One of the most popular portfolio protection trading strategies is the Constant Proportion Portfolio Insurance (CPPI). This strategy maximizes the exposure in stock at each rebalancing time while ensuring that the portfolio level never drops under the so-called floor.

# Modeling Financial Transaction Tax

The currently proposed financial transaction tax is nothing else than a proportional transaction cost. The proposed value for the EU (especially France) is 0.1% for stocks and 0.01% for derivatives. In the following, we will denote this cost factor by alpha.

# Traditional CPPI – no transaction costs

Using ThetaML, the CPPI strategy is easily defined. For simplicity, we leave out the interest-rate since currently, it is about zero anyways:

model CPPI import S "Stock price process" import m "Multiplier" import F "Bond floor" import Pi_0 "Initial Portfolio value" import T "Maturity time" import n "Number of time steps" export Pi "Portfolio value in EUR" d = (m * (100-F))/S Pi = Pi_0 loop n S_old = S % Let time pass for a time-step Theta T/n Pi = Pi + d*(S-S_old) d = max( 0,(m*(Pi-F))/S ) end end

# CPPI and Transaction Costs

Since the Financial Transaction Tax is nothing else than proportional transaction cost, we can just decrease the portfolio after each rebalancing by transaction costs factor alpha times change in stock position.

That means we can write:

model CPPI import S "Stock price process" import m "Multiplier" import F "Bond floor" import Pi_0 "Initial Portfolio value" import T "Maturity time" import n "Number of time steps" import alpha "Proportional transaction costs" export Pi "Portfolio value in EUR" d = (m * (100-F))/S Pi = Pi_0 loop n S_old = S % Let time pass for a time-step Theta T/n Pi = Pi + d*(S-S_old) d_old = d d = max( 0,(m*(Pi-F))/S ) % Reduce portfolio value by transaction costs Pi = Pi - abs(alpha * (d-d_old) * S) end end

# Sample Case

Let’s consider a specific case of a CPPI strategy:

m "Multiplier" = 5 F "Bond floor" = 8000 Pi_0 "Initial Portfolio value" = 10000 T "Maturity time" = 10 (10 years) n "Number of time steps" = 250 * 10 (daily) alpha "Proportional transaction costs" = 0.001 (0.1%)

and a market model with zero interest-rate and a geometric brownian motion with 20% volatility as stock price process.

That means, we look at a 10 year CPPI strategy with daily rebalancing and 10.000 EUR initial investment.

Using Theta Suite to evaluate ThetaML by Monte Carlo Simulation -without transaction costs- we get

- Mean: EUR 27,096
- Median: EUR 8,186

We can see that without transaction costs, there is a 2.5% chance of making EUR 100,000 and more from the initial EUR 10,000 portfolio. But, about 50% of the portfolios lose the whole stock exposure and fall on the EUR 8,000 guarantee level (Bond Floor).

With transaction costs we get

- Mean: EUR 19,526 (-28%)
- Median: EUR 8,111 (-0.2%)

In the graph above we can see that with a financial transaction tax of 0.1% per trade, the 2.5% of the best paths earn about EUR 50,000 and more. This is half of what this strategy makes without this tax. The sample mean is down 28% and the median is almost unchanged. Consequently, the up-side potential of this CPPI strategy is hit dramatically by a 0.1% financial transaction tax.

# CPPI and Exchange Traded Funds

Last week, I attended an interesting seminar by Phillipe Bertrand comparing CPPI and Leveraged Exchange Traded Funds (LETF). His slides are available here. He shows that CPPI and LETFs can be very similar and some deliver almost the same returns. I suspect that the reduced upside potential of the CPPI strategy will also hit the Leveraged Exchange Traded Funds. But, I leave the analysis to you, the reader: It is not hard to model LETFs with transaction costs.

# Conclusion

The introduction of a financial transaction tax will hit the up-side potential of constant proportional portfolio insurance strategies considerably. In the presented case, the best 2.5% of the simulations will earn only about half after the introduction of this tax resp. transaction costs. This case can be seen as somewhat extreme since the presented strategy with a multiplier of 5 and daily rebalancing over 10 years is very aggressive. But, it shows that a portfolio protection which is desirable for most investors can be hit dramatically by the introduction of a financial transaction tax.

## Soeren Gerlach

Hi Andreas,

as I was just working recently in Munich on a CPPI-product on the IT side implementation your study shows exactly what has been discussed internally many times: The amount of fees seem to have a major impact on the overall performance, possibly even negating the possitive effect of the products “I” feature… The product setup was kind of portfolio management using a CPPI strategy, where the porfolio is built mainly from ETFs. Additionally the client was sold a gap option to protect him on gap risk that would occur due to the lagged rebalancing:

– A yearly fee based on the NAV of the portfolio

– A yearly fee (option premium) for the gap option, based on the NAV of the porfolio

– hidden: Transaction costs for the ETF transactions

– hidden: ETF management fees

– hidden: Index underperformance costs because not market, but provider indices where used (e.g. like Deutsche Bank’s “dbIQ” index family)

– maybe hidden: Kickbacks with ETF providers for the bank because of the usage of certain ETFs (this of course is part of the ETF’s TER but would likely increase it slightly).

However, this were just IT side internal discussions. Presenting these questions to the businees never resulted in an answer. Obviously they were more interested in the fees, then the clients portfolio performance.

Personally after gaining the knowledge about all this, I would not recommend to buy such a product.

On the other hand we assumed, that the average portfolio would need a rebalancing only once or twice a month, so a daily rebalancing is way too much.

Best regards,

Sören