ETFs: replication, tracking error, leveraged ETFs and UCITS.

A fund that trades like a stock

An ETF (exchange-traded fund) is an investment fund whose shares are bought and sold on an exchange during the session, like a stock. Almost all of them replicate an index. It has two prices:

• Net asset value (NAV): what the fund's portfolio is worth, minus its expenses, divided by the number of shares. It is calculated once a day with the closing prices. During the session, many exchanges publish a continuous estimate (iNAV).
• Market price: the price at which the shares cross on the exchange. It is set by supply and demand.

That the two prices almost coincide is no accident: it is guaranteed by an arbitrage mechanism that only a few intermediaries can use. The fund's costs are summarized in the TER (total expense ratio), an annual percentage that is deducted from the net asset value day by day. And the dividends of the portfolio are paid out to the investor (distributing share class) or reinvested (accumulating share class).

The data for this lesson, etf_indice_diario.csv, is seven years of a fictitious index, ISX, and of four series of products that track it: a physical ETF with a TER of 0.07% (its net asset value and its exchange price) and two daily-reset ETFs, one 2x and one inverse (x−1).

import numpy as np
import pandas as pd

etf = pd.read_csv("etf_indice_diario.csv", parse_dates=["fecha"],
                  index_col="fecha")
print(len(etf), "sessions")
etf.head(3)

1760 sessions
            indice  etf_vl  etf_precio  etf_2x_precio  etf_inverso_precio
fecha
2019-01-02  991.65  99.165       99.16       100.0000            100.0000
2019-01-03  992.89  99.287       99.24       100.2407             99.8858
2019-01-04  998.27  99.831       99.80       101.3178             99.3553

ISX is a net total return index: it adds the dividends of its stocks, after withholding tax, as if they were reinvested. It is the correct benchmark for an accumulating ETF. Comparing it with a price index, which does not add dividends, would make the ETF look as if it beats its index (lesson 1.2.1).

Creation and redemption: the authorized participant

An ordinary fund issues and redeems shares with each investor. An ETF only does so with authorized participants (APs): large intermediaries with a contract with the fund manager. Everyone else buys and sells already existing shares on the exchange.

flowchart LR
  AP[Participant] -- basket --> G[ETF manager]
  G -- shares --> AP
  AP -- sells and buys --> B[Exchange]
  B --> I[Investors]

• Creation. The participant delivers the basket of index stocks (or cash) to the fund manager and receives a large block of new shares (a “creation unit”), which it sells on the exchange.
• Redemption. The participant buys shares on the exchange, delivers them to the fund manager and receives the basket.

That two-way path is an arbitrage. If the ETF trades above its net asset value (premium), the participant has an incentive to create shares with the basket and sell them dear; if it trades below (discount), to buy them cheap on the exchange and redeem them. Both operations push the price toward the net asset value. That is why a liquid ETF trades within a few basis points of its portfolio:

prima = 1e4 * (etf["etf_precio"] / etf["etf_vl"] - 1)  # basis points
print(prima.describe().round(2))
prima.nsmallest(3).round(1)

count    1760.00
mean       -0.68
std         9.24
min      -113.78
25%        -3.37
50%         0.11
75%         3.51
max        16.81
dtype: float64
fecha
2020-03-17   -113.8
2020-03-31   -106.5
2020-04-03   -102.4
dtype: float64

In half of the sessions the price is between −3.4 and +3.5 basis points from the net asset value. The median is 0.1. The extremes are concentrated in a few weeks: on March 17, 2020 the ETF closed at a discount of 113.8 basis points, 1.14%.

The figure shows the range of the premium or discount in each month. 76 of the 84 months fit in a ±15 basis point band; March and April 2020 break out, with discounts of more than 100 basis points. It is not that the mechanism fails: it is that the arbitrage costs more. When the basket's stocks have wide spreads and intermediaries shrink their balance sheets, the participant only redeems if the discount covers that cost. In bond ETFs, whose bonds trade much less than stocks, the same mechanism lets larger discounts through when the market comes under strain.

prima.abs().groupby(prima.index.year).mean().round(2)

fecha
2019    4.05
2020    9.06
2021    3.86
2022    3.97
2023    4.16
2024    3.88
2025    4.13
dtype: float64

In 2020, the average size of the premium or discount is more than double that of any other year. For a trading system, the consequence is direct: the price of an ETF in a crisis may not be the value of what it contains. A backtest that uses the net asset value as the execution price, or the other way round, gets it wrong precisely on the days that matter most.

Physical and synthetic replication

There are two ways of giving a fund the return of an index:

In synthetic replication, the fund buys a basket of assets (which may have nothing to do with the index) and signs a total return swap with a counterparty: the fund hands over the return of its basket and receives that of the index.

flowchart LR
  F[Fund] -- return of its basket --> C[Counterparty]
  C -- index return --> F
  F --> K[Basket as collateral]

The risk is that the counterparty does not pay. The UCITS Directive limits it: the exposure to one counterparty in OTC derivatives cannot exceed 10% of the fund's assets if it is a credit institution, and 5% in other cases (Article 52). In practice, the swap is reset when its value approaches that limit. Physical replication has its own smaller counterparty risk, when it lends its stocks in exchange for a fee.

The same directive sets diversification limits: a UCITS fund cannot invest more than 5% of its assets in securities of a single issuer, which can be raised to 10% as long as those above 5% do not add up to more than 40%. A very concentrated index cannot be replicated as is in a UCITS; there are capped versions of the index for that.

Tracking difference and tracking error

Two measures summarize how well an ETF follows its index, and they are not the same:

The tracking difference (TD) is the fund's return minus the index's over a period, for example a calendar year. It is what the ETF costs the investor. The tracking error (TE) is the annualized standard deviation of the daily differences: it measures how irregular the replication is, not how much is lost.

fin = etf.groupby(etf.index.year).last()
inicio = fin.shift()
inicio.iloc[0] = etf.iloc[0]  # 2019 starts at the first session
anual = 100 * (fin / inicio - 1)

td = pd.DataFrame({"indice": anual["indice"],
                   "etf_vl": anual["etf_vl"]})
td["dif_replica"] = td["etf_vl"] - td["indice"]
print(td.round(2))
print(f"Mean: {td['dif_replica'].mean():.3f} points a year")

       indice  etf_vl  dif_replica
fecha
2019    26.57   26.36        -0.21
2020     3.75    3.66        -0.08
2021    28.64   28.46        -0.18
2022   -21.56  -21.58        -0.02
2023    18.83   18.81        -0.03
2024    22.89   22.69        -0.20
2025    26.54   26.37        -0.17
Mean: -0.128 points a year

Every year the fund returns somewhat less than its index: between 0.02 and 0.21 points, 0.128 on average. The TER alone explains 0.07 points; the rest is replication error, which adds in some years and subtracts in others. That is why the tracking difference of a single year is not enough to judge an ETF: several are needed.

r = etf.pct_change().dropna()
dif = r["etf_vl"] - r["indice"]
te = 100 * np.sqrt(252) * dif.groupby(dif.index.year).std()
print(te.round(3))
print(f"Total: {100 * np.sqrt(252) * dif.std():.3f} % annualized")

fecha
2019    0.073
2020    0.078
2021    0.078
2022    0.080
2023    0.080
2024    0.085
2025    0.084
dtype: float64
Total: 0.080 % annualized

A tracking error of 0.08% annualized is very low: the fund follows the index almost day by day. Its tracking difference is, however, always negative: the cost is small but systematic. For a long-term investor, the tracking difference is the figure that matters; for someone who uses the ETF to hedge a portfolio over a few days, the tracking error.

Two practical details when comparing with real data. First, tracking difference is measured on the net asset value, not on the exchange price, which adds each day's premium or discount (the etf_precio column). Second, the index has to be the same one the fund declares: price, gross or net total return, in the same currency.

Leveraged and inverse ETFs: the daily reset

A 2x leveraged ETF promises twice the daily return of its index; an inverse one, the daily return with its sign flipped. To achieve it, the fund manager resets its exposure every day at the close (with swaps or futures): if the fund is worth 100, the next day it needs an exposure of 200 to the index. That word, daily, changes everything. Two days are enough to see it:

dias = pd.Series([0.10, -1 / 11])  # +10% and then −9.09%
for nombre, L in [("index", 1), ("x2", 2), ("inverse", -1)]:
    valor = 100 * (1 + L * dias).prod()
    print(f"{nombre:8s} {valor:7.2f}")

index     100.00
x2         98.18
inverse    98.18

The index rises 10% and falls 9.09%: it returns to 100. The 2x rises 20% (to 120) and then falls 18.18% from 120: it ends at 98.18. The inverse falls 10% and then rises 9.09% from 90: 98.18 as well. Neither of the two recovers the starting point. The return over a period is the product of the daily returns, and $\prod (1 + L r_t) \ne 1 + L \left(\prod (1 + r_t) - 1\right)$ as soon as the days have different signs.

With seven years of data, the period from January 2022 to June 2023 has the worst for these products: a lot of volatility and little net trend.

tramo = etf.loc["2022-01-03":"2023-06-30"]
rent = 100 * (tramo.iloc[-1] / tramo.iloc[0] - 1)
print(rent.round(1))
print(f"Twice the index: {2 * rent['indice']:.1f}")

indice                13.8
etf_vl                13.7
etf_precio            13.7
etf_2x_precio          9.9
etf_inverso_precio   -14.0
dtype: float64
Twice the index: 27.6

The index gains 13.8%. Whoever expected double from the 2x, 27.6%, gets 9.9%: less than the index itself. The inverse loses 14.0%, somewhat more than the index gains. The header figure plots the three paths: the 2x drifts away from “twice the index” with each swing and never recovers.

How much of the difference is volatility and how much is cost? The 2x fund pays, on top of the TER (0.95% a year in this data), the financing of the extra exposure: it borrows what it puts into the index above its net assets, at the three-month rate. The two can be separated by rebuilding a cost-free 2x from the daily returns of the index:

ri = tramo["indice"].pct_change().dropna()
x2_sin_costes = 100 * ((1 + 2 * ri).prod() - 1)
print(f"Twice the index:  {2 * rent['indice']:5.1f} %")
print(f"x2 without costs: {x2_sin_costes:5.1f} %")
print(f"x2 actual:        {rent['etf_2x_precio']:5.1f} %")

sigma = ri.std() * np.sqrt(252)
anos = len(ri) / 252
print(f"Index volatility: {sigma:.1%} over {anos:.2f} years")

Twice the index:   27.6 %
x2 without costs:  16.9 %
x2 actual:          9.9 %
Index volatility: 26.4% over 1.48 years

Of the 17.7 points that separate the promised 27.6% from the actual 9.9%, 10.7 are eaten by volatility (the cost-free 2x already stops at 16.9%) and the other 7 by financing and the fee. The index volatility over the period, 26.4% annualized for 1.48 years, is high: with half the volatility, the decay would be a quarter as large.

The approximation that explains it comes from working with logarithms. For a product with leverage $L$ and an index with volatility $\sigma$, over $T$ years and without costs:

For $L = 2$, the decay term is $\sigma^2 T$; for $L = -1$, too ($\frac{1 + 1}{2} = 1$). The decay grows with the square of volatility and linearly with time. In a market with a strong trend and little volatility, the effect reverses: the 2x can return more than double, because each day it reinvests the previous gain.

UCITS versus the US: PRIIPs and the KID

The ETFs sold to retail investors in the EU are almost always UCITS (OICVM in Spanish terminology): funds harmonized by Directive 2009/65/EC, with diversification, liquidity and leverage limits, and a passport to be marketed throughout the Union from the member state in which they are domiciled. US ETFs are governed by US regulation and are not UCITS.

The practical difference for a Spanish investor comes from another rule, Regulation (EU) 1286/2014 (PRIIPs). Its core rules are three:

• Article 5.1. Before making it available to retail investors, the manufacturer of a packaged investment product has to draw up a key information document (KID) and publish it on its website.
• Article 6.4. The document is short: a maximum of three sides of A4-sized paper, with risk, performance scenarios and costs in a common format that makes it possible to compare products.
• Article 13.1. Whoever advises on the product or sells it to a retail investor has to provide them with the KID in good time before they are bound.

An ETF is a packaged product. UCITS had a transitional exemption (Article 32) that was extended until December 31, 2022; since January 1, 2023 they also use the PRIIPs KID (Directive 2021/2261). US-domiciled ETFs do not usually have a KID: their manager does not market them in the EU and has no obligation to draw it up. Without a KID, the European broker cannot comply with Article 13 and cannot sell them to a retail client.

flowchart TB
  O[Buy order] --> C{Retail?}
  C -- no --> E[Executed]
  C -- yes --> K{Has a KID?}
  K -- yes --> E
  K -- no --> R[Broker rejects it]

The regulation applies to sales to retail investors. A client classified as professional under MiFID II, with the requirements that entails, can buy an ETF without a KID. For a trading system the consequence is one of data and design: a backtest with a US ETF (SPY, for example) can be a good experiment, but to trade it from a European retail account you have to find the UCITS equivalent, which has another ticker, sometimes another currency, another exchange, other trading hours and, if it is accumulating, does not pay dividends.

What this changes in your data

• Price or net asset value. An ETF has two series. Execution is simulated with the price; replication is measured with the net asset value. In a crisis they can diverge by 1% or more.
• The right index. To measure an accumulating ETF, a net total return index; against a price index, the ETF seems to beat it.
• Daily-reset products. Their series is not built by multiplying the index's: it is compounded day by day, with financing and fees.
• The tradable universe. What can be bought from an EU retail account is not what appears on US screens; a backtest universe should be that of the products that will be traded (lesson 1.3.8).
• Dividends and share classes. A distributing ETF has ex-dates like a stock (lesson 1.2.1); an accumulating one does not.

Exercises

1. Calculate the return of the inverse ETF in each year and compare it with minus the index return. In which year is the difference largest and why?

inv = pd.DataFrame({"inverso": anual["etf_inverso_precio"],
                    "menos_indice": -anual["indice"]})
inv["diferencia"] = inv["inverso"] - inv["menos_indice"]
inv.round(1)

       inverso  menos_indice  diferencia
fecha
2019     -19.5         -26.6         7.1
2020     -11.4          -3.7        -7.6
2021     -24.3         -28.6         4.3
2022      24.1          21.6         2.5
2023     -12.0         -18.8         6.8
2024     -12.0         -22.9        10.9
2025     -20.3         -26.5         6.3

2. Compare year by year the return of the 2x with that of a cost-free 2x built from the daily returns of the index. How much do financing and the fee cost each year?

ri_todo = etf["indice"].pct_change().fillna(0)
sin_costes = (1 + 2 * ri_todo).groupby(etf.index.year).prod()
coste = pd.DataFrame({
    "x2_sin_costes": 100 * (sin_costes - 1),
    "x2_real": anual["etf_2x_precio"],
})
coste["coste"] = coste["x2_real"] - coste["x2_sin_costes"]
coste.round(1)

       x2_sin_costes  x2_real  coste
fecha
2019            58.0     53.2   -4.8
2020            -0.7     -2.0   -1.3
2021            62.1     60.2   -1.8
2022           -42.3    -44.2   -1.9
2023            34.8     26.9   -7.9
2024            49.0     40.3   -8.7
2025            50.2     42.8   -7.4

3. A Spanish retail client wants to buy a US ETF that replicates an index of technology stocks. Their broker rejects the order. Explain the reason and what alternatives they have.