Monte Carlo Simulations of Portfolios Allocated with Structured Products: A method to see the effect on risk and return for long time horizons

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Department of Physics

Monte Carlo Simulations of Portfolios Allocated with Structured Products

A method to see the effect on risk and return for long time horizons

January 12, 2018

Supervisor Kristofer Eriksson

kristofer.eriksson@nordea.com Examiner

Martin Rosvall

martin.rosvall@umu.se

Student Malin Fredriksson maalinfredriksson@hotmail.se

Master’s Thesis, Engineering Physics, 30 credits

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Abstract

Structured products are complex non-linear financial instruments that make it difficult to calculate their future risk and return. Two categories of structured products are Capital Protected and Participation notes, which are built by bonds and options. Since the structured products are non- linear, it is difficult to asses their long-term risk today. This study, conducted at Nordea Markets, focuses on the risk of structured products and how the risk and return in a portfolio changes when we include structured products into it. Nordea can only calculate the one-year risk with their current risk advisory tool, which makes long time predictions difficult. To solve this problem, we have simulated portfolios and structured products over a five-year time horizon with the Monte Carlo method. To investigate how the structured product allocations behave in different conditions, we have developed three test methods and a ranking program. The first test method measures how different underlying assets changes the risk and return in the portfolio allocations. The second test method varies the drift, volatility, and correlation for both the underlying asset and the portfolio to see how these parameters changes the risk and return. The third test method simulates a crisis market with high correlations and low drift. All these tests go through the ranking program, the most important part, where the different allocations are compared against the original portfolio to decide when the allocations perform better. The ranking is based on multiple risk measures, but the focus in this study is at using Expected Shortfall for risk while the expected return is used for ranking the return.

We used five different reference portfolios and six different structured products with specific pa- rameters in an example run where the ranking program and all three test methods are used. We found that the properties of the reference portfolio and the structured product’s underlying are significant and affect the performance the most. In the example run it was possible to find prefer- able cases for all structured products but some performed better than others. The test methods revealed many aspects of portfolio allocation with structured products, such as the decrease in portfolio risk for Capital Protected notes and increase in portfolio return for Participation notes.

Our ranking program proved to be useful in the sense that it simplifies the result interpretations.

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Contents

1 Introduction 1

1.1 Objective . . . 1

2 Non-Linear Instruments 2 2.1 Options . . . 2

2.2 Structured Products . . . 5

2.2.1 Capital Protected Notes . . . 5

2.2.2 Participation Notes . . . 6

3 Modeling 8 3.1 Linear Instruments . . . 8

3.2 Options . . . 9

3.3 Structured Products . . . 9

3.4 Portfolios . . . 10

3.5 Standard Monte Carlo Method . . . 10

3.6 Risk Measures . . . 10

3.6.1 Standard Deviation . . . 10

3.6.2 Sharpe Ratio . . . 11

3.6.3 Value-at-Risk (VaR) . . . 11

3.6.4 Expected Shortfall (ES) . . . 12

4 Methodology 13 4.1 Risk Measures . . . 14

4.2 Test Methods . . . 15

4.2.1 Varying Underlying Asset . . . 15

4.2.2 Market Variations . . . 16

4.2.3 Crisis Market . . . 16

4.3 Example Run . . . 16

5 Results and Discussion 18 5.1 Capital Protected Notes . . . 18

5.2 Participation Notes . . . 19

5.3 Conclusions . . . 19

6 Further Development 21

Bibliography 22

Appendix A Result tables from Example Run I

A.1 Find Best Underlying . . . II A.2 Market Variations . . . IV A.3 Crisis Market . . . X

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Dictionary

Linear asset Financial instrument, such as a stock, fund or bond, where the payoff has a linear dependence of the price

Non-linear asset Financial instrument, such as an option or a structured product, where the payoff has a non-linear dependence of the price

Portfolio Collection of financial instruments Option A contract on an underlying asset Call option The right to buy the underlying asset

Put option The right to sell the underlying asset Long position The holder of the contract

Short position The writer of the contract

Floor Downside risk protection of negative payoff Cap Upside limit of positive payoff

Drift Trend in the price over time for an asset Volatility A measure of deviation for the price trend

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Chapter 1

Introduction

In the wake of the financial crisis of 2008 came the second edition of Markets in Financial Instru- ments Directive, MiFID II, from the European Commission. The rules of MiFID II will be applied to investment firms and regulated markets in the European Union by the 3rd of January 2018 [1].

The directive states, among other things, that the risk of investing must be apparent. Calculating the risk can be a complex task when handling portfolios containing non-linear instruments, such as structured products. This thesis have been written at Nordea where I have worked on a study that has been a part of a bigger project within the bank. This study focuses on the risk of structured products and how the risk and return in a portfolio changes when we include structured products into it.

Structured products are complex financial instruments which can be constructed in many ways.

One way to create a structured product is to combine bonds and options. Options are non-linear instruments compared to stocks and bonds, which are linear, and hence simpler. The complexity of the structured products makes it difficult to calculate their risk and expected returns. Nordea’s risk advisory tool cannot fully handle the risk calculations when including a structured product in a portfolio. Additionally, the calculated risk in the advisory tool is only given for a one year time horizon which is not optimal if the product’s expiry is several years.

It is possible to scale one-year portfolio risk over time with the square-root rule√

t if the portfolio returns are independent and identically distributed (i.i.d.) over time [2]. Structured products with options have early in this study have shown indications of non i.i.d. properties over time and to use the square-root scaling technique on a portfolio containing these structured products would therefore not be a suitable approach [3].

Since time scaling the risk from the advisory tool might not be appropriate we need to look at the portfolio risk in another way. Therefore, I have, together with my project partner Emelie Aspholm at Nordea Markets, simulated portfolios including structured products with options over their maturity time. We have developed test methods to evaluate the combinations of products and portfolios in a thorough way. By varying the parameters, of both a reference portfolio and the underlying of the structured products, and the weights of the structured products we have looked for cases when the portfolio is improved by including a structured product.

1.1 Objective

The goal of the study is to find when a structured product improves a portfolio. This may seem as a simple statement but behind it lies a big amount of aspects to consider. What measure decides if the portfolio is ”improved” and how much improvement is needed for the allocation to be a good investment? In addition to this, there are an infinite number of possible portfolios and ways to construct the structured products. The problem at hand is to find a method that allows us to see the effects on the risk and returns in portfolios when including structured products, both in static and changing environments.

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Chapter 2

Non-Linear Instruments

The structured products that we are looking at in this study are financial instruments which are constructed of a linear and a non-linear part. The linear part is a zero-coupon bond and represents a risk-free investment. The non-linear part can be different kinds of derivatives depending on what properties it should have. Such derivatives may be one or several options.

2.1 Options

An option is a contract on an asset which gives the holder of the contract the right, but not the obligation, to buy (call option) or sell (put option) the asset. A strike price is determined when the option is written. The strike price, K, is the price limit of the underlying asset and determines if the contract will be exercised or not. The asset is often referred to as "the underlying" and can for example be a stock, an equity index or a rate. The price of the underlying asset is referred to as the spot price and denoted with S.

Some options are of European or American form. American option contracts can be exercised at any time until expiry while European option contracts only are exercised at the pre-determined expiry date of the contract. For the case of this paper we are only considering the European kind of options.

Depending on if you are the holder of the option contract (long position) or the writer (short position) your payoff exposure against the price of the underlying asset is different. The different payoff exposures of long and short positions of call and put options can be seen in Fig. 2.1 [4]. In the plots of the payoff, we have ignored the option premium. The premium is paid by the holder and received by the writer of an option which would make the payoff exposures shift down for long positions and shift up for short positions. When the options are included in a structured product the premium is no longer of interest in the payoff plots, for the sake of this study.

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0.8 0.9 1 1.1 1.2

−0.2 0 0.2

K

Underlying Short Call

Spot price

Payoff

(a)

0.8 0.9 1 1.1 1.2

−0.2 0 0.2

K

Underlying Long Call

Spot price

Payoff

(b)

0.8 0.9 1 1.1 1.2

−0.2 0 0.2

K

Underlying Short Put

Spot price

Payoff

(c)

0.8 0.9 1 1.1 1.2

−0.2 0 0.2

K

Underlying Long Put

Spot price

Payoff

(d)

Figure 2.1: The option’s exposure to the price of the underlying asset in (a) a short call, (b) a long call, (c) a short put and (d) a long put. Long position means holding the contract and short position means writing the contract. A call option gives the right to buy the underlying asset while a put options gives the right to sell the underlying asset.

It is possible to combine different positions and strike prices in calls and puts on the same underlying to achieve more advanced payoff exposures. To keep track of the strike prices it is useful to categorize the strikes as in-the-money (ITM), at-the-money (ATM) and out-of-the-money (OTM).

How the strike prices are related to the spot price of the underlying depends on if we have a call or put option. All the combinations are shown in Tab. 2.1 [5].

Table 2.1: The relation of strike and spot prices for ITM, ATM, OTM call and put options.

Call option Put option

In-the-money (ITM)

Strike price is less than underlying’s spot price

K < S

Strike price is greater than underlying’s spot price

K > S At-the-money

(ATM)

Strike price is equal to underlying’s spot price

K = S

Strike price is equal to underlying’s spot price

K = S Out-of-the-money

(OTM)

Strike price is greater than underlying’s spot price

K > S

Strike price is less than underlying asses spot price

K < S

There is an option combination called call spread. The call spread is made up of a long ATM call combined with a short OTM call option. This combination creates a protection against down-side risk (floor) but also an up-side limit (cap). How the payoff exposure towards the underlying asset’s

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spot price look can be seen in Fig. 2.2a. It is also possible to replicate holding the underlying asset with a spread by having a long position in an ATM call and a short position in an ATM put.

This replication can be thought of as a ”synthetic stock” but we refer to it as a Booster option.

The exposure to the underlying of the Booster can be seen in Fig. 2.2b. We will come back to the Booster option further on and look at some additional features to it [6].

0.8 0.9 1 1.1 1.2

−0.2 0 0.2

K1 K2

Underlying Call Spread

Spot price

Payoff

(a)

0.8 0.9 1 1.1 1.2

−0.2 0 0.2

K

Underlying Booster

Spot price

Payoff

(b)

Figure 2.2: Exposure to the price of the underlying asset for (a) a call spread option and (b) a Booster option.

Another option type is the Barrier option. The typical attribute of a Barrier option is that it has some sort of constraint on the price of the underlying asset, a barrier. There are many different ways in which the barrier can work but in this study we are focusing on an expiry dependent barrier. If the spot price of the underlying reaches a pre-determined barrier value B at expiry some event will occur. This event can for example be that the option becomes worthless or that the payoff structure changes. We are looking at a case with payoff structure like a call option but where the down-side protection disappears if the spot price falls below the barrier B. The option behaves as a call option until the down-side barrier is reached and then the exposure drops down to the underlying’s down-side. This can be created with a long position in an ATM call and a short position in a European Down-and-In (DI) ATM put. A European DI option means that if the spot price of the underlying goes down below the DI barrier at expiry the option is active. Above the barrier the DI put option is non-active at expiry. So, if the spot price is above the DI barrier at expiry we only have the ATM call and if the spot price falls below the DI barrier at expiry it acts like a ATM put. What this exposure looks like can be seen in Fig. 2.3 [7].

0.8 0.9 1 1.1 1.2

−0.2 0 0.2

K B

Underlying DI Barrier

Spot price

Payoff

Figure 2.3: Exposure to the price of the underlying asset for a Down-and-In barrier option.

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2.2 Structured Products

The structured products that we are looking at are issued on a primary market. There is also a secondary market for these products, where the product can be traded after the issue date. The individual instruments in the product are mostly traded over-the-counter (OTC) and can not be traded on an exchange. Nordea acts as issuer, organizer and distributor of the structured products they offer to their customers.

Say that the issue price of a structured product is 10 000 SEK. The issuer of the product buys bonds for an amount that grow with a certain rate to be 10 000 SEK at expiry of the product. This rate is not risk-free since it includes the risk of a credit event for the issuer, in our case Nordea.

The remaining amount of our 10 000 SEK, the market share, is used to purchase options. This concept is illustrated for an example of a structured product in Fig. 2.4. Which options we buy determine the properties of the product. Depending on the amount of options in the product we can reach different leverage effects, also called the Participation Ratio. The Participation Ratio can be described as the degree of participation in the underlying asset. A Participation Ratio of 1.0 means that we get the same return at expiry as if we hold the underlying asset, given that the product has the same payoff structure as the underlying.

Start Expiry

Bond Market share

Fee

Nominal amount Possible return

Figure 2.4: Schematic figure for a Capital Protected structured product, for which the owner is guaranteed to receive the nominal amount at expiry.

Nordea has four standardized categories of structured products; Capital Protected notes, Yield Enhancement notes, Participation notes and Credit Linked notes. The structured products within these categories make up a major part of the structured products issued by Nordea. In this paper, we are focusing on the Capital Protected and Participation notes.

2.2.1 Capital Protected Notes

Capital Protected Notes are, just as the name indicates, products that provide protection against down-side risk. So, if the issue price of the structured product is 10 000 SEK we are sure of receiving at least 10 000 SEK (the nominal amount) at expiry, provided that no credit event will take place at Nordea. This can be achieved by buying call or call spread options with the market share. When the rate is low, such as today, there is often a need of an overcharge of the product in order to finance the option purchase. An overcharge is typically 5-10% of the issue price and is not guaranteed at expiry. However, the overcharge increases the Participation Ratio of the product.

An illustration of a Capital Protected note with overcharge can be found in Fig. 2.5.

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Start Expiry Bond

Market share Fee Overcharge

Nominal amount Possible return

Figure 2.5: Schematic figure for a Capital Protected structured product with overcharge, for which the owner is guaranteed to receive the nominal amount at expiry but not the overcharge.

Nordea has three standard products within this category; Uncapped Capital Protected note, Capped Capital Protected note and Min Redemption note. The Uncapped Capital Protected note is constructed with long position in ATM call options and the Capped Capital Protected note with long positions in a call spread option. The Min Redemption note consists of three parts; a long ATM call, a short ATM put and a long OTM put. We will have equal amount of the short ATM put and the long OTM put. The price of an OTM put is less expensive than the ATM put since you get the downside protection earlier with an ATM put. This price difference makes it possible to be long in more ATM calls which increases the Participation Ratio on the up-side.

Graphic representations of the payoff exposure to the underlying for all three Capital Protected notes can be seen in Fig. 2.6.

0.8 1 1.2

−0.2 0 0.2

K Underlying

UCP

Spot price

Payoff

(a)

0.8 1 1.2

−0.2 0 0.2

K1 K2

Underlying CCP

Spot price

Payoff

(b)

0.8 1 1.2

−0.2 0 0.2

K1

K2

Underlying MR

Spot price

Payoff

(c)

Figure 2.6: Exposure to the price of the underlying asset for (a) a Uncapped Capital Protected Note and (b) a Capped Capital Protected Note and (c) a Min Redemption Note.

The Min Redemption note is not considered as Capital Protected in Sweden since it has a down- side risk, even though the risk is limited. In Finland the standards are different and the Min Redemption is therefore classified and issued as a Capital Protected note there.

2.2.2 Participation Notes

The Participation notes comes with a down-side risk, contrary to the Capital Protected notes, and are suitable when the investor expects the market to go up. With the Participation notes there is no overprice and no guaranteed amount at expiry.

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Nordea has three standard Participation notes; Uncapped Booster, Uncapped Booster with Buffer and Capped Booster. We have mentioned the Booster before and talked about it as a synthetic stock. In this context, we will have a long position in more ATM calls than we have short positions in ATM puts which will increase the Participation Ratio on the up-side. The Capped Booster is created in a similar fashion but with a long position in a call spread instead of a call. This call spread creates a cap on the up-side, where we get an increased Participation Ratio. The Uncapped Booster with Buffer is created with a long ATM call and a short DI put with a European barrier, as we saw in the option section. By shorting DI puts we can buy more ATM calls which create an increased Participation Ratio on the upside. The three notes are graphically represented in Fig. 2.7.

0.8 1 1.2

−0.2 0 0.2

K Underlying Booster

Spot price

Payoff

(a)

0.8 1 1.2

−0.2 0 0.2

K1K2 Underlying Cap. Booster

Spot price

Payoff

(b)

0.8 1 1.2

−0.2 0 0.2

K B

Underlying Booster w. B.

Spot price

Payoff

(c)

Figure 2.7: Exposure to the price of the underlying asset for (a) a Uncapped Booster Note, (b) a Capped Booster Note and (c) a Booster with Buffer Note.

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Chapter 3

Modeling

Now it is time to go into the modelling of these instruments. We will start with the linear instru- ments, such as the bond and the underlying asset, and then move on to the modelling of options, structured products and portfolios.

3.1 Linear Instruments

The linear instruments that we are looking at in this thesis are bonds, funds, stocks and indexes.

The bond is assumed to grow with a risk-free rate while funds, stocks and indexes are stochastic processes. In other words, the price process of the bond is deterministic. The price depend on the price at time 0, B(0), and the yearly risk-free spot rate rf for the time period at hand. For the sake of modelling we are considering the rate to be risk-free but keep in mind that it is not the case in reality. At a future time t the bond is worth

B(t) = B(0)e(rf+c)t (3.1)

where c is the credit spread of the issuer. The credit spread stands for the risk of a credit event, for example that Nordea, the issuer, goes bankrupt. The customer is rewarded for taking this extra risk and the credit spread is added to the rate. This extra risk for the customer increases the value of the bond over time, if there are no credit events. The credit spread is in fact a stochastic process but we model it as a static process in this study. Some rates are negative in todays market but we are only considering positive and non-zero rates in this paper.

For a linear risky asset, such as a stock, the price at time t, S(t), is a stochastic process modelled by a geometric Brownian motion

S(t) = S(0)e(µ−12σ2)t+σW (t) (3.2) where the drift, µ, is the trend of the random walk and the volatility, σ, describes the instability of the process. The exponent is normally distributed, due to the random Brownian Motion W (t), and we get a log normal distributed price of the asset. S(t) satisfies the stochastic differential equation

dS(t) = µS(t)dt + σS(t)dW (t), (3.3)

which is a more intuitive way of interpret the path of a stock.

The simple return R(t) of an asset is calculated using the price at t, S(t), and the price at an earlier point in time S(t − 1) with the formula

R(t) = S(t)

S(t − 1)− 1, (3.4)

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which can be transformed into log returns to create normally distributed returns r(t) using

r(t) = ln[R(t) + 1] = ln[ S(t)

S(t − 1)]. (3.5)

The way of calculating the price process in Eq. 3.2 can be used for stocks, funds and indexes. We are using the simple returns in Eq. 3.4 for all financial instruments included in the study [8].

The correlation between two financial instruments is calculated on their log returns, r(t). The correlation ρ of assets A and B is calculated with the correlation coefficient

ρA,B=

Pn

i=1(rA,i− ¯rA)(rB,i− ¯rB) pPn

i=1(rA,i− ¯rA)2Pn

i=1(rB,i− ¯rB)2. (3.6)

3.2 Options

We have mentioned many different options in the previous section but most of them are just constructed by combinations of call and put options. The payoff exposures against the underlying can be mathematically explained by the call and put options payoff functions. The payoff is determined by the spot price at expiry, S(T ), and the strike price K of the option using Eq. 3.7

C(T, S(T )) = max{S(T ) − K, 0}

P (T, S(T )) = max{K − S(T ), 0}. (3.7)

To calculate the price of a European call or put option at t < T we use the Black-Scholes formula in Eq. 3.8

C(t, S(t)) = S(t) exp{−δ(T − t)}N [d1(t, S(t))] − K exp{−rf(T − t)}N [d2(t, S(t))]

P (t, S(t)) = K exp{−rf(T − t)}N [d2(t, S(t))] − S(t) exp{−δ(T − t)}N [d1(t, S(t))] (3.8) (d1(t, S(t)) = 1

σ

T −t{ln(S(t)K ) + (rf1

2σ2)(T − t)}

d2(t, S(t)) = d1(t, S(t)) − σ√ T − t

where N is the cumulative distribution function, δ is the dividend yield, rf is the risk-free rate, σ is the volatility of the option, T is the expiry, S(t) is the spot price of the underlying at time t and K is the strike price. To calculate the price of options that consist of call and put combinations, we combine the price equations in the same manner. The price of an call spread at time t is CS(t) = C(t, K1) − C(t, K2) where K1and K2are the different strike prices for the long and short call options [8].

For the DI put option considered in this study, the payoff PBI at expiry is calculated as

PBI(T, S(T )) =

(K − S(T ) if S(T ) ≤ B

0, if S(T ) > B (3.9)

where B is the barrier value.

3.3 Structured Products

In scope of this thesis, the structured products contains of a bond and a set of options, as mentioned before. We know the initial price of the structured product in advance so the focus in this paper lies on the changes of risk and return until expiry when the market parameters change. The market parameters we vary are the drift and volatility of the underlying asset, the drift and volatility of a reference portfolio and the correlation between these two instruments. The dividend yield, rate

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and credit spread of the bond as well as the strike prices of the options are held constant. Other parameters that affect the result of a structured product are the fee, the Participation Ratio of the options and for how long we will hold the product. The holding period ends at expiry and is determined in advance. We assume that each option have the same expiry as the product.

The return of the structured product in our case is just the return of its parts; the return of the bond and the return of the options. During the time until expiry we use Eq. 3.8 to calculate the return of the options. At expiry, we use Eq. 3.7 and Eq. 3.9 to calculate the payoff of the options.

3.4 Portfolios

A portfolio can be explained as a collection of financial instruments. In this paper, we are looking at a portfolio that either contains a fund and an index or a fund and a structured product. The fund alone is used as a reference portfolio to which we add either the index or the structured product.

Even though we say that we have two instruments in the portfolio, the instruments included can have different weights. The weights wi indicates how the value of the instruments are distributed.

The return of a portfolio is calculated as in Eq. 3.10.

Rp=

n

X

i=1

wiRi (3.10)

3.5 Standard Monte Carlo Method

One way to simulate instruments is to use the Standard Monte Carlo (SMC) method. The following steps create n samples and approximates the option payoff at expiry:

1. Generate n independent N (0, 1) random numbers Z1, Z2, .., Zn

2. Let Si(T ) = S(0)e(µ−12σ2)T +σ

T Zi

3. Let Ci= e−µTmax{Si(T ) − K, 0}, or Pi= e−µTmax{K − Si(T ), 0} for put option 4. C ≈ ˆCn =n1Pn

i=1Ci

When doing SMC in the multidimensional case we perform a Cholesky factorization of the covari- ance matrix Σ to find a matrix A such that AAT = Σ. This Cholesky factor is then used to create the correlated random variables X = AZ, where Z are a vector of i.i.d. N (0, 1) random variables.

This method can also be done for several time steps but then we use the Black-Scholes formula in Eq. 3.8 for the option prices in step 3 of the SMC method [9].

3.6 Risk Measures

The risk of a financial instrument is calculated on a set of its returns. There are many ways to measure how risky a financial instrument is but which risk measure to use is not always obvious.

We are looking at four measures; Standard Deviation, Sharpe Ratio, Value-at-Risk (VaR) and Expected Shortfall (ES). The Sharpe Ratio is more of a performance measure than a risk measure but it depends on the standard deviation. The modelling is based on that we have simulated samples of returns for our instruments.

3.6.1 Standard Deviation

When we use standard deviation as a risk measure we look at the dispersion of the return distri- bution around the mean value. Mean value of the simulated returns is the same as the expected return of the instrument, not to be mistaken as the mean value of historical returns. The standard deviation is calculated as

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sd(t) = v u u t

1 N − 1

N

X

i=1

(Ri(t) − ¯R(t))2, (3.11)

where ¯R(t) is the expected return at a certain point in time and the N pieces of Ri(t) are simulated return samples for the same point in time. A large standard deviation means greater risk since the values of the returns are more uncertain than for a sample with a smaller standard deviation.

It is also possible to only look at the negative side of the return distribution. This kind of risk measure if often referred to as the downside standard deviation and is calculated as

Dsd(t) = v u u t

1 N− 1

N

X

i=1

(Ri (t) − 0), (3.12)

where Ri (t) is the i:th element of the N negative simulated returns at a certain point in time.

3.6.2 Sharpe Ratio

The Sharpe Ratio can be seen as a measure of performance and is sometimes used when we want to compare two instruments. The ratio is the expected excess return divided by the standard deviation of the instrument. The expected excess return is the same as the expected return of the instrument minus the risk-free rate. We assume that the risk-free rate is constant during the holding period and hence has no variance. The Sharpe Ratio for an instrument at time t is expressed as

SR(t) = E[R(t) − rf] pV ar[R(t)] =

R(t) − r¯ f

sd(t) , (3.13)

which was developed by William F. Sharpe in 1966 [10].

3.6.3 Value-at-Risk (VaR)

The risk measure VaR means that for a given confidence level α ∈ (0, 1) we have that the probability of the return R(t) exceeding the value −x is at most 1 − α. The mathematical expression for this is

V aRα(R(t)) = inf {x ∈ R : P (R(t) + x < 0) ≤ 1 − α}. (3.14) In other words, with α = 99% it is 1% probability that the return R(t) falls below −x. For a log normal distributed sample of returns this value can be represented as in Fig. 3.1.

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0 1 2 3 0

0.2 0.4 0.6 0.8

VaR

Profit/Loss

PDF

Figure 3.1: The risk measure VaR of an log normal sample

For a simulated sample, we get the VaR value in an easy way. First all returns are sorted in order at the desired time point and then α:th value is chosen. For example, with 20 000 simulated returns the 95% VaR is the 1 000:th smallest return. This risk measure stands for the amount at risk and is therefore presented as a positive value [11].

3.6.4 Expected Shortfall (ES)

The VaR measure is good when the tails of the distribution are decreasing and smooth. If the distribution has fat tails or tails with ”bumps” the VaR value can be misleading. Expected Shortfall, ES, is a mean value of the tail beyond VaR. It considers all possible losses beyond the VaR value no matter what the tail looks like. The mathematical formula is written as

ESα= 1 1 − α

Z 1−α 0

V aRγ(R(t))dγ. (3.15)

The part of the distribution on which the ES is calculated is illustrated in Fig. 3.2 with the shaded area.

0 1 2 3

0 0.2 0.4 0.6 0.8

ES

Profit/Loss

PDF

Figure 3.2: The part of the distribution on which the risk measure ES is calculated, for a log normal sample.

The numerical calculation for our simulated sample of returns is the sum of all returns beyond V aRα, divided by 1 − α. As with the VaR value we are measuring ES in positive losses [11].

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Chapter 4

Methodology

A pilot study done by Krister Alvelius at Nordea Markets shows that different risk measures give different results for portfolios including structured products [12]. For this pilot study, he has written a C# code in Visual Studios which uses Monte Carlo simulations to create portfolios containing both linear instruments and structured products. This original code is used through an Excel interface where a portfolio can be tested over time for a fixed set of market parameters and a set of different weights of a structured product.

The pilot study serves as our starting point and the code is expanded with all the relevant structured products and risk measures that have been mentioned in the theory section. To be able to run variations of all parameters and to handle big volumes of data we have written an additional program, to serve as a new main program that does not need Excel, in the existing Visual Studio solution.

The most important function of the new main program is that it ranks the possible allocations toward the reference portfolio. So, if you have a reference portfolio (R) you could weigh in some percentage of a structured product (SP) or some percentage of its linear underlying asset (Lin). The output from the ranking is a text file that tells you which combination that performs best for each case. It comes with the combinations of letters R (100% reference portfolio), SP (structured product weighted into reference portfolio) and Lin (underlying asset weighted into reference portfolio) depending of the ranking. This text file is designed to be pasted into Excel where it is color coded based on the ranking result which gives a quick overview of the results. In Fig. 4.1 the color coding is explained.

Table 4.1: Explanation of the color coded ranking output.

SP R (or SP Lin) GREEN: Structured product weigh-in performs best

R SP YELLOW: Reference portfolio performed best, SP is second best

Lin SP ORANGE: Linear weigh-in is better than SP but both are better than ref. portf.

Lin R (or R Lin) RED: Both reference portfolio and linear asset perform better than SP

In the output from the ranking there is also a pair of numbers. These numbers correspond to the quotients (SP/R) and (Lin/R) of each statistical measure; risk or return value. The quotients are used to enhance the result in the color coding. The color coding is a good way to get a quick overlook and the numbers says how big the differences are. An example of ranking risk in the output is [Lin R (1.044) (0.959)] which is a red box in the excel file. The first number (1.044) says that SP has 4.4 percentage larger risk than R while the second number (0.959) says that Lin has 4.1 percentage units smaller risk. The ranking works such that low risks are ranked best and that high returns are ranked best. An example of a green box when ranking on return is therefore [SP R (1.056) (0.973)]. A part of a color coded ranking can be found in Appendix refapp:res, Fig. A.1.

For the upside performance, the expected return (ER) of the portfolio is used for ranking but for the downside performance the choice is not as simple. It is possible to choose which risk measure to use for downside ranking, depending on your own preferences.

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4.1 Risk Measures

The ranking code can handle all the mentioned risk measures but it is important to know how they work for non-linear instruments, especially the structured products that we are looking at in this study. To find which risk measure to focus on in this study we tested the structured products for all of them to investigate the strengths and weaknesses of the risk measures.

Standard Deviation The standard deviation is a common measure which is easy to interpret for linear instruments. When the payoff function of an instrument is non-linear, as for most of the structured products in this study, the standard deviation gives misleading results. The standard deviation for the same non-linear instrument is different depending on its expected return, i.e. which part of the payoff function that gets the most influence.

An increasing Participation Ratio on the upside (as for an Uncapped Booster) causes an increased risk with this measure. Structured products with a cap (as for a Capped Booster) gets a decreasing risk for a high drift with this measure. If an Uncapped Booster and a Capped Booster have the same underlying, their downside payoff looks the same but with the standard deviation as risk measure, the upside has a big impact which causes misleading results.

Sharpe Ratio Since the Sharpe ratio is a function of the standard deviation it inher- its the disadvantages of it. The Sharpe ratio is usually a good way to compare the performance of different linear instruments but for non- linear instruments the comparison is misleading. If we could construct two Booster notes on the same underlying but with different Participa- tion Ratio on the upside, we would always choose the one with highest Participation Ratio. When we look at the Sharpe ratio for two such products they are estimated to perform equally. The Sharpe ratio as a performance measure does not reflect the performance of these products because it depends on the standard deviation. This is another example on how the standard deviation is not suitable for non-linear instruments.

Downside St. Dev. The problem with the ordinary standard deviation is that it states an increased risk when a product has high upside Participation Ratio. If the upside is left out and we only look at the spread of the negative returns (the downside) the problem caused by different Participation Ratios is eliminated. We have a pure downside risk measure. The issue with this measure is that we have no indications of the probability of getting a negative return. So, to use the downside standard deviation as a risk measure we also need the downside probability. The downside standard deviation is better suited than expected shortfall for detecting the effects of the partial floor of a Booster with buffer.

Value-at-Risk VaR is a common measure of downside risk for linear instruments. It works best for return distributions with smooth decreasing tails which the non-linear instruments not always have. If there are many extreme values in the last part of the tail of the distribution VaR does not take these into account. This shortcoming of VaR can cause misleading results when used as a risk measure for the structured products in this study.

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Expected Shortfall If we use ES instead of VaR we are not as sensitive to the return dis- tribution since ES is the mean of the remaining tail. If there are some extreme values that causes a spike in the tail ES will reflect these. Com- pared to standard deviation, ES is only considering the downside risk which makes it independent of the upside behavior of a structured prod- uct. A problem with ES is that it does not reflect the partial floor of a Booster with Buffer as good as the downside standard deviation does.

From all these alternatives, we choose ES to be our standard risk measure for this study. It is a pure downside risk measure and is not sensitive to the shape of the return distribution. Since ES is based on a fixed percentage of the distribution it is easy to compare different instruments, a perk that downside standard deviation does not possesses.

4.2 Test Methods

To find suitable portfolio allocations with structured products we have developed a program code that can perform tests where different parameters, such as drift, volatility and correlation, are varied for each simulation run. One run means that one weight combination of a structured product and a reference portfolio, with fixed yearly parameters, is simulated for the whole holding period.

The aim is to find cases where a structured product improves a portfolio but the term ”improve”

is a matter of opinion. The two categories of structured products in this study behave different and are therefore separated in the matter of how an improvement is determined. Participation notes have upside Participation Ratio on more than 1.0 and should therefore increase the expected return when weighted into a portfolio while the downside risk should remain relatively unchanged.

Capital Protected notes have a downside floor and should therefore decrease the downside risk when weighted into a portfolio while the upside should remain relatively unchanged. These expectations are stated in Tab. 4.2 where the term ”relatively unchanged” is set to a maximum of 5% from the reference portfolio, for the scope of this study. It makes no difference for the output from the tests, the change limit is only a tool for analyzing the results.

Table 4.2: Classifications of the improvements for the two categories of structured products on upside performance (Expected Return) and downside risk performance (Expected Shortfall). The changes are referred to as percentage units from 100% reference portfolio.

Category Upside change (Exp. Ret.) Downside change (ES) Capital Protected Notes Max 5% worse Improve reference portfolio Participation Notes Improve reference portfolio Max 5% worse

The idea is that the user of the program has the possibility to test any combination of a structured product and a portfolio, at any number of levels of weigh-ins and at any time horizon. To make the tests of suitable portfolio allocations as comprehensive as possible we constructed three different test methods. All the test methods uses the ranking program on its results to produce quickly analyzed results.

4.2.1 Varying Underlying Asset

The first test method lets us examine which combination of drift and volatility of the structured product’s underlying that suits the reference portfolio best when weighted in. There are no limits on how many different underlyings to test but to keep the data set manageable we recommend to keep the combinations to a maximum of 50, for example ten different drifts and five different volatilities. In our test, we use 12 different drifts within 1-14 and three volatilities within 15-20, a total of 36 combinations, and a fixed correlation at 0.5.

Besides showing which underlying that suits each case the best, this test method also gives an indication on how the different structured products perform compared to each other. The lower

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the drift that is needed for an allocation to be preferable, the better the structured product is.

Underlyings with high drift, low volatility and low correlation are usually preferred overall so it is interesting to look for structured products that improve a portfolio on opposite terms.

All percentage changes that are stated in the test methods are relative changes from an initial value. When it comes to drift, it is the drift of the underlying asset of the structured product that changes and not the drift of the product itself.

4.2.2 Market Variations

In the second test method, we start with an initial drift, volatility and correlation for the underlying asset and the reference portfolio. From this starting point, we let the market parameters vary between each simulation in three different ways:

1. The drift, of both the underlying and the reference portfolio, is varied with −150% and +50%

while the volatility and correlation are static.

2. The correlation is varied between 0.1 and 0.9 while the drift and volatility are held static.

3. The volatility, of both the underlying and the reference portfolio, is varied with −50% and +50% while the drift, correlation and Participation Ratio of the structured products are held static.

These are the intervals of the market variations we use in the tests but they can easily be altered in the code. An important aspect is to test for negative drifts, hence to go lower than −100% in the drift variation.

4.2.3 Crisis Market

The third test method simulates a market crash with a correlation close to 1.0 and a large decrease in drift. In our study, we let the drift of both the underlying and the reference portfolio decrease with −600% with a correlation of 0.95, but these parameter choices are optional in the code.

4.3 Example Run

To try our ranking and test methods we did an example run with realistic structured products and reference portfolios. The parameters were chosen in agreement with our project leader at Nordea Markets, Fredrik Stenberg. The product specific parameters are listed in Tab. 4.3 and the common parameters are listed in Tab. 4.4.

Table 4.3: The structured products and their different properties; issue price factor, strike prices, fee and Participation Ratio factor at volatility 17.48 of the underlying.

Structured Product Issue Price K1 K2 Fee Participation Ratio

Uncapped Capital Protected 1.10 1 - 0.03 0.996

Capped Capital Protected 1.05 1 1.30 0.03 0.902

Min Redemption 1 1 0.90 0.03 0.649

Uncapped Booster 1 1 - 0.05 1.945

Capped Booster 1 1 1.30 0.05 3.360

Booster with Buffer 1 1 0.70 0.05 1.468

Table 4.4: Common parameters for all structured products Expiry Yield Rate Credit Spread

5 3% 0.5% 0.2%

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We have approximated Nordea’s popular Stratega funds with 11-year average data from MSCI World Index (drift: 5.94, vol.: 17.48) and Global Aggregate Fund Index (drift: 3.44, vol.: 5.47) to use as our reference portfolios. The number of the reference portfolio corresponds to index part, for example reference portfolio R30 has 30% index (MSCI World) and 70% fund (Global Aggregate).

The yearly fees for the reference portfolios are the current fees for the Stratega funds. All five reference portfolios and their parameters are listed in Tab. 4.5.

Table 4.5: Funds used as reference portfolio and their yearly parameters Reference Portfolio Drift Volatility Fee

R10 3.69% 6.67% 0.98%

R30 4.19% 9.07% 1.39%

R50 4.69% 11.48% 1.61%

R70 5.19% 13.88% 1.62%

R100 5.94% 17.48% 1.79%

In the market variation and crisis market tests we use an initial underlying with similar drift and volatility as the R100 portfolio; drift 6 and volatility 17.48. The underlying itself has no fee in these tests which might create misleading results when it is used as an in-weighed linear asset in the portfolio. The loop vector of weights to weigh in linear assets and structured products in the example run is [0,1,2,3,4,5,7,10,20,30,50,70,100].

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Chapter 5

Results and Discussion

In the example run we tested weights of the structured products from 1% to 100% but our focus for comparing the products have been at 10% weigh-in. As an overview of the Find Best Underlying test method, we are listing the lowest drift needed to weigh in the structured product in each portfolio at 10% weight, when the volatility of the underlying is 17.48. The overview can be seen in Tab. 5.1. This overview shows that the Uncapped Capital Protected and the Min Redemption notes are identical while the Capped Capital Protected note need higher drift to be a preferred alternative. All structured products within the Participation category behave very similar in this test and a higher drift is needed when the reference portfolio has a higher share of indexes.

Table 5.1: Minimum drift the underlying asset needs for its structured product to improve a reference portfolio when weighted in 10%. The underlying asset has a volatility of 17.48.

Structured Product R10 R30 R50 R70 R100

Uncapped Capital Protected 4 4 5 5 6

Capped Capital Protected 7 7 8 9 10

Min Redemption 4 4 5 5 6

Uncapped Booster 7 4 4 4 4

Capped Booster 7 4 4 4 5

Booster with Buffer 6 4 4 5 5

I have chosen to look at only one drift level of the results and three levels of volatility in the Varying Underlying Asset test method to limit the extent of the example run. For all three test methods, I have only gathered the 10% weigh-in results for the same reason. These limited results from the test methods can be found in Appendix A, Tab. A.1.1-A.3.1.

5.1 Capital Protected Notes

The best performing product within this category in the Varying Underlying Asset test method is the Uncapped Capital Protected note. It has the most preferable cases within the category.

The downside, measured in ES, is improved by Uncapped Capital Protected note while the upside, measured in Expected Return, is never worsened more than 5% units. The Min Redemption note performs second best, something that was expected since it is very similar to the Uncapped Capital Protected note. For the Capped Capital Protected note there is almost no preferable case since this product worsen the Expected Return of the portfolio more than 5% units in nearly every case.

All products within the category perform best for the lower level of volatility.

In the Market Variations test method, we can see that the Uncapped Capital Protected note and the Min Redemption note show the same tendencies for parameter changes. They both perform better in the portfolios when the drift decreases and when the volatility increases, which could be considered as an advantage. The Capped Capital Protected note also perform better for decreasing

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drift but the trend for volatility changes is not as clear. The return is worsened for increasing volatility while the risk improves. All three products within the category are rather insensitive to changes in correlation which is a strength for the category.

In a crisis market the Capital Protected notes improve the portfolios as expected. The improvement in risk is however never larger than the weigh-in percentage. So, for a weigh-in of 10% you can not expect an improvement larger than that on the downside. In the crisis test it is the Capped Capital Protected note that perform best within the category, most likely due to the lack of both overcharge and downside.

5.2 Participation Notes

Among the Participation notes there are many preferable cases in the Varying Underlying Asset test method. All three products have approximately the same amount of preferable cases but the Uncapped Booster note show the highest improvement in Expected Return in the portfolios.

The Capped and Uncapped Booster notes have basically the same downside performance while the Booster with Buffer note show slightly better improvement in ES for the portfolios, which is expected due to the partial floor of Booster with Buffer. Both the Capped and Uncapped Booster note perform best for the lower level of volatility while this is not as apparent for Booster with Buffer.

When the Participation notes are tested in a varied market all products show improved performance for increasing drift in all reference portfolios, except in R10. The Capped Booster note performs better for low volatilities while the Uncapped Booster and Booster with Buffer favor no specific volatility. Their returns improve for high volatility while their risks get worse. The return for all Participation notes is rather unchanged when the correlation differs but the risk is favored by low correlation.

The Participation notes are not to prefer in a crisis market. For the level of largest decrease in drift, none of the products are preferable. In the R100 portfolio the changes are small but the R10 portfolio is worsened both on its up- and downside by all products within the category.

5.3 Conclusions

We managed to produce a program for the a long-time risk horizon that could handle many different scenarios. Now, we can look at 5-year risk instead of depending on 1-year risk. The test methods have the ability to show many sides of the structured products in varying environments, which was one of our goals. Effects on the risk and return from caps, floors and barriers well as effects from changes in leverage are possible to detect. The program produces a big amount of data but with the ranking and its color coding the analysis becomes more efficient. It is easy to make quick assessments and evaluate the results, and hopefully, it will be a useful tool in risk calculations for portfolio allocations when structured products are considered.

The example run show that the test methods and the ranking program have the ability to deliver results that reflects the products properties, such as floors and Participation Ratios. The results show that the Capital Protected notes, which have floors, decrease the losses in a portfolio for decreasing drift and that the Participation notes, which have high Participation Ratios, improve portfolios most for increasing drift.

These properties of the two categories were used as the improvement requirements but with a allowed margin of 5% impairment on the products less favorable side, considering risk and return separately. Our findings strengthen the choice of improvement requirements and the error margin is not a fixed value, it can be selected by the user. The exact numbers of improvement are mostly affected by the parameters of the underlying asset and the reference portfolio, they are not product specific for other circumstances.

Besides from the more expected results, we found that the performance of the Capital Protected notes is not affected by a changed correlation. This result is interesting and will hopefully be useful

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in future work with portfolio allocations. The numeric results from the test methods and ranking program would be even more interesting to interpret for an actual investment opportunity using parameters from real products and portfolios.

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Chapter 6

Further Development

The potential with these test methods and the ranking code is good but there are still a lot that could be done to develop them. In this study the parameters are held constant in the simulations during the holding time of the products. A way to make the simulations closer to reality is to use stochastic volatility and drift parameters. This requires some development of the code but it should improves the relevance of the simulations. The focus of this study is to develop a method to see the effects of structured products on a large scale. This has resulted in extensive output files which take time to summarize. The results might be even more useful if they are represented graphically in a slimmer format.

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Bibliography

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[2] A. Meucci. "Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management". GARP Risk Professional, pp. 49-51, Apr. 2010.

[3] M. Fredriksson and E. Aspholm. "Portfolio Allocation with Structured Products". Nordea Markets (Internal Report), Stockholm, Dec. 2017.

[4] J.C. Hull. "Mechanics of option markets" in Options, Futures and Other Derivatives. 8th ed.

Upper Saddle River: Person, 2011, pages. 194-213

[5] J. Bhanushali. "Options: What is ATM, ITM, OTM?" Internet:

https://www.indiainfoline.com/article/general-blog/

options-what-is-atm-itm-otm-117110700618_1.html, Nov. 7, 2017. [Jan. 5, 2018]

[6] J.C. Hull. "Trading strategies involving options" in Options, Futures and Other Derivatives.

8th ed. Upper Saddle River: Person, 2011, pages. 234-252

[7] J.C. Hull. "Exotic options" in Options, Futures and Other Derivatives. 8th ed. Upper Saddle River: Person, 2011, pages. 574-596

[8] T. Bjork. "Arbitrage Pricing" in Arbitrage Theory in Continuous Time 2nd ed. Oxford: Ox- ford University Press, 2004, pages. 88-110.

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Appendix A

Result tables from Example Run

An extract of a color coded ranking result from the example run is shown in Fig. A.1. We can see how the ranking result for all statistic measures; Expected Return (Av), standard deviation (StDev), Sharpe Ratio (Sharpe), Value-at-Risk at 1% (VaR-0.01), Expected Shortfall at 1% (ES- 0.01) and downside standard deviation (DS StDev). To the left of the colored fields are the values of the varied parameter, the drift in this case. Next to them are the weights for the structured product (or linear asset) in the portfolio. In this example in Fig. A.1 it is easy to see that the reference portfolio dominates performance when come to the return since every field is marked red.

At the same time, it is the structured product that dominates all risk measures except the Sharpe Ratio.

Figure A.1: Example of a color coded ranking

The following tables are based on such color coded ranking results, as the one in Fig.A.1, from each test method. The numbers in the result tables comes from the first parenthesis with numbers which represents how the in-weighted structured product perform in comparison to the 100% reference portfolio. In the tables the positive and negative values stands for if the in-weighted structured product is better or worse than the reference portfolio, not if the actual values are bigger or smaller.

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Since risk and return values are ranked differently it is easier to only interpret a positive value as a good performance and a negative value as a less good performance. ER stands for Expected Return and ES for Expected Shortfall, as mentioned earlier.

A.1 Find Best Underlying

Table A.1.1: Performance of the Uncapped Capital Protected in the Find Best Underlying test method. Portfolio weigh-in of 10% and drift of underlying on 6 and correlation 0.5.

Reference Portfolio

Volatility 15 Volatility 17.48 Volatility 20

ER ES ER ES ER ES

R10 2.4 6.7 0.1 6.6 −1.5 6.5

R30 2.0 7.7 −0.3 7.6 −1.8 7.6

R50 0.8 8.2 −1.2 8.1 −2.6 8.1

R70 −0.8 8.5 −2.5 8.4 −3.7 8.4

R100 −2.2 8.7 −3.6 8.7 −4.6 8.6

Table A.1.2: Performance of the Capped Capital Protected, compared to 100% reference portfolio, in the Find Best Underlying test method. Portfolio weigh-in of 10% and drift of underlying on 6 and correlation 0.5.

Reference Portfolio

Volatility 15 Volatility 17.48 Volatility 20

ER ES ER ES ER ES

R10 −4.6 8.5 −5.6 8.4 −6.2 8.3

R30 −4.7 8.9 −5.7 8.9 −6.3 8.9

R50 −5.3 9.2 −6.1 9.1 −6.7 9.1

R70 −6.0 9.3 −6.7 9.2 −7.2 9.2

R100 −6.6 9.4 −7.2 9.4 −7.6 9.3

Table A.1.3: Performance of the Min Redemption, compared to 100% reference portfolio, in the Find Best Underlying test method. Portfolio weigh-in of 10% and drift of underlying on 6 and correlation 0.5.

Reference Portfolio

Volatility 15 Volatility 17.48 Volatility 20

ER ES ER ES ER ES

R10 1.0 6.6 −0.9 6.6 −2.2 6.3

R30 0.7 7.7 −1.2 7.6 −2.5 7.5

R50 −0.3 8.2 −2.0 8.1 −3.2 8.0

R70 −1.8 8.5 −3.2 8.4 −4.2 8.3

R100 −3.0 8.7 −4.2 8.6 −5.1 8.6

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Table A.1.4: Performance of the Uncapped Booster, compared to 100% reference portfolio, in the Find Best Underlying test method. Portfolio weigh-in of 10% and drift of underlying on 6 and correlation 0.5

Reference Portfolio

Volatility 15 Volatility 17.48 Volatility 20

ER ES ER ES ER ES

R10 20.7 −3.1 18.3 −5.8 16.7 −8.4

R30 19.7 1.1 17.4 −0.8 15.8 −2.5

R50 16.9 3.0 14.8 1.5 13.3 0.2

R70 12.9 4.1 11.1 2.8 9.9 1.6

R100 9.4 5.1 7.9 4.0 6.9 3.0

Table A.1.5: Performance of the Capped Booster, compared to 100% reference portfolio, in the Find Best Underlying test method. Portfolio weigh-in of 10% and drift of underlying on 6 and correlation 0.5.

Reference Portfolio

Volatility 15 Volatility 17.48 Volatility 20

ER ES ER ES ER ES

R10 15.4 −3.1 13.5 −5.8 12.3 −8.3

R30 14.6 1.1 12.8 −0.7 11.6 −2.5

R50 12.2 3.1 10.6 1.6 9.5 0.2

R70 8.9 4.1 7.5 2.8 6.6 1.7

R100 6.1 5.1 4.9 4.0 4.1 3.0

Table A.1.6: Performance of the Booster with Buffer, compared to 100% reference portfolio, in the Find Best Underlying test method. Portfolio weigh-in of 10% and drift of underlying on 6 and correlation 0.5.

Reference Portfolio

Volatility 15 Volatility 17.48 Volatility 20

ER ES ER ES ER ES

R10 12.2 −1.1 12.6 −4.2 12.6 −7.3

R30 11.5 2.6 11.8 0.5 11.9 −1.7

R50 9.5 4.3 9.7 −2.6 9.8 0.9

R70 6.6 5.2 6.8 3.8 6.8 2.3

R100 4.1 6.0 4.3 4.8 4.3 3.6

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A.2 Market Variations

Table A.2.1: Performance of the Uncapped Capital Protected, compared to 100% reference port- folio, in the Market Variations test method. Portfolio weigh-in of 10% and initial underlying with drift 6, volatility 17.48 and correlation 0.5.

Reference Portfolio

Drift Correlation Volatility

Change ER ES Value ER ES Change ER ES

R10

−150% 6.0 7.8 0.3 0.1 7.6 −50% −4.5 −2.5

−97% 7.6 7.5 0.5 0.1 6.6 −20% −1.9 5.3

−57% 2.8 7.2 0.7 0 6.2 +20% 2.0 7.3

+50% 0.9 5.8 0.9 0 6.1 +50% 5.0 7.8

R30

−150% 6.7 8.2 0.3 −0.3 8.4 −50% −4.7 3.9

−97% 8.3 8.1 0.5 −0.3 7.6 −20% −2.1 7.0

−57% 9.1 7.9 0.7 −0.3 7.3 +20% 1.6 8.0

+50% 0.1 7.5 0.9 −0.4 7.3 +50% 4.5 8.4

R50

−150% 7.1 8.5 0.3 −1.2 8.8 −50% −5.2 5.7

−97% 8.6 8.4 0.5 −1.2 8.1 −20% −2.9 7.7

−57% 9.4 8.3 0.7 −1.2 7.9 +20% 0.5 8.4

+50% −1.0 8.2 0.9 −1.3 7.8 +50% 3.1 8.7

R70

−150% 7.3 8.3 0.3 −2.5 9.0 −50% −5.94 6.6

−97% 8.6 8.5 0.5 −2.5 8.4 −20% −3.9 8.0

−57% 2.9 8.5 0.7 −2.5 8.2 +20% −1.0 8.6

+50% −2.3 8.5 0.9 −2.6 8.2 +50% 1.2 8.8

R100

−150% 7.5 8.8 0.3 −3.6 9.2 −50% −6.5 7.4

−97% 8.7 8.7 0.5 −3.6 8.7 −20% −4.9 8.4

−57% 0.3 8.7 0.7 −3.7 8.5 +20% −2.5 8.8

+50% −3.5 8.8 0.9 −3.7 8.4 +50% −0.5 8.9

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Table A.2.2: Performance of the Capped Capital Protected, compared to 100% reference portfolio, in the Market Variations test method. Portfolio weigh-in of 10% and initial underlying with drift 6, volatility 17.48 and correlation 0.5.

Reference Portfolio

Drift Correlation Volatility

Change ER ES Value ER ES Change ER ES

R10

−150% 8.3 8.9 0.3 −5.6 9.3 −50% −4.6 4.5

−97% 10.1 8.7 0.5 −5.6 8.4 −20% −5.4 7.8

−57% −2.4 8.6 0.7 −5.6 8.0 +20% −5.9 8.7

+50% −6.0 8.3 0.9 −5.6 7.9 +50% −6.3 8.9

R30

−150% 8.6 9.1 0.3 −5.7 9.5 −50% −4.8 7.4

−97% 10.1 9.0 0.5 −5.7 8.9 −20% −5.4 8.6

−57% 1.4 8.9 0.7 −5.7 8.6 +20% −6.0 9.1

+50% −6.3 9.0 0.9 −5.7 8.6 +50% −6.4 9.2

R50

−150% 8.7 9.2 0.3 −6.1 9.6 −50% −5.3 8.2

−97% 10.1 9.1 0.5 −6.1 9.1 −20% −5.8 8.9

−57% 1.6 9.1 0.7 −6.1 8.9 +20% −6.4 9.2

+50% −6.7 9.3 0.9 −6.1 8.9 +50% −6.8 9.3

R70

−150% 8.8 9.3 0.3 −6.7 9.7 −50% −6.0 8.5

−97% 10.1 9.2 0.5 −6.7 9.2 −20% −6.4 9.1

−57% −2.3 9.2 0.7 −6.7 9.1 +20% −6.9 9.3

+50% −7.1 9.4 0.9 −6.7 9.0 +50% −7.3 9.4

R100

−150% 8.9 9.7 0.3 −7.2 9.7 −50% −6.6 8.9

−97% 10.1 9.3 0.5 −7.2 9.4 −20% −7.0 9.3

−57% −3.8 9.3 0.7 −7.2 9.2 +20% −7.4 9.4

+50% −7.6 9.5 0.9 −7.2 9.2 +50% −7.7 9.5

(32)

Table A.2.3: Performance of the Min Redemption, compared to 100% reference portfolio, in the Market Variations test method. Portfolio weigh-in of 10% and initial underlying with drift 6, volatility 17.48 and correlation 0.5.

Reference Portfolio

Drift Correlation Volatility

Change ER ES Value ER ES Change ER ES

R10

−150% 6.1 7.6 0.3 −0.9 8.0 −50% −2.8 0

−97% 7.9 7.3 0.5 −0.9 6.4 −20% −1.8 5.3

−57% 2.5 7.0 0.7 −0.9 5.8 +20% 0.2 7.1

+50% −0.6 6.0 0.9 −1.0 5.7 +50% 1.9 7.7

R30

−150% 6.8 8.1 0.3 −1.2 8.7 −50% −3.0 5.2

−97% 8.6 7.9 0.5 −1.2 7.6 −20% −2.1 7.0

−57% 8.6 7.7 0.7 −1.2 7.1 +20% −0.1 7.9

+50% −1.3 7.7 0.9 −1.3 7.0 +50% 1.5 8.3

R50

−150% 7.1 8.3 0.3 −2.0 9.0 −50% −3.7 6.7

−97% 8.8 8.2 0.5 −2.0 8.1 −20% −2.9 7.7

−57% 8.9 8.1 0.7 −2.1 7.7 20% −1.1 8.3

+50% −2.3 8.3 0.9 −2.1 7.6 50% 0.4 8.6

R70

−150% 7.3 8.3 0.3 −3.2 9.2 −50% −4.6 7.4

−97% 8.8 8.4 0.5 −3.2 8.4 −20% −3.9 8.1

−57% 2.5 8.4 0.7 −3.2 8.0 +20% −2.4 8.5

+50% −3.3 8.6 0.9 −3.3 8.0 +50% −1.1 8.7

R100

−150% 7.6 8.7 0.3 −4.2 9.3 −50% −5.5 8.0

−97% 8.9 8.6 0.5 −4.2 8.6 −20% −4.8 8.4

−57% 0 8.6 0.7 −4.3 8.3 +20% −3.6 8.8

+50% −4.4 8.9 0.9 −4.3 8.3 +50% −2.4 8.9

(33)

Table A.2.4: Performance of the Uncapped Booster, compared to 100% reference portfolio, in the Market Variations test method. Portfolio weigh-in of 10% and initial underlying with drift 6, volatility 17.48 and correlation 0.5.

Reference Portfolio

Drift Correlation Volatility

Change ER ES Value ER ES Change ER ES

R10

−150% −6.6 −4.2 0.3 18.3 −0.4 −50% 13.5 −12.9

−97% −0.8 −4.7 0.5 18.3 −5.8 −20% 16.3 −7.0

−57% 27.7 −5.1 0.7 18.2 −10.3 +20% 20.4 −5.1

+50% 18.8 −6.8 0.9 18.1 −14.2 +50% 23.5 −4.4

R30

−150% −3.6 −1.4 0.3 17.4 3.2 −50% 12.7 −0.6

−97% 2.5 −1.4 0.5 17.4 −0.8 −20% 15.4 −0.8

−57% 46.2 −1.2 0.7 17.3 −4.0 +20% 19.4 −0.7

+50% 16.6 0.2 0.9 17.2 −6.8 +50% 22.5 −0.7

R50

−150% −2.1 0.1 0.3 14.8 4.8 −50% 10.5 2.8

−97% 3.5 0.4 0.5 14.8 1.5 −20% 13.0 1.8

−57% 47.5 0.8 0.7 14.7 −1.1 +20% 16.6 1.3

+50% 13.7 2.8 0.9 14.6 −3.4 +50% 19.4 1.1

R70

−150% −1.4 1.1 0.3 11.1 5.7 −50% 7.5 4.4

−97% 3.5 1.5 0.5 11.1 4.0 −20% 9.6 3.2

−57% 27.9 1.9 0.7 11.0 0.5 +20% 12.6 2.5

+50% 10.5 4.2 0.9 11.0 −1.5 +50% 15.0 2.1

R100

−150% −0.3 2.0 0.3 7.9 6.4 −50% 4.8 5.9

−97% 4.1 2.5 0.5 7.9 4.0 −20% 6.6 4.5

−57% 20.4 3.0 0.7 7.8 2.1 +20% 9.2 3.6

+50% 7.3 5.3 0.9 7.8 0.4 +50% 11.3 3.0

(34)

Table A.2.5: Performance of the Capped Booster, compared to 100% reference portfolio, in the Market Variations test method. Portfolio weigh-in of 10% and initial underlying with drift 6, volatility 17.48 and correlation 0.5.

Reference Portfolio

Drift Correlation Volatility

Change ER ES Value ER ES Change ER ES

R10

−150% −5.3 −4.2 0.3 13.5 −0.3 −50% 22.1 −12.6

−97% 2.9 −4.7 0.5 13.5 −5.8 −20% 16.9 −7.0

−57% 27.7 −5.1 0.7 13.5 −10.3 +20% 10.1 −5.1

+50% 10.9 −6.7 0.9 13.6 −14.2 +50% 5.2 −4.4

R30

−150% −2.6 −1.4 0.3 12.7 3.4 −50% 21.0 −0.4

−97% 5.1 −1.2 0.5 12.8 −0.7 −20% 16.1 −0.7

−57% 46.2 −1.2 0.7 12.8 −4.0 +20% 9.5 −0.7

+50% 9.3 0.4 0.9 12.8 −6.8 +50% 4.7 −0.7

R50

−150% −1.2 0.1 0.3 10.5 4.9 −50% 18.0 3.1

−97% 5.8 0.4 0.5 10.6 1.6 −20% 13.5 1.9

−57% 47.3 0.8 0.7 10.6 −1.1 +20% 7.6 1.4

+50% 7.2 3.0 0.9 10.6 −3.4 +50% 3.3 1.1

R70

−150% −0.6 1.1 0.3 7.5 5.8 −50% 13.9 4.6

−97% 5.8 1.5 0.5 7.5 2.8 −20% 10.1 3.3

−57% 27.9 1.9 0.7 7.5 0.5 +20% 5.0 2.5

+50% 4.8 4.3 0.9 7.6 −1.5 +50% 1.3 2.1

R100

−150% 0.5 2.0 0.3 4.8 6.5 −50% 10.2 6.1

−97% 6.1 2.5 0.5 4.9 4.0 −20% 7.0 4.6

−57% 20.4 3.0 0.7 4.9 2.1 +20% 2.7 3.6

+50% 2.5 5.5 0.9 4.9 0.4 +50% −0.4 3.0

(35)

Table A.2.6: Performance of the Booster with Buffer, compared to 100% reference portfolio, in the Market Variations test method. Portfolio weigh-in of 10% and initial underlying with drift 6, volatility 17.48 and correlation 0.5.

Reference Portfolio

Drift Correlation Volatility

Change ER ES Value ER ES Change ER ES

R10

−150% −4.2 −4.2 0.3 12.5 1.5 −50% 9.0 6.0

−97% 2.5 −4.6 0.5 12.6 −4.2 −20% 11.5 −3.9

−57% 22.6 −4.8 0.7 12.5 −10.0 +20% 13.4 −4.3

+50% 12.4 −4.0 0.9 12.4 −14.2 +50% 14.8 −4.2

R30

−150% −1.6 −1.4 0.3 11.8 4.6 −50% 8.4 8.6

−97% 4.8 −1.3 0.5 11.8 0.5 −20% 10.8 1.5

−57% 38.5 −1.0 0.7 11.8 −3.7 +20% 12.7 −0.1

+50% 10.7 2.1 0.9 11.7 −6.8 +50% 14.0 −0.5

R50

−150% −0.3 0.1 0.3 9.7 5.9 −50% 6.6 9.4

−97% 5.5 0.5 0.5 9.7 2.6 −20% 8.8 3.6

−57% 39.5 1.0 0.7 9.7 −0.8 +20% 10.5 1.9

+50% 8.4 4.3 0.9 9.6 −3.4 +50% 11.7 1.2

R70

−150% 0.2 1.1 0.3 6.8 6.6 −50% 4.1 9.6

−97% 5.5 1.6 0.5 6.8 3.8 −20% 6.0 4.8

−57% 22.8 2.1 0.7 6.8 0.8 +20% 7.5 3.0

+50% 5.9 5.4 0.9 6.7 −1.4 +50% 8.5 2.2

R100

−150% 1.3 2.0 0.3 4.3 7.2 −50% 2.0 9.8

−97% 5.9 2.6 0.5 4.3 4.8 −20% 3.6 5.8

−57% 16.2 3.2 0.7 4.2 2.3 +20% 4.8 4.0

+50% 3.4 6.3 0.9 4.2 0.4 +50% 5.7 3.1

Figure

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