Chapter 01

Easy

Returns, Risk, and the Opportunity Set

00 · Symbol Glossary

$R$R — simple net return

$R = P_1/P_0 - 1$ . The P&L over the period divided by starting value. Portfolio theory adds these across assets; it does not add them across time.

$G$G — gross return

$G = P_1/P_0 = 1+R$ . The growth factor on one unit of numeraire. Gross returns multiply across dates; net returns do not.

$r$r — log return

$r = \log G = \log(1+R)$ , natural log. Log returns add across dates. They do not form a portfolio by value-weighting.

$w_i$w sub i — portfolio weight

The fraction of portfolio value held in asset $i$ at the start of the period. Fully invested books satisfy $\sum_i w_i = 1$ . A negative weight is a short.

$R_p$R sub p — portfolio simple return

$R_p = \sum_i w_i R_i$ , the value-weighted sum of simple returns. This is the object whose mean and variance later chapters optimize.

$\mu$mu — expected return

$\mu = \mathbb{E}[R]$ , the probability-weighted average of the simple return. Read "mu". The vertical coordinate of the opportunity set.

$\sigma^2$sigma squared — variance

$\sigma^2 = \mathrm{Var}(R) = \mathbb{E}[(R-\mu)^2]$ . The second central moment. Units are return squared, not return.

$\sigma$sigma — volatility

$\sigma = \sqrt{\mathrm{Var}(R)}$ . Dispersion of the simple return in the same units as $\mu$ . The horizontal coordinate of the opportunity set.

$R_f$R sub f — risk-free rate

The simple return on the asset whose payoff over this one period is contracted, so its variance in the numeraire is zero. A constant, not a random variable, in this chapter.

$R-R_f$excess return

Simple return minus the risk-free rate. Shifts the mean by $R_f$ and, because $R_f$ is constant, leaves the variance unchanged. The capital market line built from excess returns is Chapter 05.

01 · One Period, One Numeraire

Portfolio theory begins before any optimizer. Fix a horizon — one year, one quarter, one rebalance — and a numeraire, the unit in which wealth is scored. An asset is then a random end-of-period value, and the only question this chapter answers is how to turn that value into a return, how to score a book of returns, and how to place each asset on a plane whose axes are expected return and volatility.

The probability required is the discrete-RV calculus already in hand. If a return takes finitely many values $x_k$ with probabilities $p_k$ , then

\mathbb{E}[R] = \sum_k x_k\, p_k, \qquad \mathrm{Var}(R) = \sum_k (x_k - \mathbb{E}[R])^2 p_k

provided those sums are finite. Linearity of expectation holds whether or not the summands are independent. Variance is not linear. This chapter uses those two facts and does not rebuild them. Not here: the variance of a sum of assets (Chapter 02), the weights that minimize it (Chapter 03), and the line from mixing cash with a risky book (Chapter 05).

02 · Simple and Gross Returns

Definition — Simple Return and Gross Return

Given a start price $P_0>0$ and an end price $P_1$ , the gross return and the simple net return are

G = \frac{P_1}{P_0}, \qquad R = G - 1 = \frac{P_1 - P_0}{P_0}

$G$ is the factor that takes one unit of numeraire into end-of-period wealth. $R$ is the profit per unit of starting value. A price that doubles has $G=2$ and $R=1$ , not $R=2$ .

Prices may include a distribution: if the asset pays a cash amount $D$ over the period and the end price is the ex-distribution price, the honest gross return is $(P_1+D)/P_0$ . Dropping $D$ measures the price change, not the holding-period return. Every formula below assumes $R$ is already the total return.

Example — Equity index, one-year total return

An index starts at $100$ and finishes the year at $120$ , with no separate distribution. Then

G = \frac{120}{100} = 1.20, \qquad R = 1.20 - 1 = 0.20

One unit of numeraire became $1.20$ units. The desk's P&L line is $+0.20$ , or 20 percent of starting NAV, not $+1.20$ .

❌ Booking the gross factor as the P&L

The same index is reported as a "return of $1.20$ " and a performance fee is charged on $120\%$ of NAV.

Why it breaks: $1.20$ is $G$ , the wealth relative. The net profit per unit of capital is $R=G-1=0.20$ . The identity $G=1+R$ is a change of units, not a rounding convention.

Consequence: fees, excess returns, and any later comparison with $R_f$ are scaled by the wrong object. Quote $G$ when wealth multiplies. Quote $R$ when P&L is a rate. Limited liability gives $R\ge -1$ ; a financed short need not. The bound is about the instrument, not the definition.

03 · Log Returns and Compounding

Definition — Log Return

The log return, or continuously compounded return, is

r = \log G = \log(1+R)

with $\log$ the natural logarithm. It is defined for $G>0$ , i.e. $R>-1$ .

Two properties decide when $r$ is the right object and when $R$ is.

Across dates, gross returns multiply and log returns add. If $G_t$ is the gross return in period $t$ , then over $T$ periods

G(0,T) = \prod_{t=1}^{T} G_t, \qquad r(0,T) = \sum_{t=1}^{T} r_t = \log G(0,T)

The multi-period simple return is $R(0,T)=G(0,T)-1$ . It is not the sum of the single-period simple returns. The cross terms are the whole gap:

(1+R_1)(1+R_2) = 1 + R_1 + R_2 + R_1 R_2

Across assets at one date, the portfolio identity runs the other way. Value weights combine simple returns. They do not combine log returns. Section 04 writes the identity; the failure mode sits below so the two conventions do not get swapped.

For small $R$ , the Taylor expansion $\log(1+R) = R - R^2/2 + R^3/3 - \cdots$ says the log return sits below the simple return by about half the squared return. At $|R|$ of a few percent the gap is a few basis points. At a 20 percent year it is not a rounding error.

Step-by-step — Two years of the index, from 100

Year-one simple return: $P_0=100$ , $P_1=120$ , so $R_1 = 120/100 - 1 = 0.20$ and $G_1=1.20$ . The $0.20$ is end price minus start price, divided by the start price.

Year-two simple return: the index then falls 4 percent, so $R_2=-0.04$ , $G_2=0.96$ , and $P_2 = 120 \times 0.96 = 115.2$ .

Compound the gross factors: $G(0,2) = 1.20 \times 0.96 = 1.152$ . The multi-period simple return is $R(0,2) = 1.152 - 1 = 0.152$ . Wealth is $115.2$ on a start of $100$ .

Compare with the sum of net returns: $R_1+R_2 = 0.20 + (-0.04) = 0.16$ . The gap $0.152 - 0.16 = -0.008$ equals the cross term $R_1 R_2 = (0.20)(-0.04)$ . Summing net returns overstates the two-year P&L by 80 basis points.

Add the log returns instead: $\log(1.20) \approx 0.1823$ and $\log(0.96) \approx -0.0408$ . The sum is $\approx 0.1415$ , and $\log(1.152) \approx 0.1415$ .

Example — The quadratic gap at a 20 percent year

For the up year alone, $\log(1.20)\approx 0.1823$ while $R - R^2/2 = 0.20 - 0.02 = 0.18$ . Treating 20 percent simple and 20 percent log as the same number is off by about 1.8 percentage points on that year. The next Taylor term is $R^3/3 = 0.008/3 \approx 0.0027$ , which accounts for most of the $0.0023$ still missing from $0.18$ .

❌ Summing simple returns across a path

A two-year track record is summarized as $0.20 + (-0.04) = 0.16$ , and that figure is entered as the holding-period return on a start-NAV of $100$ .

Why it breaks: the holding-period simple return is $P_2/P_0 - 1 = 0.152$ , because gross returns multiply. The sum of net returns drops the cross term $R_1 R_2$ .

Consequence: reported wealth is $116$ instead of $115.2$ . Use $\prod G_t - 1$ , or sum log returns and exponentiate. Scaling a one-period mean by the number of periods, or a volatility by the square root of that number, is a stacking assumption from time series, not a definition in this chapter.

04 · The Portfolio Return Is a Weighted Sum

Definition — Value Weight and Portfolio Simple Return

Let asset $i$ have start value $V_i$ , and let $V = \sum_i V_i$ be portfolio NAV, with $V\neq 0$ . The value weight is $w_i = V_i/V$ . The portfolio's simple return over the period is

R_p = \sum_i w_i R_i, \qquad \sum_i w_i = 1

when every holding, including residual cash, is included in the sum. Weights are fixed at the start of the period: they are numbers, not random variables. A weight that changes with the path is a trading rule, and it is outside this chapter.

End value is $\sum_i V_i(1+R_i) = V(1+\sum_i w_i R_i)$ , so the identity is accounting. The same line is why log returns are the wrong input: $\log(1+R_p)=\log(\sum_i w_i G_i)$ , and the log of a sum is not the sum of logs. A negative $V_i$ is a short; $\sum_i w_i=1$ still means the book is fully funded. Which shorts a mandate allows is Chapter 10.

Example — A 60/40 book on a realized year

Equity NAV is $60$ , bond NAV is $40$ , total $V=100$ . Realized simple returns are $R_E = 0.20$ and $R_B = 0.03$ .

w_E = 0.60, \quad w_B = 0.40, \quad R_p = 0.60\cdot 0.20 + 0.40\cdot 0.03 = 0.120 + 0.012 = 0.132

End NAV is $60\cdot 1.20 + 40\cdot 1.03 = 72 + 41.2 = 113.2$ , and $113.2/100 - 1 = 0.132$ . The weighted sum matches the NAV change because both compute the same dollars.

❌ Value-weighting log returns

Two names, equal value weights, simple returns $+0.10$ and $-0.10$ . The portfolio simple return is

R_p = \tfrac{1}{2}(0.10) + \tfrac{1}{2}(-0.10) = 0

so end wealth equals start wealth. Averaging the log returns instead gives

\tfrac{1}{2}\log(1.10) + \tfrac{1}{2}\log(0.90) \approx \tfrac{1}{2}(0.09531 - 0.10536) = -0.005025

Why it breaks: a portfolio weight multiplies a simple return because end NAV is the sum of dollar holdings. $\log(1+R_p)=\log(\sum_i w_i G_i)$ , which equals $\sum_i w_i \log G_i$ only in degenerate cases. Here the left side is $\log(1)=0$ and the right side is about $-0.005$ .

Consequence: exponentiating the averaged logs invents a loss of about half a percent, $\exp(-0.005025)\approx 0.995$ , on a book that finished flat. Log returns add across time. Simple returns, weighted by value, add across the book.

A weight vector that does not sum to one

If the named assets sum to $0.90$ , the missing $0.10$ is a position. Cash earning $R_f$ contributes $0.10\, R_f$ . A residual that is not named does not define a portfolio return.

05 · Expected Return

Definition — Expected Simple Return

The expected return of an asset is the expectation of its simple return,

\mu = \mathbb{E}[R]

On a finite scenario set, with scenario $k$ carrying probability $p_k$ and return $x_k$ ,

\mu = \sum_k p_k\, x_k

The probabilities are part of the model. They are not replaced by "the base case," and they are not replaced by equal weights unless the model says the scenarios are equally likely.

Because weights are known at the start of the period, linearity applies directly to the portfolio:

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

Dependence among the $R_i$ does not appear. Two books with the same weights and the same asset means have the same expected return, however the assets co-move. Co-movement shows up in the variance, which is Chapter 02's subject. The mean is blind to it.

Step-by-step — Expectation of the two-scenario index

Write the model. Over the year the index returns $0.20$ or $-0.04$ , each with probability $1/2$ . The two probabilities sum to $1$ .

Weight each outcome: $(1/2)\cdot 0.20 = 0.10$ and $(1/2)\cdot(-0.04) = -0.02$ . Each term is the scenario return times its probability, not the return alone.

Sum: $\mu = 0.10 + (-0.02) = 0.08$ . The $0.08$ is not a scenario that occurs. Neither atom equals 8 percent.

Example — A 60/40 expected return, before any covariance

Take the same index, so $\mathbb{E}[R_E]=0.08$ , and a bond book with $\mathbb{E}[R_B]=0.03$ . Weights $0.60$ and $0.40$ are fixed.

\mathbb{E}[R_p] = 0.60\cdot 0.08 + 0.40\cdot 0.03 = 0.048 + 0.012 = 0.060

The $0.048$ is 60 percent of the equity mean; the $0.012$ is 40 percent of the bond mean. Nothing in the calculation asks whether the bond sells off in the equity down-scenario. That question changes $\mathrm{Var}(R_p)$ . It does not change $0.060$ .

❌ Replacing the expectation by the base case

A three-scenario year is specified as follows. Expansion, probability $0.50$ , return $0.18$ . Base, probability $0.30$ , return $0.06$ . Recession, probability $0.20$ , return $-0.16$ . The investment memo states the expected return as "the base case, 6 percent."

Why it breaks: $\mathbb{E}[R] = 0.50\cdot 0.18 + 0.30\cdot 0.06 + 0.20\cdot(-0.16) = 0.090 + 0.018 - 0.032 = 0.076$ . The base case is the mode of the scenario set only if one confuses "most discussed" with "highest probability," and even the true mode is not the mean unless the distribution is symmetric around it. An equal-weighted average of the three returns, $(0.18+0.06-0.16)/3 = 0.08/3 \approx 0.027$ , is a third wrong number: it replaces $(0.50, 0.30, 0.20)$ by $(1/3, 1/3, 1/3)$ .

Consequence: a hurdle, a premium, or a comparison with $R_f$ is run against an input the scenario set does not imply. The expectation is the probability-weighted sum, here $0.076$ .

06 · Variance and Volatility

Definition — Variance and Volatility

For a simple return with $\mathbb{E}[R^2]<\infty$ and $\mu=\mathbb{E}[R]$ ,

\sigma^2 = \mathrm{Var}(R) = \mathbb{E}[(R-\mu)^2] = \mathbb{E}[R^2] - \mu^2, \qquad \sigma = \sqrt{\sigma^2}

$\sigma^2$ is the variance. $\sigma$ is the volatility. Both are computed on simple returns over the stated horizon. Volatility is in return units; variance is in return-squared units and is not plotted on the same axis as $\mu$ .

The first central moment vanishes for any random variable that has a mean: $\mathbb{E}[R-\mu]=0$ . A risk number built from unsquared deviations is identically zero and carries no information. Squaring, or any equivalent even penalty, is what makes dispersion visible. The computational line $\mathbb{E}[R^2]-\mu^2$ is the same object, and it is the usual numerical check.

Step-by-step — Variance of the two-scenario index

Start from the mean already computed. Outcomes $0.20$ and $-0.04$ , each probability $1/2$ , and $\mu = 0.08$ from Section 05.

Center: $0.20 - 0.08 = 0.12$ and $-0.04 - 0.08 = -0.12$ . The two deviations are equal in size because the scenarios sit symmetrically around $0.08$ .

Square: $0.12^2 = 0.0144$ and $(-0.12)^2 = 0.0144$ . The sign of the down move does not survive the square.

Average the squares: $\sigma^2 = (1/2)\cdot 0.0144 + (1/2)\cdot 0.0144 = 0.0144$ , so $\sigma = \sqrt{0.0144} = 0.12$ . Check: $\mathbb{E}[R^2] = (1/2)(0.04 + 0.0016) = 0.0208$ and $\mu^2 = 0.0064$ , and $0.0208 - 0.0064 = 0.0144$ .

Example — Same variance, a different shape

A second book has the same mean $0.08$ and the same variance $0.0144$ , with a different law: return $0.14$ with probability $0.80$ , and return $-0.16$ with probability $0.20$ .

Deviations from $0.08$ are $0.14-0.08=0.06$ and $-0.16-0.08=-0.24$ . Then

\sigma^2 = 0.80\cdot(0.06)^2 + 0.20\cdot(-0.24)^2 = 0.80\cdot 0.0036 + 0.20\cdot 0.0576 = 0.00288 + 0.01152 = 0.0144

and $\sigma=0.12$ again. Section 09 uses this pair on purpose. Matching $(\sigma,\mu)$ does not match the distribution.

❌ Quoting mean absolute deviation as volatility

On the skewed book, the average absolute deviation is

0.80\cdot|0.06| + 0.20\cdot|-0.24| = 0.048 + 0.048 = 0.096

and the risk report labels $0.096$ as "volatility."

Why it breaks: volatility in this subject is $\sqrt{\mathbb{E}[(R-\mu)^2]}$ , which is $0.12$ , not $\mathbb{E}[|R-\mu|]$ , which is $0.096$ . On the symmetric two-point book the two numbers happen to agree — both deviations are $0.12$ — so that book hides the distinction.

Consequence: the skewed book looks quieter than the symmetric book on an absolute-deviation report, and identical to it on a variance report. Downstream chapters compare assets by $\sigma$ . Feed them the square-root second moment.

One boundary. $\mathrm{Var}(w_1 R_1 + w_2 R_2)$ is not $w_1\mathrm{Var}(R_1)+w_2\mathrm{Var}(R_2)$ . The missing cross term is covariance, derived in Chapter 02. Until then a portfolio has an expected return as soon as the means and the weights are known, and it does not yet have a volatility.

07 · Excess Return and the Risk-Free Rate

Definition — Risk-Free Rate and Excess Return

Over the chosen period, an asset is risk-free in the numeraire when its simple return equals a known constant $R_f$ . There is nothing left to vary: $\mathrm{Var}(R_f)=0$ because a constant has variance zero. The excess return of a risky asset is

R^{e} = R - R_f

Subtracting a constant shifts the mean and does not change distances to the mean:

\mathbb{E}[R-R_f] = \mu - R_f, \qquad \mathrm{Var}(R-R_f) = \mathrm{Var}(R)

Excess return and total return carry the same volatility. $R_f$ is not assumed to be zero, and a T-bill is risk-free only for this one period in this numeraire. The gap $\mu-R_f$ is the compensation offered for bearing $\sigma$ . Lining that gap up across assets, and the tangent obtained by mixing cash with a risky book, is the capital market line in Chapter 05.

Example — Excess return on the two-scenario index

Keep $\mu=0.08$ , $\sigma=0.12$ , and set $R_f=0.02$ for the same year. The excess mean is $0.08-0.02=0.06$ . Scenario excess returns are $0.20-0.02=0.18$ and $-0.04-0.02=-0.06$ .

\mathbb{E}[R-R_f] = \tfrac{1}{2}\cdot 0.18 + \tfrac{1}{2}\cdot(-0.06) = 0.09 - 0.03 = 0.06

Deviations from $0.06$ are $0.18-0.06=0.12$ and $-0.06-0.06=-0.12$ . Those are the same deviations as in Section 06, so the variance is still $0.0144$ and the volatility is still $0.12$ . Cash changed the level. It did not change the dispersion.

❌ Subtracting the risk-free rate inside the volatility

A sheet reports excess volatility as the standard deviation of the two excess returns taken raw, without centering, or as $\sigma - R_f = 0.12 - 0.02 = 0.10$ .

Why it breaks: variance is already centered. $\mathrm{Var}(R-R_f)=\mathbb{E}[(R-\mu)^2]$ . Subtracting $R_f$ from $\sigma$ mixes a return with a volatility. The uncentered root-mean-square of the excess returns is different again: $(1/2)(0.18)^2+(1/2)(-0.06)^2=0.0180$ , whose square root is about $0.134$ .

Consequence: the point moves left in the $(\sigma,\mu)$ plane for no change in dispersion around $\mu$ . Excess return shifts the vertical coordinate by $-R_f$ . The horizontal coordinate stays.

08 · The Opportunity Set

Definition — Opportunity Set of the Given Assets

Fix the horizon and the numeraire. The opportunity set of a menu of assets, before mixing, is the set of points

\{(\sigma_i,\ \mu_i)\}

one point per asset, in the half-plane $\sigma\ge 0$ . The risk-free asset is the point $(0,\ R_f)$ on the vertical axis.

"Before mixing" is the scope limit. A portfolio with weights $w$ is some other point $(\sigma_p,\ \mathbb{E}[R_p])$ . The vertical coordinate is $\sum_i w_i \mu_i$ and is available now. The horizontal coordinate is not a weighted average of the $\sigma_i$ . Drawing the line segment between two asset points, and calling every point on the segment a portfolio, invents volatilities the assets do not deliver. The attainable curve is Chapter 02's calculation and Chapter 04's frontier.

What can be read off the points alone is dominance.

Definition — Mean-Variance Dominance

Asset A dominates asset B when $\mu_A \ge \mu_B$ and $\sigma_A \le \sigma_B$ , and at least one inequality is strict. An investor who prefers a higher mean and a lower variance at this horizon will not hold a dominated asset when the dominating asset is available on its own.

Dominance is a partial order. Most pairs are incomparable: one has the higher mean, the other the lower volatility. Choosing among the undominated points needs a utility (Chapter 09) or an explicit optimization (Chapter 03). Dominance only deletes points that lose on both axes.

Example — Five points, two that lose outright

One-year coordinates, volatility then expected simple return:

Asset	$\sigma$	$\mu$
Cash	$0$	$0.020$
IG credit	$0.04$	$0.035$
Equity index	$0.12$	$0.080$
Commodity	$0.22$	$0.060$
Legacy fund	$0.10$	$0.015$

Cash dominates the legacy fund: higher mean ( $0.020>0.015$ ) and lower volatility ( $0<0.10$ ). Credit dominates the legacy fund as well ( $0.035>0.015$ and $0.04<0.10$ ). Equity dominates the commodity: $0.080>0.060$ and $0.12<0.22$ .

Credit against equity is incomparable. Equity offers eight percent against credit's three and a half, and it charges twelve percent volatility against four. Cash against equity is the same kind of pair: two percent with no volatility, versus eight percent with twelve. Nothing in the menu deletes either point.

❌ Connecting the dots and calling it a frontier

The equity point $(0.12,\ 0.08)$ and the credit point $(0.04,\ 0.035)$ are joined by a straight line. The midpoint is plotted at $\sigma = 0.08$ and $\mu = 0.0575$ .

Why it breaks: the mean of a 50/50 book is $0.5\cdot 0.08 + 0.5\cdot 0.035 = 0.0575$ , because expectation is linear. The volatility $\sigma(0.5 R_E + 0.5 R_C)$ equals the average of the volatilities only for special dependence, not in general. Both coordinates must also be moments of the same simple return over the same horizon — a monthly $\sigma$ against an annual $\mu$ is not a point in this plane.

Consequence: the segment is not a set of attainable portfolios. Rank assets as points. Do not interpolate $\sigma$ .

09 · Why the First Two Moments

Mean and variance are what $\mathbb{E}[U(W)]$ sees under either of two restrictions. Quadratic utility $U(W)=W-bW^2/2$ , with $b>0$ on the relevant range, makes $\mathbb{E}[U(W)]$ an affine function of $\mathbb{E}[W]$ and $\mathbb{E}[W^2]$ . A smooth utility stopped after the second derivative of its Taylor expansion leaves the same pair and drops the remainder. End wealth is start wealth times $1+R_p$ , so moments of $R_p$ carry the same information once start NAV is fixed. Chapter 03 takes that commitment as given. Chapter 09 derives it from utility. Chapter 12 puts the remainder back.

The two books in Section 06 share the point $(0.12,\ 0.08)$ . So does the mirror image: return $0.32$ with probability $0.20$ and $0.02$ with probability $0.80$ . The mean is $0.2\cdot 0.32 + 0.8\cdot 0.02 = 0.080$ . Deviations are $+0.24$ and $-0.06$ , so the variance is $0.2\cdot 0.0576 + 0.8\cdot 0.0036 = 0.0144$ . One point, three contracts.

Example — What the plane keeps and what it drops

The shared point keeps an expected simple return of $0.08$ and a volatility of $0.12$ . It drops the left tail. At a loss threshold of zero, that probability is $1/2$ on the symmetric book, $0.20$ on the crash book, and $0$ on the lottery book. A mandate written as "same $\mu$ , same $\sigma$ , therefore same risk" has identified the coordinates and declined the distribution.

❌ Treating a matched mean and volatility as a matched law

The crash book is substituted for the symmetric index inside a mandate that constrains only $\mu$ and $\sigma$ . Both print $0.08$ and $0.12$ , so the constraint is signed off.

Why it breaks: equality of the first two moments is equality of $\mathbb{E}[R]$ and $\mathbb{E}[(R-\mu)^2]$ . It does not imply equality of $\mathbb{E}[(R-\mu)^3]$ , nor of $\mathbb{P}(R\le -0.10)$ . On the symmetric book the return is $-0.04$ or $0.20$ . On the crash book $\mathbb{P}(R=-0.16)=0.20$ .

Consequence: a drawdown, a margin call, or a payoff that is flat above a strike and steep below it will not score the two books equally. Use the opportunity set to discard dominated points and to state the mean-variance tradeoff. Do not read a point as a full description of $R$ .

10 · Exercises

EXERCISE 1.1

Convert each leg to a gross factor before multiplying. The sum of the two net returns is a different number, and the gap has a name.

Year one: $G_1 = 90/75 = 1.20$ , so $R_1 = 0.20$ . Year two: $G_2 = 81/90 = 0.90$ , so $R_2 = -0.10$ . The two-year gross factor is $G_1 G_2 = 1.20 \times 0.90 = 1.08$ , and $R(0,2) = 1.08 - 1 = 0.08$ . Equivalently, $81/75 - 1 = 1.08 - 1 = 0.08$ . The sum of simple returns is $0.20 + (-0.10) = 0.10$ , which exceeds the holding-period return by $0.02$ . That gap is the cross term $R_1 R_2 = (0.20)(-0.10) = -0.02$ , and $0.10 + (-0.02) = 0.08$ . Log returns add: $\log(1.20)+\log(0.90)=\log(1.08)$ .

A price goes $75 \to 90 \to 81$ over two years, with no separate distribution. Compute each year's simple return, the two-year simple return, and the sum of the two yearly simple returns. Identify the cross term that separates the sum from the holding-period return.

EXERCISE 1.2

Weights are start-of-period values divided by total NAV. The expected portfolio return is the same weighted sum applied to the two means.

Values are $25 \times 40 = 1{,}000$ and $50 \times 72 = 3{,}600$ , so NAV is $4{,}600$ . Weights: $1000/4600 = 10/46 = 5/23 \approx 0.2174$ in the first name and $3600/4600 = 36/46 = 18/23 \approx 0.7826$ in the second. They sum to $1$ . Realized portfolio return: $(5/23)(0.08) + (18/23)(-0.05) = (0.40 - 0.90)/23 = -0.50/23 \approx -0.02174$ . Expected return, same weights: $(5/23)(0.06) + (18/23)(0.01) = (0.30 + 0.18)/23 = 0.48/23 \approx 0.02087$ . The realized figure uses the realized asset returns; the expected figure uses the means. Covariance is not required for either.

A book holds 25 shares of asset A at a start price of $40$ , and 50 shares of asset B at a start price of $72$ . Realized simple returns are $0.08$ and $-0.05$ . Separately, the means are $\mathbb{E}[R_A]=0.06$ and $\mathbb{E}[R_B]=0.01$ . Compute the value weights, the realized portfolio simple return, and $\mathbb{E}[R_p]$ .

EXERCISE 1.3

Center before squaring. Then form the excess return in each scenario and center that variable at $\mu - R_f$ , not at zero.

$\mu = (1/2)(0.12) + (1/2)(-0.04) = 0.06 - 0.02 = 0.04$ . Deviations: $0.12 - 0.04 = 0.08$ and $-0.04 - 0.04 = -0.08$ . Squares: $0.0064$ each. Variance: $(1/2)(0.0064)+(1/2)(0.0064)=0.0064$ . Volatility: $\sqrt{0.0064}=0.08$ . Check: $\mathbb{E}[R^2]=(1/2)(0.0144 + 0.0016)=0.0080$ , and $\mu^2=0.0016$ , so $0.0080-0.0016=0.0064$ . Excess outcomes at $R_f=0.01$ : $0.11$ and $-0.05$ . Excess mean: $0.04-0.01=0.03$ , and $(1/2)(0.11)+(1/2)(-0.05)=0.055-0.025=0.03$ . Deviations from $0.03$ are again $0.08$ and $-0.08$ , so $\mathrm{Var}(R-R_f)=0.0064=\mathrm{Var}(R)$ . Subtracting a constant moved the mean and left the variance unchanged.

A single-period return equals $0.12$ or $-0.04$ with equal probability $1/2$ . The risk-free simple return over the same period is the constant $R_f=0.01$ . Compute $\mu$ , $\sigma^2$ , and $\sigma$ . Then compute $\mathbb{E}[R-R_f]$ and $\mathrm{Var}(R-R_f)$ , and state what the constant did to each.

EXERCISE 1.4

Dominance needs both a weakly better mean and a weakly better volatility, with at least one strict. The midpoint's mean is a weighted sum; its volatility is not settled by the two sigmas.

Cash dominates the starter fund: $0.03>0.02$ and $0<0.09$ . Equity does not dominate credit, and credit does not dominate equity: equity has the higher mean ( $0.09>0.04$ ) and the higher volatility ( $0.15>0.05$ ). Cash dominates neither risky asset, because both have means above $0.03$ . The 50/50 mean is $0.5\cdot 0.09 + 0.5\cdot 0.04 = 0.065$ . The 50/50 volatility is not $0.5\cdot 0.15 + 0.5\cdot 0.05 = 0.10$ in general; that average ignores covariance, which Chapter 02 supplies. Plotting the midpoint at $\sigma=0.10$ draws a portfolio the two numbers do not determine.

On a one-year horizon the menu is cash at $(\sigma,\mu)=(0,\ 0.03)$ , credit at $(0.05,\ 0.04)$ , equity at $(0.15,\ 0.09)$ , and a starter fund at $(0.09,\ 0.02)$ . Which points are dominated, and by which assets? For a 50/50 mix of credit and equity, compute the expected return, and explain why $\sigma=0.10$ is not yet known.

11 · Chapter Summary

Term	What it is
Gross return $G$	$P_1/P_0$ , the wealth factor; multiplies across dates
Simple return $R$	$G-1$ , P&L per unit of start value; the portfolio input
Log return $r$	$\log(1+R)$ ; adds across dates; is not value-weighted
Portfolio return $R_p$	$\sum_i w_i R_i$ with value weights summing to one
Expected return $\mu$	$\mathbb{E}[R]$ ; linear in the assets, blind to co-movement
Volatility $\sigma$	$\sqrt{\mathrm{Var}(R)}$ , in return units, same horizon as $\mu$
Excess return	$R-R_f$ ; mean shifts by $R_f$ , variance does not
Opportunity set	The points $(\sigma_i,\mu_i)$ of the menu, before mixing
Dominance	Better mean and better volatility; a partial order, not a choice
Two-moment limit	Matching $(\sigma,\mu)$ leaves skew and tail probabilities free

Next: Chapter 02 — Diversification and Covariance, which supplies the cross term so that a weighted book has a volatility, and shows why that volatility is not the weighted average of the asset volatilities.