Lesson 1 of 6 11–14% of the exam

By Daniil Tarakanov Updated Oct 22, 20262027 curriculum

Portfolio returns over time: buy-and-hold, drift and rebalancing

Each period, a portfolio earns the weighted average of its assets' returns, using weights from the start of that period. Left alone, winners grow into bigger weights (portfolio drift), so compound the period returns, or weight each asset's cumulative return by its starting weight.

5 min cram, 26 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: Holding period return, Arithmetic and geometric mean returns

On the exam

Calculate, interpret, and evaluate the expected return, variance, standard deviation, covariance, and correlation of portfolio returns

Calculate, interpret and evaluate portfolio returns over time. Expect a period return from start-of-period weights, the weights after a period of drift, a buy-and-hold cumulative return, the geometric versus the arithmetic average, and whether rebalancing would have helped.

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 does a portfolio earn in one period?

The portfolio return in a period is the weighted average of the assets’ returns, with each weight measured at the start of the period. The weights are the beginning-of-period weights : each holding’s share of the portfolio’s value at the end of the previous period. They add to 1.

Portfolio return in a period Memorize

r_{P,t} = \sum_{i=1}^{N} w_{i,t-1}\, r_{i,t}

Each asset’s return $r_{i,t}$ is its total return for the period, including any income: $(P_{i,t} - P_{i,t-1} + \text{Div}_{i,t}) \div P_{i,t-1}$ , as in the holding period return lesson.

One running example carries the lesson. An investor puts GBP 100,000 into a mix of 70% in an equity fund and 30% in a bond fund, and never trades. The funds return:

Year	Equity fund	Bond fund
1	+18%	+3%
2	−12%	+6%
3	+22%	−2%

What happens to the weights if you do nothing?

They drift. An asset that beats the rest of the portfolio ends the period with a bigger weight, and one that lags ends with a smaller one. That is portfolio drift : the weights change without any trade. The new weight is the asset’s value divided by the portfolio’s value.

Weight after one period Use it

w_{i,t} = \dfrac{w_{i,t-1}\,(1 + r_{i,t})}{1 + r_{P,t}}

Worked example Three years without rebalancing

Follow the GBP 100,000 through the three years. Each year’s return uses that year’s starting weights.

Year 1. Equity grows to 70,000 × 1.18 = 82,600 and bonds to 30,000 × 1.03 = 30,900, a total of 113,500. The new weights are 82,600 ÷ 113,500 = 72.78% equity and 27.22% bonds.
Year 2. $r_{P,2}$ = 0.7278 × (−12%) + 0.2722 × 6% = −8.73% + 1.63% = −7.10%. Values: 72,688 and 32,754, a total of 105,442. Weights 68.94% and 31.06%.
Year 3. $r_{P,3}$ = 0.68936 × 22% + 0.31064 × (−2%) = 15.1659% − 0.6213% = 14.54% (with the weights rounded to 0.6894 and 0.3106 the sum would show 14.55%). Values: 88,679.36 and 32,098.92, a total of GBP 120,778.28. Weights 73.42% and 26.58%.

The mix was never traded, yet its equity weight moved from 70% to 72.78%, then 68.94%, then 73.42%.

Year 1's new weight on your calculator

0.7×1.18=STO1 0.826
0.3×1.03+RCL1= 1.135, the portfolio's growth
1/x×RCL1= 0.7278

STO store · RCL recall a stored number · 1/x reciprocal

0.7ENTER1.18×STO1 0.826
0.3ENTER1.03×+ 1.135, the portfolio's growth
1/xRCL1× 0.7278

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number

What did the portfolio earn over all three years?

For a buy-and-hold return , weight each asset’s cumulative return by its starting weight. Compounding the portfolio’s three period returns gives the same answer. This assumes no money moved between assets and that any distributions were reinvested in the asset that paid them.

Buy-and-hold cumulative return Memorize

r_{P,T} = \sum_{i=1}^{N} w_{i,0}\left[\prod_{t=1}^{T}(1 + r_{i,t}) - 1\right]

The result is the cumulative portfolio return . The geometric average portfolio return turns it into a constant annual rate.

Geometric average portfolio return Memorize

\bar{r}_{gP} = \left[\prod_{t=1}^{T}(1 + r_{P,t})\right]^{1/T} - 1 = (1 + r_{P,T})^{1/T} - 1

Worked example The three years, buy-and-hold and geometric

Equity fund. 1.18 × 0.88 × 1.22 − 1 = 26.68% (26.6848%).
Bond fund. 1.03 × 1.06 × 0.98 − 1 = 7.00% (6.9964%).
Portfolio. 0.70 × 26.6848% + 0.30 × 6.9964% = 18.68% + 2.10% = 20.78%.
Check. GBP 120,778.28 ÷ 100,000 − 1 = 20.78%, and 1.1350 × 0.9290 × 1.1454 − 1 = 20.78% too.
Geometric average. 1.207783^(1/3) − 1 = 6.50% a year (6.4951%).

Cumulative return, then the geometric average

1.18×0.88×1.22−1=×0.7=STO1 0.1868, the equity part
1.03×1.06×0.98−1=×0.3+RCL1= 0.2078
1.207783y^x31/x= 1.065
−1= 0.065

STO store · RCL recall a stored number · y^x power · 1/x reciprocal

1.18ENTER0.88×1.22×1−0.7×STO1 0.1868, the equity part
1.03ENTER1.06×0.98×1−0.3×RCL1+ 0.2078
1.207783ENTER31/xy^x 1.065
1− 0.065

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number · 1/x reciprocal · y^x power

The same tool works for a portfolio: enter the portfolio’s period returns.

Rearrange itGeometric mean: find it, or find the missing year

The question asks for

A portfolio returns 20.00%, -10.00% and 5.00% in three years. Its geometric mean annual return is closest to:

Year 1 return%Year 2 return%Year 3 return%

r ˉ_{G} = [(1 + r_{1}) (1 + r_{2}) (1 + r_{3})]^{1/3} - 1

(1.2 × 0.9 × 1.05)^(1/3) − 1 = 4.28%

Geometric mean4.28%

On your calculator

1.2×0.9×1.05=the growth of 1
y^x31/x=the cube root
−1=

y^x power · 1/x reciprocal

1.2ENTER0.9×1.05×the growth of 1
31/xy^xthe cube root
1−

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

The arithmetic average portfolio return is the plain mean of the period returns. It is not the cumulative return divided by T.

Arithmetic average portfolio return Memorize

\bar{r}_P = \frac{1}{T}\sum_{t=1}^{T} r_{P,t}

Some sources also call the plain sum of the period returns (about 20.95% here, from the unrounded period returns) a “total arithmetic return”. It ignores compounding, so it is not what the investor earned. Ending wealth always comes from the product.

Exam tip Wealth multiplies, so compound the period returns. Averages and sums of returns overstate what was earned.

Four ways to state a portfolio’s return over several periods, and what each one measures:

Portfolio returns over several periods

Portfolio return, T periodsweights at the start of each period

Period return Memorize Σ w(i,t−1) × r(i,t)
Buy-and-hold cumulative Memorize Σ starting weight × asset cumulative return = product of (1 + r_P,t) − 1
Geometric average Memorize (1 + cumulative)^(1/T) − 1; matches ending wealth
Arithmetic average Memorize simple mean of period returns; higher, does not compound

Does rebalancing help?

Not automatically. Rebalancing trades the portfolio back to its target weights, which sells part of what just did well and buys what lagged. It pays when the leader then lags (returns reverse). It costs when the leader keeps leading (the trend continues). Its real job is to keep the portfolio’s risk where the investor wants it.

For the same three years rebalanced to 70/30 at every year-end, the period returns are 13.50%, −6.60% and 14.80%. The ending value is GBP 121,698.33, a cumulative 21.70%, with a geometric average of 6.76% and an arithmetic average of 7.23%. Here rebalancing earned about GBP 920 more than buy-and-hold, because it cut equity before the year-2 fall.

Whether it helps depends on what happens next. Take two years, 70/30, GBP 100,000. In both cases year 1 is equity +20%, bonds +2%, which leaves GBP 114,600 with the equity weight at 73.30%.

Worked example Trend continues

Year 2: equity +15%, bonds +1%. Buy-and-hold: 100,000 × (0.7 × 1.20 × 1.15 + 0.3 × 1.02 × 1.01) = GBP 127,506.00. Rebalanced to 70/30 after year 1: 114,600 × (0.7 × 1.15 + 0.3 × 1.01) = GBP 126,976.80. Leaving it alone wins by GBP 529.20.

Buy-and-hold, then rebalanced

0.7×1.2×1.15=STO1 0.966
0.3×1.02×1.01+RCL1=×100000= 127,506, buy-and-hold
114600×(0.7×1.15+(0.3×1.01))= 126,976.8, rebalanced

STO store · RCL recall a stored number

0.7ENTER1.2×1.15×STO1 0.966
0.3ENTER1.02×1.01×RCL1+100000× 127,506, buy-and-hold
0.7ENTER1.15×0.3ENTER1.01×+114600× 126,976.8, rebalanced

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Rebalancing back to target costs trading, so a common recommendation is a systematic, pre-defined rebalancing frequency (for example once a year), which keeps the portfolio aligned with its strategy and risk without trading on every move.

InteractiveLeft alone for three periods: how do the weights drift?

Try this 0 of 3

Make the equity weight pass 80% without any trade
Rebalance or not? tab: find returns where rebalancing loses money against doing nothing
Make the arithmetic average positive and the cumulative return negative

Starting weight in the equity fund70%

Returns each period (−40% to +60%)

Period 1: equity fund%Period 1: bond fund%Period 2: equity fund%Period 2: bond fund%Period 3: equity fund%Period 3: bond fund%

Equity fund weight
Bond fund weight
Portfolio value (GBP)

Period 1 return, r_P13.50%

Period 2 return, r_P−7.10%

Period 3 return, r_P14.54%

Buy-and-hold cumulative return20.78%

Geometric average6.50%

Arithmetic average6.98%

Sum of period returns (not the cumulative return)~~20.95%~~

GBP 100,000 ends at GBP 120,778.28. The equity weight went from 70% to 73.42% with no trade.

Show working

Period 1: r_P = 0.7 × 18% + 0.3 × 3% = 13.50% (weights from the start of the period). Equity weight after it: 0.7 × 1.18 ÷ 1.135 = 72.78%.

Period 2: r_P = 0.7278 × −12% + 0.2722 × 6% = −7.10% (weights from the start of the period). Equity weight after it: 0.7278 × 0.88 ÷ 0.929004 = 68.94%.

Period 3: r_P = 0.6894 × 22% + 0.3106 × −2% = 14.54% (weights from the start of the period). Equity weight after it: 0.6894 × 1.22 ÷ 1.145448 = 73.42%.

Cumulative (buy-and-hold): 1.135 × 0.929004 × 1.145448 − 1 = 20.78%. Value: GBP 120,778.28.

Geometric average: (1 + 0.207783)^1/3 − 1 = 6.50%. Arithmetic average: (13.50% + −7.10% + 14.54%) ÷ 3 = 6.98%, which does not compound.

Adding the period returns gives 20.95%: not the cumulative return.

Cumulative return on your calculator

1.135×0.929004×1.145448=1.207783, growth of 1
−1=0.207783, the cumulative return

1.135ENTER0.929004×1.145448×1.207783, growth of 1
1−0.207783, the cumulative return

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

Geometric average on your calculator

1.135×0.929004×1.145448=y^x(1÷3)=1.065
−1=0.065

y^x power

1.135ENTER0.929004×1.145448×31/xy^x1.065
1−0.065

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Left alone, winners take over the portfolio. Compound period returns to see what you earned.

Portfolio drift and style drift

Portfolio drift is about weights. Style drift is about strategy: the portfolio moves away from what it set out to be. Drift in weights can cause it, but it also happens with no portfolio drift at all, when the holdings themselves change. A small company that a fund bought as a small-cap holding can grow into a large company, and the fund’s style shifts while its weights stay put. Sometimes portfolio drift happens to offset style drift, but relying on that is risky.

How drift plays out depends on how the portfolio starts:

Starting weights	What drift does	What the portfolio becomes
Equal weights	the best performers grow into bigger weights	tilted toward fast-growing or more volatile assets
Unequal weights	the largest positions grow and dominate	momentum-driven, with concentration risk

When the largest positions keep growing, a few holdings come to dominate the portfolio. That is concentration risk : one bad outcome now moves the whole portfolio.

Two kinds of drift

Drift

Portfolio drift Know the idea weights move with relative performance, no trades
Style drift Know the idea strategy moves away from plan; can happen with no weight change
- Remedy systematic, pre-defined rebalancing frequency

Learning Module 3 shows the same effect in equal-weighted and market-capitalization-weighted indexes.

Memorize this

MemorizePortfolio returns over time

Measure	Formula
Period return	$r_{P,t} = \sum_i w_{i,t-1} r_{i,t}$ (start-of-period weights)
Weight after a period	$w_{i,t-1}(1 + r_{i,t}) \div (1 + r_{P,t})$
Buy-and-hold, T periods	$\sum_i w_{i,0} \times$ asset cumulative return $= \prod_t (1 + r_{P,t}) - 1$
Geometric average	$(1 + r_{P,T})^{1/T} - 1$
Arithmetic average	$\frac{1}{T}\sum_t r_{P,t}$ , higher, does not compound

Memory hook Weights look back one period. Winners grow, losers shrink. Wealth multiplies, never adds.

Know it cold Rebalancing is not automatically better: it wins when returns reverse, loses when the trend continues.

Portfolio returns over several periods

Portfolio return, T periodsweights at the start of each period

Period return Memorize Σ w(i,t−1) × r(i,t)
Buy-and-hold cumulative Memorize Σ starting weight × asset cumulative return = product of (1 + r_P,t) − 1
Geometric average Memorize (1 + cumulative)^(1/T) − 1; matches ending wealth
Arithmetic average Memorize simple mean of period returns; higher, does not compound

Two kinds of drift

Drift

Portfolio drift Know the idea weights move with relative performance, no trades
Style drift Know the idea strategy moves away from plan; can happen with no weight change
- Remedy systematic, pre-defined rebalancing frequency

Portfolio return in a period

r_{P,t} = \sum_{i=1}^{N} w_{i,t-1}\, r_{i,t}

Quick example

Start-of-period weights 55% in A and 45% in B. A returns 9%, B returns −3%. 0.55 × 9 + 0.45 × (−3) = 4.95 − 1.35 = 3.6%.

On your calculator

0.55×9+(0.45×3+|-)= 3.6

+|- change sign

0.55ENTER9×0.45ENTER3CHS×+ 3.6

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Weight after one period

w_{i,t} = \dfrac{w_{i,t-1}\,(1 + r_{i,t})}{1 + r_{P,t}}

Quick example

Start 50/50. A returns 20%, B returns 4%. w_A = 0.5 × 1.20 ÷ (0.5 × 1.20 + 0.5 × 1.04) = 0.60 ÷ 1.12 = 53.57%.

On your calculator

0.5×1.2=STO1 0.6
0.5×1.04+RCL1= 1.12, the portfolio's growth
1/x×RCL1= 0.5357

STO store · RCL recall a stored number · 1/x reciprocal

0.5ENTER1.2×STO1 0.6
0.5ENTER1.04×+ 1.12, the portfolio's growth
1/xRCL1× 0.5357

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number

Buy-and-hold cumulative return

r_{P,T} = \sum_{i=1}^{N} w_{i,0}\left[\prod_{t=1}^{T}(1 + r_{i,t}) - 1\right]

Quick example

Start 30% in X and 70% in Y. Over three years X gains 40% in total and Y gains 10%. 0.3 × 40 + 0.7 × 10 = 19.0%.

On your calculator

0.3×40+(0.7×10)= 19

0.3ENTER40×0.7ENTER10×+ 19

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

Geometric average portfolio return

\bar{r}_{gP} = \left[\prod_{t=1}^{T}(1 + r_{P,t})\right]^{1/T} - 1 = (1 + r_{P,T})^{1/T} - 1

Quick example

Portfolio returns 12%, −5% and 7%. (1.12 × 0.95 × 1.07)^(1/3) − 1 = 1.13848^(1/3) − 1 = 4.42% a year. The arithmetic average, (12 − 5 + 7) ÷ 3, is 4.67%.

On your calculator

1.12×0.95×1.07= 1.1385
y^x31/x= 1.0442
−1= 0.0442

y^x power · 1/x reciprocal

1.12ENTER0.95×1.07× 1.1385
31/xy^x 1.0442
1− 0.0442

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Arithmetic average portfolio return

\bar{r}_P = \frac{1}{T}\sum_{t=1}^{T} r_{P,t}

Quick example

Portfolio returns 12%, −5% and 7%. (12 − 5 + 7) ÷ 3 = 4.67%, above the geometric average of 4.42%.

On your calculator

12−5+7=÷3= 4.67

12ENTER5−7+3÷ 4.67

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

How it is asked

Weight after drift

Start 60/40; A +26%, B −2%. A's weight now?

0.60 × 1.26 ÷ (0.756 + 0.392) = 65.85%. Traps: 60% (no drift), 75.6% (forgot to divide by the portfolio's growth).
Buy-and-hold over several years

Start 40/60; X gains 45% in total, Y 12%.

0.4 × 45 + 0.6 × 12 = 25.2%. Traps: 28.5% (equal weights), 31.8% (weights swapped).
Geometric versus arithmetic

Portfolio returns 11%, −7%, 14%. Annual growth?

(1.11 × 0.93 × 1.14)^(1/3) − 1 = 5.58%. Traps: 6.00% (arithmetic), 5.89% (cumulative ÷ 3).
Would rebalancing have helped?

Stocks beat bonds in both years.

No: rebalancing after year 1 sold the asset that kept winning.

Exam traps

Using end-of-period weights (or the original weights in every period) for a portfolio that was not rebalanced.
Computing the new weight as w × (1 + r) without dividing by the portfolio's growth.
Reporting the arithmetic average, or the sum of period returns, as what the investor earned.
Dividing the cumulative return by T and calling it the average.
Assuming rebalancing always adds return.

Checkpoint

3 exam-style questions. About 90 seconds each.

1An equally weighted portfolio is never rebalanced. Over time it will most likely become tilted toward the assets that have:
2A portfolio starts the year with 65% in asset A and 35% in asset B and is not rebalanced. During the year A returns 24% and B returns −4%. At the end of the year the weight of A is closest to:
3A portfolio returns 15%, −10% and 8% in three successive years. Its geometric average annual return is closest to:

Key takeaways

Period portfolio return = Σ start-of-period weight × asset return.
Without rebalancing, weights drift toward the winners: new weight = w(1 + r) ÷ (1 + r_P).
Buy-and-hold cumulative return = Σ starting weight × asset cumulative return = product of (1 + r_P) − 1.
Report the geometric average; the arithmetic average and the sum of period returns overstate what was earned.
Rebalancing controls risk; it adds return only when returns reverse.

Portfolio return in a period

r_{P,t} = \sum_{i=1}^{N} w_{i,t-1}\, r_{i,t}

Quick example

Start-of-period weights 55% in A and 45% in B. A returns 9%, B returns −3%. 0.55 × 9 + 0.45 × (−3) = 4.95 − 1.35 = 3.6%.

On your calculator

0.55×9+(0.45×3+|-)= 3.6

+|- change sign

0.55ENTER9×0.45ENTER3CHS×+ 3.6

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Weight after one period

w_{i,t} = \dfrac{w_{i,t-1}\,(1 + r_{i,t})}{1 + r_{P,t}}

Quick example

Start 50/50. A returns 20%, B returns 4%. w_A = 0.5 × 1.20 ÷ (0.5 × 1.20 + 0.5 × 1.04) = 0.60 ÷ 1.12 = 53.57%.

On your calculator

0.5×1.2=STO1 0.6
0.5×1.04+RCL1= 1.12, the portfolio's growth
1/x×RCL1= 0.5357

STO store · RCL recall a stored number · 1/x reciprocal

0.5ENTER1.2×STO1 0.6
0.5ENTER1.04×+ 1.12, the portfolio's growth
1/xRCL1× 0.5357

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number

Buy-and-hold cumulative return

r_{P,T} = \sum_{i=1}^{N} w_{i,0}\left[\prod_{t=1}^{T}(1 + r_{i,t}) - 1\right]

Quick example

Start 30% in X and 70% in Y. Over three years X gains 40% in total and Y gains 10%. 0.3 × 40 + 0.7 × 10 = 19.0%.

On your calculator

0.3×40+(0.7×10)= 19

0.3ENTER40×0.7ENTER10×+ 19

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

Geometric average portfolio return

\bar{r}_{gP} = \left[\prod_{t=1}^{T}(1 + r_{P,t})\right]^{1/T} - 1 = (1 + r_{P,T})^{1/T} - 1

Quick example

Portfolio returns 12%, −5% and 7%. (1.12 × 0.95 × 1.07)^(1/3) − 1 = 1.13848^(1/3) − 1 = 4.42% a year. The arithmetic average, (12 − 5 + 7) ÷ 3, is 4.67%.

On your calculator

1.12×0.95×1.07= 1.1385
y^x31/x= 1.0442
−1= 0.0442

y^x power · 1/x reciprocal

1.12ENTER0.95×1.07× 1.1385
31/xy^x 1.0442
1− 0.0442

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Arithmetic average portfolio return

\bar{r}_P = \frac{1}{T}\sum_{t=1}^{T} r_{P,t}

Quick example

Portfolio returns 12%, −5% and 7%. (12 − 5 + 7) ÷ 3 = 4.67%, above the geometric average of 4.42%.

On your calculator

12−5+7=÷3= 4.67

12ENTER5−7+3÷ 4.67

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

All Quantitative Methods formulas, every lesson

Practice exam-style questions on Portfolio Return and Risk

Next lesson Portfolio expected return, variance and the role of correlation

Questions about this lesson?

Where this comes back

Portfolio expected return, variance and the role of correlation Portfolio Return and Risk
Basics of Portfolio Planning and Construction Portfolio Construction

Questions candidates ask

What is portfolio drift?

The change in a portfolio's weights caused by the assets' different returns, with no trades. An asset that beats the portfolio grows into a bigger weight. Over time the portfolio can carry more risk, or a different style, than the investor chose.

How do you calculate a buy-and-hold portfolio return over several years?

Multiply each asset's starting weight by its cumulative return and add them up. Compounding the portfolio's yearly returns gives the same answer. Starting 70/30 with cumulative returns of 26.68% and 7.00% gives 20.78%.

Does rebalancing increase returns?

Not by itself. It sells recent winners and buys laggards, so it helps when returns reverse and hurts when the trend continues. Its main purpose is to keep the portfolio's risk in line with the target.

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