of 30.00% and a correlation of 0.55 with the market, whose standard deviation is 15.00%. The asset's beta is closest to:

Correlation with the market, ρ_i,mStandard deviation of the asset, σ_i%Standard deviation of the market, σ_m%

β_{i} = \frac{ρ _{i, m} σ _{i}}{σ _{m}}

0.55 × 30 ÷ 15 = 1.1

Beta (from the correlation)1.1

On your calculator

0.55×30÷15=percent units cancel

0.55ENTER30×15÷percent units cancel

ENTER enter (on the HP, puts the number on the stack)

Rearrange itCapital asset pricing model: solve for the return, beta, the market return or the risk-free rate

The question asks for

The risk-free rate is 3.50% and the expected market return is 9.50%. A stock has a beta of 1.1. Its expected return under the capital asset pricing model is closest to:

Risk-free rate, r_f%Expected market return, E[r_m]%Beta, β_i

E [r_{i}] = r_{f} + β_{i} (E [r_{m}] - r_{f})

3.5 + 1.1 × (9.5 − 3.5) = 10.10%

Expected return of the asset, E[r_i]10.10%

On your calculator

1.1×(9.5−3.5)+3.5=in percent

9.5ENTER3.5−1.1×3.5+in percent

ENTER enter (on the HP, puts the number on the stack)

A question can also run the CAPM backwards.

Three lines look alike and do different jobs.

Line	x-axis	Holds for	Slope
Capital allocation line	σ of the mix	a risk-free asset plus any one risky portfolio	that portfolio’s Sharpe ratio
Capital market line	σ of the mix	a risk-free asset plus the market portfolio (efficient portfolios only)	the market’s Sharpe ratio
Security market line	beta (or σ_i,m)	every asset and portfolio, efficient or not	the market risk premium (per unit of beta)

Three lines, three jobs

Risk–return lines

Capital allocation line Memorize risk-free + one risky portfolio; slope = its Sharpe ratio
Capital market line Memorize risk-free + market portfolio; efficient portfolios only
Security market line Memorize expected return vs beta; every asset

Which risk is paid

Total risk σ_i

Systematic (market) risk Memorize measured by beta; earns the market risk premium
Non-systematic (security-specific) risk Memorize diversifiable; earns nothing

How is beta estimated?

Beta estimation by regression uses the asset’s excess returns on the market’s excess returns. The slope is the historical beta, and the intercept is the alpha .

Beta regression Use it

r_i - r_f = \alpha_i + \beta_i \left(r_m - r_f\right) + \varepsilon_i

In practice, people use weekly or monthly returns over one to five years. A longer window gives a more stable estimate but can miss recent changes in the business. The choice of market index (a broad national or regional index) changes the answer. The CAPM predicts α = 0 in an efficient market, and persistent non-zero alphas found in studies are one of the model’s main criticisms. The CAPM also assumes the risk and return relation is a straight line.

Learning Module 10 tests numbers like these properly.

What are the limits?

The limitations of portfolio theory are worth knowing by name:

Returns are not normal: they show skewness and fat tails.
Correlations are not constant, and they rise in stress.
Markets are not perfectly efficient.
Estimates of expected returns and covariances are uncertain, and small errors move the optimal weights a lot.
History may not repeat.
Trading has costs.
Taxes differ for income and gains, and tax-loss harvesting can matter. Building tax-aware portfolios is beyond this module.

Memorize this

MemorizeFrom the capital allocation line to the CAPM

Line or measure	Formula
Capital allocation line	$E[r_p] = r_f + \frac{E[r_P] - r_f}{\sigma_P}\sigma_p$
Weight for a target return	$w_P = \frac{r_p - r_f}{E[r_P] - r_f}$ ; $\sigma_p = w_P\sigma_P$
Capital market line	$E[r_p] = r_f + \frac{E[r_m] - r_f}{\sigma_m}\sigma_p$
Beta	$\frac{\sigma_{i,m}}{\sigma_m^2} = \frac{\rho_{i,m}\sigma_i}{\sigma_m}$
Capital asset pricing model (security market line)	$E[r_i] = r_f + \beta_i(E[r_m] - r_f)$

Memory hook Capital allocation and capital market lines: x = σ. Security market line: x = β. σ_p ÷ σ_m is beta only on the capital market line.

Know it cold Slope of a capital allocation line = Sharpe ratio, constant along the line. Weight above 100% = borrowing. Only systematic risk is priced.

Three lines, three jobs

Risk–return lines

Capital allocation line Memorize risk-free + one risky portfolio; slope = its Sharpe ratio
Capital market line Memorize risk-free + market portfolio; efficient portfolios only
Security market line Memorize expected return vs beta; every asset

Which risk is paid

Total risk σ_i

Systematic (market) risk Memorize measured by beta; earns the market risk premium
Non-systematic (security-specific) risk Memorize diversifiable; earns nothing

Return of a risk-free and risky mix

E[r_p] = w_f\, r_f + (1 - w_f)\, E[r_P]

Quick example

40% in a 3% risk-free asset, 60% in a risky portfolio with E[r] of 11%. 0.4 × 3 + 0.6 × 11 = 7.8%.

On your calculator

0.4×3+(0.6×11)= 7.8

0.4ENTER3×0.6ENTER11×+ 7.8

ENTER enter (on the HP, puts the number on the stack)

Risk of a risk-free and risky mix

\sigma_p = (1 - w_f)\, \sigma_P

Quick example

60% in a risky portfolio with σ of 16% and 40% in the risk-free asset. 0.6 × 16 = 9.6%.

On your calculator

0.6×16= 9.6

0.6ENTER16× 9.6

ENTER enter (on the HP, puts the number on the stack)

Capital allocation line

E[r_p] = r_f + \dfrac{E[r_P] - r_f}{\sigma_P}\, \sigma_p

Quick example

r_f 3%, risky portfolio E[r] 11% and σ 16%. At σ_p = 9.6%: 3 + (8 ÷ 16) × 9.6 = 7.8%. The slope, 8 ÷ 16 = 0.50, is the Sharpe ratio.

On your calculator

(11−3)÷16×9.6+3= 7.8

11ENTER3−16÷9.6×3+ 7.8

ENTER enter (on the HP, puts the number on the stack)

Risky weight for a target return

w_P = \dfrac{r_p - r_f}{E[r_P] - r_f}, \quad \sigma_p = w_P\, \sigma_P

Quick example

r_f 2%, risky portfolio 8%, target 10%. w = (10 − 2) ÷ (8 − 2) = 133.3%, so borrow 33.3% of your own money.

On your calculator

(10−2)÷(8−2)= 1.3333

10ENTER2−8ENTER2−÷ 1.3333

ENTER enter (on the HP, puts the number on the stack)

Capital market line

E[r_p] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m}\, \sigma_p

Quick example

r_f 3%, E[r_m] 9%, σ_m 15%, σ_p 9%. 3 + (6 ÷ 15) × 9 = 6.6%.

On your calculator

(9−3)÷15×9+3= 6.6

9ENTER3−15÷9×3+ 6.6

ENTER enter (on the HP, puts the number on the stack)

Beta

\beta_i = \dfrac{\sigma_{i,m}}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i\,\sigma_m}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i}{\sigma_m}

Quick example

ρ = 0.5, σ_i = 24%, σ_m = 16%: β = 0.5 × 24 ÷ 16 = 0.75. From a covariance: σ_i,m = 0.0288 with σ_m = 16% gives 0.0288 ÷ 0.0256 = 1.125.

On your calculator

0.5×24÷16= 0.75
0.0288÷0.16x²= 1.125

x² square

0.5ENTER24×16÷ 0.75
0.0288ENTER0.16ENTER×÷ 1.125

ENTER enter (on the HP, puts the number on the stack)

Security market line

E[r_i] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m^2}\, \sigma_{i,m}

Quick example

r_f 2%, E[r_m] 8%, σ_m 20%, σ_i,m = 0.052. (8 − 2) ÷ 0.04 × 0.052 + 2 = 7.8 + 2 = 9.8%. The same as a beta of 1.3.

On your calculator

(8−2)÷0.2x²×0.052+2= 9.8

x² square

8ENTER2−0.2ENTER×÷0.052×2+ 9.8

ENTER enter (on the HP, puts the number on the stack)

Capital asset pricing model

E[r_i] = r_f + \beta_i \left(E[r_m] - r_f\right)

Quick example

r_f 2.5%, E[r_m] 8.5%, β 0.75. 2.5 + 0.75 × (8.5 − 2.5) = 7.0%.

On your calculator

0.75×(8.5−2.5)+2.5= 7

8.5ENTER2.5−0.75×2.5+ 7

ENTER enter (on the HP, puts the number on the stack)

Beta regression

r_i - r_f = \alpha_i + \beta_i \left(r_m - r_f\right) + \varepsilon_i

Quick example

Regress a stock's monthly excess returns on the market's. A slope of 1.35 and an intercept of −0.05% a month mean a beta of 1.35 and an alpha of −0.05% a month.

How it is asked

Weight and risk on the capital allocation line

r_f 3%, risky 11% (σ 20%), target 17%.

w = 14 ÷ 8 = 175%; σ = 35%. Trap: 30.9% (weight 17 ÷ 11).
Capital market line return

r_f 2.5%, E[r_m] 9%, σ_m 20%, optimal portfolio σ 14%.

2.5 + (6.5 ÷ 20) × 14 = 7.05%. A beta of 0.7 gives the same through the capital asset pricing model.
Beta from correlation

ρ 0.65, σ_i 28%, σ_m 20%.

0.65 × 28 ÷ 20 = 0.91. Traps: 1.4 (σ_i ÷ σ_m), 0.46 (inverted).
Back out the market return

β 1.25, capital asset pricing model return 11.25%, r_f 3%.

3 + 8.25 ÷ 1.25 = 9.6%. Trap: 6.6% (the premium only).

Exam traps

Using σ_i ÷ σ_m as beta for a single asset. That is true only on the capital market line.
Plugging the market return instead of the market risk premium into the capital asset pricing model.
Applying the capital market line to a single stock. Use the security market line.
Forgetting that a weight above 100% means borrowing, or scaling σ by the target return ratio instead of the risky weight.
Treating the market risk premium as guaranteed.
Mixing up the capital allocation line (any risky portfolio), the capital market line (the market) and the security market line (beta on the x-axis).

Checkpoint

3 exam-style questions. About 90 seconds each.

1The line that plots expected return against beta for every security, efficient or not, is the:
2The risk-free rate is 2% and the market portfolio has an expected return of 7% and a standard deviation of 18%. A stock has a standard deviation of 27% and a correlation of 0.80 with the market. Its expected return under the capital asset pricing model is closest to:
3The risk-free rate is 4%. A risky portfolio on the efficient frontier has an expected return of 10% and a standard deviation of 18%. An investor who wants an expected return of 13% by combining the two, borrowing at the risk-free rate if needed, will have a portfolio standard deviation closest to:

Key takeaways

Risk-free plus a risky portfolio: E[r_p] = w_f r_f + (1 − w_f)E[r_P] and σ_p = (1 − w_f)σ_P, a straight line whose slope is the Sharpe ratio.
Weight in the risky portfolio for a target: (r_p − r_f) ÷ (E[r_P] − r_f). Above 100% means borrowing.
With the market portfolio the line is the capital market line. Two funds, the market and the risk-free asset, are enough for everyone.
Beta = σ_i,m ÷ σ_m² = ρσ_i ÷ σ_m. Capital asset pricing model: E[r_i] = r_f + β_i(E[r_m] − r_f). Only systematic risk is priced.
Beta is estimated by regressing excess returns. The capital asset pricing model predicts an intercept (alpha) of zero.

Return of a risk-free and risky mix

E[r_p] = w_f\, r_f + (1 - w_f)\, E[r_P]

Quick example

40% in a 3% risk-free asset, 60% in a risky portfolio with E[r] of 11%. 0.4 × 3 + 0.6 × 11 = 7.8%.

On your calculator

0.4×3+(0.6×11)= 7.8

0.4ENTER3×0.6ENTER11×+ 7.8

ENTER enter (on the HP, puts the number on the stack)

Risk of a risk-free and risky mix

\sigma_p = (1 - w_f)\, \sigma_P

Quick example

60% in a risky portfolio with σ of 16% and 40% in the risk-free asset. 0.6 × 16 = 9.6%.

On your calculator

0.6×16= 9.6

0.6ENTER16× 9.6

ENTER enter (on the HP, puts the number on the stack)

Capital allocation line

E[r_p] = r_f + \dfrac{E[r_P] - r_f}{\sigma_P}\, \sigma_p

Quick example

r_f 3%, risky portfolio E[r] 11% and σ 16%. At σ_p = 9.6%: 3 + (8 ÷ 16) × 9.6 = 7.8%. The slope, 8 ÷ 16 = 0.50, is the Sharpe ratio.

On your calculator

(11−3)÷16×9.6+3= 7.8

11ENTER3−16÷9.6×3+ 7.8

ENTER enter (on the HP, puts the number on the stack)

Risky weight for a target return

w_P = \dfrac{r_p - r_f}{E[r_P] - r_f}, \quad \sigma_p = w_P\, \sigma_P

Quick example

r_f 2%, risky portfolio 8%, target 10%. w = (10 − 2) ÷ (8 − 2) = 133.3%, so borrow 33.3% of your own money.

On your calculator

(10−2)÷(8−2)= 1.3333

10ENTER2−8ENTER2−÷ 1.3333

ENTER enter (on the HP, puts the number on the stack)

Capital market line

E[r_p] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m}\, \sigma_p

Quick example

r_f 3%, E[r_m] 9%, σ_m 15%, σ_p 9%. 3 + (6 ÷ 15) × 9 = 6.6%.

On your calculator

(9−3)÷15×9+3= 6.6

9ENTER3−15÷9×3+ 6.6

ENTER enter (on the HP, puts the number on the stack)

Beta

\beta_i = \dfrac{\sigma_{i,m}}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i\,\sigma_m}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i}{\sigma_m}

Quick example

ρ = 0.5, σ_i = 24%, σ_m = 16%: β = 0.5 × 24 ÷ 16 = 0.75. From a covariance: σ_i,m = 0.0288 with σ_m = 16% gives 0.0288 ÷ 0.0256 = 1.125.

On your calculator

0.5×24÷16= 0.75
0.0288÷0.16x²= 1.125

x² square

0.5ENTER24×16÷ 0.75
0.0288ENTER0.16ENTER×÷ 1.125

ENTER enter (on the HP, puts the number on the stack)

Security market line

E[r_i] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m^2}\, \sigma_{i,m}

Quick example

r_f 2%, E[r_m] 8%, σ_m 20%, σ_i,m = 0.052. (8 − 2) ÷ 0.04 × 0.052 + 2 = 7.8 + 2 = 9.8%. The same as a beta of 1.3.

On your calculator

(8−2)÷0.2x²×0.052+2= 9.8

x² square

8ENTER2−0.2ENTER×÷0.052×2+ 9.8

ENTER enter (on the HP, puts the number on the stack)

Capital asset pricing model

E[r_i] = r_f + \beta_i \left(E[r_m] - r_f\right)

Quick example

r_f 2.5%, E[r_m] 8.5%, β 0.75. 2.5 + 0.75 × (8.5 − 2.5) = 7.0%.

On your calculator

0.75×(8.5−2.5)+2.5= 7

8.5ENTER2.5−0.75×2.5+ 7

ENTER enter (on the HP, puts the number on the stack)

Beta regression

r_i - r_f = \alpha_i + \beta_i \left(r_m - r_f\right) + \varepsilon_i

Quick example

Regress a stock's monthly excess returns on the market's. A slope of 1.35 and an intercept of −0.05% a month mean a beta of 1.35 and an alpha of −0.05% a month.

All Quantitative Methods formulas, every lesson

Practice exam-style questions on Portfolio Return and Risk

Next module: Simulating Prices and Returns Historical, bootstrap or Monte Carlo: the four-step engine

Questions about this lesson?

Where this comes back

Simple Linear Regression in Finance Quantitative Methods
The capital market line Portfolio Risk and Return: Part II (CAPM and Performance Measures)
The CAPM and the security market line Portfolio Risk and Return: Part II (CAPM and Performance Measures)
The CAPM, the Market Model and Factor Models Equities

Questions candidates ask

What is the difference between the capital allocation line (CAL), the capital market line (CML) and the security market line (SML)?

The capital allocation line combines a risk-free asset with any risky portfolio, against σ. The capital market line is the CAL through the market portfolio. The security market line plots expected return against beta and prices every asset.

How do you calculate beta from correlation?

β = ρσ_i ÷ σ_m. A stock with σ 30% and correlation 0.55 with a market of σ 15% has β = 0.55 × 30 ÷ 15 = 1.10. σ_i ÷ σ_m alone is beta only for a portfolio perfectly correlated with the market.

What does a weight above 100% on the capital allocation line mean?

The investor borrows at the risk-free rate and puts more than all of their own money into the risky portfolio. Return and risk both rise along the same line, so the Sharpe ratio does not change.

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