Realized Returns With Changing Portfolio Weights

From Asset Returns to a Portfolio Return

A portfolio return describes the change in value of the portfolio as a whole over a stated period. Its result depends on both the returns earned by the assets and how much of the portfolio was invested in each asset at the start of that period. Those weights generally change as asset values change, even if the investor makes no trades.

This distinction is useful in practice. An investor may begin a period with a target allocation, such as 40% in one asset and 60% in another. If the first asset performs better, it will represent a larger share of the portfolio afterward. The target weights and the ending weights describe different things.

Definition: For a portfolio with assets $1,\ldots,n$, let $w_{i,0}$ be the fraction of the portfolio’s beginning value invested in asset $i$, and let $R_i$ be that asset’s realized return over the period. If the portfolio is fully invested, the beginning weights satisfy $\sum_{i=1}^{n} w_{i,0}=1$. The portfolio’s realized return is $$ R_P=\sum_{i=1}^{n} w_{i,0}R_i. $$

This formula assumes that the weights refer to values at the beginning of the period and that the asset returns cover the same period. It also assumes the portfolio’s investment performance is being measured without external deposits or withdrawals during the period. Trading costs, taxes, and cash flows are excluded unless they are explicitly included in the calculation. A return can be stated as a decimal or a percentage: for example, $0.10=10\%$.

The asset returns in this calculation are realized outcomes, not forecasts. If asset returns are uncertain, the return earned in a particular period still depends on the returns that actually occurred in that period. The formula does not require a probability distribution.

Why Beginning Weights Give the Correct Return

The weighted-return formula follows directly from adding the dollar gains and losses on the assets. Suppose asset $i$ has beginning value $V_{i,0}$, and the total beginning portfolio value is $V_0=\sum_i V_{i,0}$. Its beginning weight is $w_{i,0}=V_{i,0}/V_0$, and its ending value, before any trades or external cash flows, is $V_{i,0}(1+R_i)$.

\begin{aligned} V_1 &=\sum_{i=1}^{n}V_{i,0}(1+R_i)\\ &=\sum_{i=1}^{n}V_{i,0}+\sum_{i=1}^{n}V_{i,0}R_i\\ &=V_0\left(1+\sum_{i=1}^{n}w_{i,0}R_i\right). \end{aligned}

Therefore, the portfolio return is its ending value divided by its beginning value, minus one:

R_P=\frac{V_1-V_0}{V_0}=\sum_{i=1}^{n}w_{i,0}R_i.

The derivation also explains why the weights are beginning weights: they determine the dollar amount exposed to each asset’s return over the period. Ending weights result from those returns and are not the proportions of starting capital that earned them.

Worked Example: A One-Period Return and the Resulting Weights

A hypothetical investor begins a quarter with $20,000: $8,000 in a growth fund and $12,000 in a short-term bond fund. During the quarter, the growth fund returns 25% and the bond fund returns 0%. Calculate the portfolio’s realized return and its ending weights. Ignore fees, taxes, and external cash flows.

Method. First calculate the beginning weights:

w_{\text{growth},0}=\frac{8{,}000}{20{,}000}=0.40, \qquad w_{\text{bond},0}=\frac{12{,}000}{20{,}000}=0.60.

Use these starting weights with the returns for the quarter:

R_P=(0.40)(0.25)+(0.60)(0)=0.10=10\%.

As a dollar check, the growth fund ends at $8{,}000(1.25)=\$10{,}000$, and the bond fund ends at $12{,}000(1)=\$12{,}000$. Total ending value is $\$22{,}000$, so the portfolio return is $(22{,}000-20{,}000)/20{,}000=0.10=10\%$, matching the weighted calculation.

The ending weights are based on the ending values:

w_{\text{growth},1}=\frac{10{,}000}{22{,}000}=45.4545\%, \qquad w_{\text{bond},1}=\frac{12{,}000}{22{,}000}=54.5455\%.

They sum to 100.0000%, as required. Common mistake avoided: using the 45.4545% ending growth-fund weight to calculate the quarter’s return would give $(0.454545)(0.25)+(0.545455)(0)=11.3636\%$, which is wrong. That weight arose only after the growth fund earned its return. The starting weights, not the ending weights, determine the quarter’s portfolio return.

Buy-and-Hold Weights After Returns

In a buy-and-hold portfolio, the investor does not trade to restore the initial allocation during the period. Each asset’s ending value reflects its own return. If $w_{i,0}$ is its beginning weight and $R_i$ is its return, then its ending value is proportional to $w_{i,0}(1+R_i)$. Dividing by the ending portfolio value gives its ending weight:

Result: With no trades or external cash flows during a period, the buy-and-hold ending weight of asset $i$ is $$ w_{i,1}=\frac{w_{i,0}(1+R_i)}{1+R_P}. $$ The denominator is $1+R_P$, since the entire portfolio’s ending value is $V_0(1+R_P)$.

This result assumes that the portfolio contains the assets being measured and that returns and weights use a consistent value basis. The ending weights sum to one because

\sum_i w_{i,1} =\frac{\sum_i w_{i,0}(1+R_i)}{1+R_P} =\frac{1+R_P}{1+R_P}=1.

A portfolio that is rebalanced is different. Rebalancing means trading at a specified time to bring weights back to chosen target weights. The trades change the amounts held; they do not undo the return already earned. In the examples below, rebalancing occurs after each period and has no transaction costs or taxes.

Worked Example: Two Periods With and Without Rebalancing

A hypothetical portfolio starts with $1,000, invested $500 each in assets A and B. In year 1, A returns 20% and B returns −10%. In year 2, A returns −10% and B returns 20%. Compare the two-year outcomes when the investor (a) buys and holds and (b) rebalances to equal weights after year 1. Ignore fees, taxes, and external cash flows.

Method for buy and hold. At the end of year 1, the values are $500(1.20)=\$600$ in A and $500(0.90)=\$450$ in B. The total is $\$1{,}050$, so the first-year portfolio return is $(1{,}050-1{,}000)/1{,}000=5\%$. The ending weights are $600/1{,}050=57.1429\%$ in A and $450/1{,}050=42.8571\%$ in B.

With no trade, year 2 begins with those dollar values. A ends at $600(0.90)=\$540$, and B ends at $450(1.20)=\$540$. Total ending value is $\$1{,}080$. The second-year return is $(1{,}080-1{,}050)/1{,}050=2.8571\%$, and the two-year cumulative return is $(1{,}080-1{,}000)/1{,}000=8\%$. As a check, compounding the annual returns gives $(1.05)(1.028571)-1=0.08=8\%$.

Method for annual rebalancing. At the end of year 1, the portfolio is worth $\$1{,}050$. Restoring equal weights requires holding $\$525$ in each asset. In year 2, A ends at $525(0.90)=\$472.50$, and B ends at $525(1.20)=\$630$. Total ending value is $\$1{,}102.50$. The second-year portfolio return is $52.50/1{,}050=5\%$, and the cumulative return is $(1.1025)-1=10.25\%$. Equivalently, the two annual returns compound as $(1.05)(1.05)-1=10.25\%$.

The results differ because the year 2 starting allocations differ: buy and hold starts year 2 with unequal weights, while rebalancing restores equal weights. Common mistake avoided: rebalancing does not mean using equal weights to recalculate the year 1 return after the fact. The year 1 return is 5% under either strategy; the trade changes the allocation for the next period.

A Multi-Asset Allocation With a Rebalance

For more than two assets, the same steps apply: calculate the realized return from the starting weights, calculate each ending dollar value, and then divide each ending value by the portfolio total to obtain buy-and-hold weights. If a rebalance is made, calculate the target dollar holdings using the portfolio value after returns.

Worked Example: Three Assets and a Target Allocation

A hypothetical investor has an $80,000 portfolio with 25% in a stock fund, 25% in a bond fund, and 50% in a cash-equivalent fund. Over the year, their returns are 20%, −10%, and 5%, respectively. Calculate the realized portfolio return, the buy-and-hold ending weights, and the dollar trades needed to restore the original allocation. Ignore fees, taxes, and external cash flows.

Method. The beginning dollar amounts are $\$20{,}000$, $\$20{,}000$, and $\$40{,}000$. Apply each return:

\begin{aligned} V_{\text{stock},1}&=20{,}000(1.20)=\$24{,}000,\\ V_{\text{bond},1}&=20{,}000(0.90)=\$18{,}000,\\ V_{\text{cash},1}&=40{,}000(1.05)=\$42{,}000. \end{aligned}

The total ending value is $\$24{,}000+\$18{,}000+\$42{,}000=\$84{,}000$. Thus the realized return is $(84{,}000-80{,}000)/80{,}000=5\%$. The weighted calculation gives the same result:

(0.25)(0.20)+(0.25)(-0.10)+(0.50)(0.05) =0.05-0.025+0.025=0.05=5\%.

Without trading, the ending weights are $24{,}000/84{,}000=28.5714\%$ in stock, $18{,}000/84{,}000=21.4286\%$ in bonds, and $42{,}000/84{,}000=50.0000\%$ in cash. These sum to 100.0000%.

To restore the original 25%, 25%, and 50% allocation, calculate target dollar values using the $\$84{,}000$ ending total: $\$21{,}000$ in stock, $\$21{,}000$ in bonds, and $\$42{,}000$ in cash. The investor sells $\$3{,}000$ of stock and buys $\$3{,}000$ of bonds; the cash holding is unchanged. The sale and purchase offset, so total portfolio value remains $\$84{,}000$. Common mistake avoided: applying the target percentages to the original $\$80{,}000$ would produce target holdings for the wrong valuation date and would fail to restore the intended allocation after the year’s return.

Practical Checks and Common Errors

A reliable calculation keeps the return period, weights, and valuation dates aligned. When the problem provides beginning weights and returns, calculate the period return with those weights. To find ending buy-and-hold weights, first update the asset values, then divide by the updated total. To find the amount to trade for a target allocation, apply the target weights to the portfolio’s current value after returns.

Using ending weights for a completed period. Ending weights are outcomes of that period’s returns; they are not the weights that earned those returns.
Adding multi-period returns instead of compounding. If a portfolio earns $R_{P,1}$ and then $R_{P,2}$, its cumulative return is $(1+R_{P,1})(1+R_{P,2})-1$, not generally $R_{P,1}+R_{P,2}$.
Confusing a target with an actual allocation. A target allocation is a chosen policy. Without a trade, performance causes actual weights to drift away from it.
Rebalancing at the wrong date. State when the trade occurs. A trade at the end of a period affects the weights for the next period, not the return just earned.
Ignoring cash flows or costs when they are present. The simple formulas here assume no external deposits or withdrawals and ignore trading costs and taxes. If such items are part of a problem, the return measurement must account for them consistently.

The central distinction is that realized return measures the change in total portfolio value over a period, while portfolio weights describe how that value is allocated at a particular time. Beginning weights determine the return for the period; ending weights show the allocation produced by that period’s returns and any trades.

Check Your Understanding

For each problem, use the stated period and portfolio assumptions. Ignore fees, taxes, and external cash flows unless stated otherwise.

A portfolio begins with $30,000 in Fund A and $70,000 in Fund B. Their realized returns are 8% and −2%. Find the portfolio return and the buy-and-hold ending weights. (Answer: Return $=1.0000\%$; ending weights $=32.0792\%$ in A and $67.9208\%$ in B.)
An investor starts with $10,000: 60% in Fund C and 40% in Fund D. Over one year, C returns 10% and D returns −5%. Find the portfolio return, ending value, and ending weights without a trade. (Answer: Return $=4.0000\%$; ending value $=\$10{,}400$; weights $=63.4615\%$ in C and $36.5385\%$ in D.)
A $1,000 portfolio begins equally divided between assets E and F. In year 1, E returns 10% and F returns −10%. The investor rebalances to equal weights at the end of year 1. In year 2, E returns −10% and F returns 10%. Find each year’s portfolio return and the two-year cumulative return under rebalancing. Compare with buy and hold. (Answer: Rebalanced returns are 0% in each year and 0% cumulatively; buy-and-hold ending value is $\$990$, for a cumulative return of −1%.)
A $100,000 portfolio starts with weights of 20% in Fund G, 30% in Fund H, and 50% in Fund J. Their returns are −10%, 5%, and 4%. Find the realized portfolio return and the ending buy-and-hold weights. (Answer: Return $=1.5000\%$; ending weights are $17.7340\%$ in G, $31.0345\%$ in H, and $51.2315\%$ in J.)
A $100 portfolio begins with 40% in Fund K and 60% in Fund L. Year 1 returns are 25% and 0%; year 2 returns are −20% and 10%. Compare the two-year cumulative return with annual rebalancing to 40% and 60% against buy and hold. (Answer: Rebalanced cumulative return $=7.8000\%$; buy-and-hold ending value $=\$106$, cumulative return $=6.0000\%$, with ending weights $=37.7358\%$ in K and $62.2642\%$ in L.)

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