Introduction to Portfolios

Sukrit Mittal Franklin Templeton Investments

Outline

Risk and return: the core trade-off
Measuring return
Measuring risk: variance
Downside risk and semi-variance
Portfolios and diversification
Two-asset portfolios
Attainable set
Special cases
Minimum variance portfolio
Exercises

1. Risk and Return: The Core Trade-Off

Finance is not about maximizing return.

It is about choosing how much risk you are willing to tolerate for a given return.

No risk, no reward — but also no free lunches.

This trade-off has been understood for centuries.

The mathematics came later.

What Do We Mean by Return?

Return answers a simple question:

How much did my investment change in value?

But even simple questions deserve precise definitions.

Ambiguity here infects everything downstream.

2. Measuring Return

Let:

$V_0$ = initial value
$V_1$ = value at the end of the period

The simple return is:

\[R = \frac{V_1 - V_0}{V_0}\]

This is the object we will work with throughout this lecture.

Random Nature of Returns

Future returns are unknown today.

Hence, return is modeled as a random variable.

This is not pessimism.

It is intellectual honesty.

3. Expected Return

The expected return summarizes the center of the return distribution.

If $R$ takes values $r_i$ with probabilities $p_i$:

\[\mathbb{E}[R] = \sum_i p_i r_i\]

It is a weighted average of possible outcomes.

Interpretation of Expected Return

Not a guaranteed outcome
Not the most likely outcome
A long-run average under repeated trials

Markets do not promise outcomes.

They only offer distributions.

4. Measuring Risk: Variance

Risk is about dispersion, not direction.

The classical measure of risk is variance.

Definition:

\[\text{Var}(R) = \mathbb{E}[(R - \mathbb{E}[R])^2]\]

Large deviations — up or down — increase variance.

Standard Deviation

The square root of variance is the standard deviation:

\[\sigma = \sqrt{\text{Var}(R)}\]

It has the same units as return.

This makes it easier to interpret and compare.

Criticism of Variance

Variance treats:

Upside surprises
Downside disasters

as equally undesirable.

Investors rarely agree with that philosophy.

This criticism is old — and justified.

5. Downside Risk and Semi-Variance

To focus on losses, we define semi-variance.

Let $m$ be a benchmark (often 0 or the mean).

\[\text{SemiVar}(R) = \mathbb{E}[\max(0, m - R)^2]\]

Only downside deviations matter.

Why Semi-Variance Is Less Popular

Harder to compute
Harder to optimize
Breaks some elegant mathematics

But conceptually, it is closer to how humans think about risk.

Beauty and realism rarely coexist.

6. Portfolios and Diversification

A portfolio is a weighted combination of assets.

Let:

$w_i$ = weight of asset $i$
$R_i$ = return of asset $i$

Portfolio return:

\[R_p = \sum_i w_i R_i\]

Diversification is the only free lunch finance ever offered.

Expected Return of a Portfolio

Expectation is linear:

\[\mathbb{E}[R_p] = \sum_i w_i \mathbb{E}[R_i]\]

No interaction terms.

Risk behaves very differently.

7. Two-Asset Portfolios

We now restrict attention to two assets.

Let:

weights: $w$ and $1-w$
returns: $R_1$, $R_2$

This simple case already contains all the essential geometry.

Return of a Two-Asset Portfolio

\[R_p = wR_1 + (1-w)R_2\]

Expected return:

\[\mathbb{E}[R_p] = w\mu_1 + (1-w)\mu_2\]

This is a straight line in $w$.

Risk will not be.

Variance of a Two-Asset Portfolio

Let:

variances: $\sigma_1^2, \sigma_2^2$
covariance: $\sigma_{12}$

Then:

\[\sigma_p^2 = w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2w(1-w)\sigma_{12}\]

This single equation explains diversification.

Role of Correlation

Define correlation:

\[\rho = \frac{\sigma_{12}}{\sigma_1\sigma_2}\]

$\rho = 1$: no diversification
$\rho < 1$: risk reduction
$\rho = -1$: perfect hedging

Correlation is more important than volatility.

8. Attainable Set

As $w$ varies, the pair:

\[(\sigma_p, \mathbb{E}[R_p])\]

traces a curve.

This curve is the attainable set of portfolios.

It summarizes all feasible risk-return combinations.

Geometry of the Attainable Set

A straight line in return space
A curve in risk-return space

The shape depends entirely on correlation.

This is geometry, not economics.

9. Special Cases

Case 1: Perfect Positive Correlation ($\rho=1$)

No diversification benefit.

Portfolio risk is a weighted average.

Case 2: Zero Correlation ($\rho=0$)

Risk is reduced, but not eliminated.

Diversification works quietly.

Case 3: Perfect Negative Correlation ($\rho=-1$)

There exists a risk-free portfolio.

Variance can be driven to zero.

This is rare — and powerful.

10. Minimum Variance Portfolio

We now minimize $\sigma_p^2$ with respect to $w$.

The solution:

\[w^* = \frac{\sigma_2^2 - \sigma_{12}}{\sigma_1^2 + \sigma_2^2 - 2\sigma_{12}}\]

This portfolio has the lowest possible risk.

Regardless of expected returns.

Interpretation

Depends only on variances and covariance
Independent of investor preferences
Forms the base of the efficient frontier

Optimization comes later.

Structure comes first.

11. Exercises

Exercise 1

Two assets have:

$\mu_1=10%$, $\sigma_1=20%$
$\mu_2=6%$, $\sigma_2=10%$
$\rho=0.3$

Compute the expected return and variance for $w=0.5$.

Exercise 2

For the same assets, vary $w$ from 0 to 1.

Sketch the attainable set in $(\sigma, \mu)$ space.

Identify the minimum variance portfolio.

Exercise 3

Construct an example where semi-variance ranks two portfolios differently than variance.

Explain which ranking you find more intuitive, and why.

Final Takeaways

Risk and return are inseparable
Expected return is linear, risk is not
Variance is convenient, not perfect
Diversification emerges from correlation
Even two assets generate rich structure

From here, modern portfolio theory begins in earnest.