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Liquidity Stress Testing in Asset Management
Part 4. A Step-by-step Practical Guide∗
Thierry Roncalli
Quantitative Research
Amundi Asset Management, Paris
thierry.roncalli@amundi.com
Amina Cherief
Quantitative Research
Amundi Asset Management, Paris
amina.cherief@amundi.com
November 2021
Abstract
This article is part of a comprehensive research project on liquidity risk in asset
management, which can be divided into three dimensions. The first dimension covers
liability liquidity risk (or funding liquidity) modeling, the second dimension focuses on
asset liquidity risk (or market liquidity) modeling, and the third dimension considers
the asset-liability management of the liquidity gap risk (or asset-liability matching).
The purpose of this research is to propose a methodological and practical framework
in order to perform liquidity stress testing programs, which comply with regulatory
guidelines (ESMA,2019a,2020a) and are useful for fund managers. The review of
the academic literature and professional research studies shows that there is a lack of
standardized and analytical models. The aim of this research project is then to fill the
gap with the goal of developing mathematical and statistical approaches, and providing
appropriate answers.
The three dimensions have been developed in the published working papers: (1)
modeling the liability liquidity risk (Roncalli et al.,2021a), (2) modeling the asset liq-
uidity risk (Roncalli et al.,2021b) and (3) managing the asset-liability liquidity risk
(Roncalli,2021c). This fourth working paper provides three examples and the com-
prehensive details to compute the redemption coverage ratio, implement reverse stress
testing and estimate the liquidation cost of the redemption portfolio. The portfolios
have been chosen in order to cover the main asset classes: large-cap stocks, small-
cap stocks, sovereign bonds and corporate bonds. Since we provide the data in the
appendix, these basic examples are easily reproducible and may help quantitative an-
alysts to understand the different steps to implement liquidity stress testing in asset
management.
Keywords: liquidity risk, stress testing, asset-liability management, redemption coverage
ratio, reverse stress testing, transaction cost, reproducible research, knowledge transfer.
JEL classification: C02, G32.
∗We are grateful to Charles Kalisz, Fatma Karray-Meziou, Nermine Moussi, Fran¸cois Pan and Margaux
Regnault for their helpful comments. This research has also benefited from the support of Amundi Asset
Management. However, the opinions expressed in this article are those of the authors and are not meant to
represent the opinions or official positions of Amundi Asset Management.
1
Liquidity Stress Testing in Asset Management
1 Introduction
Since September 2020, the European Securities and Markets Authority (ESMA) has required
asset managers to adopt a liquidity stress testing (LST) policy for their investment funds
(ESMA,2020a). More precisely, each asset manager must assess the liquidity risk factors
across their funds in order to ensure that stress testing is tailored to the liquidity risk
profile of each fund. The guidelines are described in two ESMA publications (ESMA,2019a,
2020a). However, contrary to the banking regulation on liquidity risk, those regulatory texts
do not contain any methodological aspects1. Even though they are complemented by two
other ESMA publications (ESMA,2019b,2020b) and some IMF FSAP analysis (IMF,2017,
2020), the absence of standardized models and parameter values can be a major hurdle for
implementing LST policies, especially for small asset managers2. Certainly, this situation
can be explained by the lack of maturity of this topic in the asset management industry.
In April 2020, we launched an ambitious research project in order to develop quantita-
tive models and provide practical solutions for implementing LST programs. This research
project has been built around three dimensions: liability liquidity risk, asset liquidity risk
and asset-liability risk management. It resulted in three publications, each one considering
a specific dimension: (1) modeling the liability liquidity risk (Roncalli et al.,2021a), (2)
modeling the asset liquidity risk (Roncalli et al.,2021b) and (3) managing the asset-liability
liquidity risk (Roncalli,2021c). The discussions we had with the asset management industry
show that these working papers may be viewed as too elaborate. Therefore, we have decided
to complement them with a fourth working paper, which is a step-by-step practical guide.
This working paper is equivalent to the publication of Bouveret (2017), but our research is
reproducible since all the data are provided and described in the appendix.
In order to be concise and simple, we have only focused on three LST measures or tools.
The first one is the redemption coverage ratio, which can be computed using a time to
liquidation (TTL) approach or a high-quality liquid assets (HQLA) approach. The first
approach requires us to define a redemption portfolio and a liquidation policy, whereas the
second one is based on the concept of cash conversion factor. For the latter approach, we
can use the figures provided by the Basel Committee or postulate a parametric function.
The second tool or the reverse stress testing (RST) defines two measures: the liability
RST scenario and the asset RST scenario. Finally, the third measure is the liquidity cost
associated with the redemption scenario. For that, we need to specify the unit transaction
cost function and combine it with the liquidation policy. Moreover, analyzing the redemption
cost helps to determine the liquidity risk contribution of each asset.
This paper is organized as follows. Section Two deals with equity portfolios. Using a e1
bn investment in large-cap stocks, we show how to compute the redemption coverage ratio
and implement reverse stress testing. Then, we conduct a transaction cost analysis in order
to calculate the liquidation cost of the redemption portfolio. Using a second portfolio, we
show how the transaction cost formulas are impacted by small-cap stocks. We also illustrate
how several statistics change when we consider an asset-liability stress test scenario instead
of a normal scenario. In Section Three, we do the same analysis with a bond portfolio. In
particular, we highlight the differences between stock and bond portfolios. Finally, Section
Four offers some concluding remarks.
1For instance, the redemption coverage ratio (RCR) is the main tool of LST programs. However, it
is referred to only twice. First, ESMA defines it as: “a measurement of the ability of a fund’s assets to
meet funding obligations arising from the liabilities side of the balance sheet, such as a redemption shock”
(ESMA-2020a, page 7). Second, ESMA states that “an outcome of combined asset and liability LST may
be a comparable metric or score, for example based on the RCR” (ESMA-2020a, page 20).
2However, the research of Bouveret (2017) contains the basics of liquidity stress testing for investment
funds and can be used as a beginner’s guide.
2
Liquidity Stress Testing in Asset Management
2 The case of equity portfolios
We consider the equity portfolio described in Table 13 on page 26. This portfolio corresponds
to a e1 bn investment3in the Eurostoxx 50 index at the end of October 2021. For each
stock i, we have the number of shares ωiheld by the portfolio, the price Pi, the bid and ask
quotes Pbid
iand Pask
i, the annualized volatility σiand the current daily volume vi.
2.1 Redemption coverage ratio
2.1.1 Liquidation ratio
Let q+
ibe the maximum number of shares that can be sold during a trading day for the
asset i. We note q= (q1, . . . , qn) the redemption portfolio and qi(h) the number of shares
liquidated after htrading days. Following Roncalli et al. (2021b, Equation (22), page 14),
we have:
qi(h) = min
qi−
h−1
X
k=0
qi(k)!+
, q+
i
(1)
where qi(0) = 0. The liquidation ratio LR(q;h) is then the proportion of the redemption
scenario qthat is liquidated after htrading days (Roncalli et al.,2021b, Equation (23), page
14):
LR(q;h) = Pn
i=1 Ph
k=1 qi(k)·Pi
Pn
i=1 qi·Pi
(2)
where Piis the price of the asset i.
We consider the vertical slicing approach (or pro-rata liquidation). We deduce that the
redemption portfolio is defined as (Roncalli,2021c, page 9):
qi=R·ωi(3)
The liquidation policy is given by (Roncalli et al.,2021b, Equation (9), page 4):
q+
i=x+
i·vi(4)
where x+
iis the trading limit expressed in %. In the sequel, we assume that x+
i= 10%,
which is a standard figure. In Table 1, we report the values of qi,vi,q+
iand qi(h). We
observe that we need three trading days to liquidate the redemption portfolio when the
redemption shock Ris set to 80%. In fact, most exposures are liquidated in two days, but
two stocks require three trading days: Flutter Entertainment (i= 24) and Linde (i= 35).
We verify that the mark-to-market value of the liquidated portfolio is equal to the nominal
redemption shock R=R·TNA:
V(q) =
h+
X
h=1
n
X
i=1
qi(h)·Pi=R(5)
where h+is the liquidation period. In our case, we have R=e799 999 999.60.
3The exact value of the total net assets is equal to:
TNA = V(ω) =
n
X
i=1
ωi·Pi= 999 999 999.50 e
3
Liquidity Stress Testing in Asset Management
Table 1: Liquidation of the redemption portfolio (TNA = e1 bn, R= 80%, vertical slicing)
i ωiqiviq+
iqi(1) qi(2) qi(3) P3
h=1 qi(h)
1 59 106 47 284.8 514 842 51 484.2 47 284.8 0.0 0.0 47 284.8
2 8 883 7 106.4 56 255 5 625.5 5 625.5 1 480.9 0.0 7 106.4
3 150 027 120 021.6 629 509 62 950.9 62 950.9 57 070.7 0.0 120 021.6
4 184 310 147 448.0 1 316 600 131 660.0 131 660.0 15 788.0 0.0 147 448.0
5 130 520 104 416.0 750 684 75 068.4 75 068.4 29 347.6 0.0 104 416.0
6 268 123 214 498.4 1 736 372 173 637.2 173 637.2 40 861.2 0.0 214 498.4
7 131 520 105 216.0 754 901 75 490.1 75 490.1 29 725.9 0.0 105 216.0
8 651 421 521 136.8 4 358 304 435 830.4 435 830.4 85 306.4 0.0 521 136.8
9 3 192 430 2 553 944.0 54 130 721 5 413 072.1 2 553 944.0 0.0 0.0 2 553 944.0
10 5 544 072 4 435 257.6 72 371 040 7 237 104.0 4 435 257.6 0.0 0.0 4 435 257.6
11 242 317 193 853.6 2 473 040 247 304.0 193 853.6 0.0 0.0 193 853.6
12 181 800 145 440.0 2 444 130 244 413.0 145 440.0 0.0 0.0 145 440.0
13 136 334 109 067.2 1 183 053 118 305.3 109 067.2 0.0 0.0 109 067.2
14 365 067 292 053.6 2 390 614 239 061.4 239 061.4 52 992.2 0.0 292 053.6
15 251 719 201 375.2 1 434 050 143 405.0 143 405.0 57 970.2 0.0 201 375.2
16 265 762 212 609.6 2 672 846 267 284.6 212 609.6 0.0 0.0 212 609.6
17 206 041 164 832.8 1 579 517 157 951.7 157 951.7 6 881.1 0.0 164 832.8
18 60 148 48 118.4 339 768 33 976.8 33 976.8 14 141.6 0.0 48 118.4
19 311 879 249 503.2 2 536 147 253 614.7 249 503.2 0.0 0.0 249 503.2
20 1 026 503 821 202.4 9 225 311 922 531.1 821 202.4 0.0 0.0 821 202.4
21 2 459 244 1 967 395.2 30 518 046 3 051 804.6 1 967 395.2 0.0 0.0 1 967 395.2
22 795 234 636 187.2 19 419 467 1 941 946.7 636 187.2 0.0 0.0 636 187.2
23 95 262 76 209.6 491 647 49 164.7 49 164.7 27 044.9 0.0 76 209.6
24 55 520 44 416.0 212 501 21 250.1 21 250.1 21 250.1 1 915.8 44 416.0
25 1 840 196 1 472 156.8 14 316 692 1 431 669.2 1 431 669.2 40 487.6 0.0 1 472 156.8
26 351 837 281 469.6 7 543 014 754 301.4 281 469.6 0.0 0.0 281 469.6
27 413 417 330 733.6 3 643 730 364 373.0 330 733.6 0.0 0.0 330 733.6
28 1 235 905 988 724.0 15 954 487 1 595 448.7 988 724.0 0.0 0.0 988 724.0
29 5 774 696 4 619 756.8 106 942 206 10 694 220.6 4 619 756.8 0.0 0.0 4 619 756.8
30 23 113 18 490.4 204 628 20 462.8 18 490.4 0.0 0.0 18 490.4
31 161 807 129 445.6 786 412 78 641.2 78 641.2 50 804.4 0.0 129 445.6
32 261 645 209 316.0 2 285 287 228 528.7 209 316.0 0.0 0.0 209 316.0
33 290 422 232 337.6 2 489 971 248 997.1 232 337.6 0.0 0.0 232 337.6
34 76 637 61 309.6 372 415 37 241.5 37 241.5 24 068.1 0.0 61 309.6
35 162 956 130 364.8 578 973 57 897.3 57 897.3 57 897.3 14 570.2 130 364.8
36 83 427 66 741.6 364 566 36 456.6 36 456.6 30 285.0 0.0 66 741.6
37 44 351 35 480.8 256 421 25 642.1 25 642.1 9 838.7 0.0 35 480.8
38 64 954 51 963.2 363 103 36 310.3 36 310.3 15 652.9 0.0 51 963.2
39 282 801 226 240.8 2 135 693 213 569.3 213 569.3 12 671.5 0.0 226 240.8
40 120 062 96 049.6 794 444 79 444.4 79 444.4 16 605.2 0.0 96 049.6
41 362 506 290 004.8 1 551 395 155 139.5 155 139.5 34 865.3 0.0 290 004.8
42 345 779 276 623.2 1 859 400 185 940.0 185 940.0 90 683.2 0.0 276 623.2
43 180 140 144 112.0 818 822 81 882.2 81 882.2 62 229.8 0.0 144 112.0
44 237 952 190 361.6 1 136 151 113 615.1 113 615.1 76 746.5 0.0 190 361.6
45 660 350 528 280.0 10 497 975 1 049 797.5 528 280.0 0.0 0.0 528 280.0
46 835 885 668 708.0 6 596 020 659 602.0 659 602.0 9 106.0 0.0 668 708.0
47 247 964 198 371.2 1 943 066 194 306.6 194 306.6 4 064.6 0.0 198 371.2
48 189 196 151 356.8 912 539 91 253.9 91 253.9 60 102.9 0.0 151 356.8
49 57 954 46 363.2 1 071 749 107 174.9 46 363.2 0.0 0.0 46 363.2
50 163 680 130 944.0 976 446 97 644.6 97 644.6 33 299.4 0.0 130 944.0
4
Liquidity Stress Testing in Asset Management
The liquidation contribution of the trading day his the proportion of the redemption
portfolio liquidated on day h:
LC (q;h) = Pn
i=1 qi(h)·Pi
Pn
i=1 qi·Pi
(6)
By construction, we verify that the sum of liquidation contributions is equal to the liquida-
tion ratio:
LR(q;h) =
h
X
k=1 LC (q;k) (7)
In Table 2, we report the liquidation contribution LC (q;h) and the liquidation ratio LR(q;h)
for different values of the redemption rate. When Ris equal to 90%, we liquidate 72.41%
the first day, 26.06% the second day and 1.53% the third day.
Table 2: Liquidation ratio in % (TNA = e1 bn, vertical slicing)
R5% 10% 25% 50% 75% 90%
LC (q; 1) 100.00 100.00 100.00 96.43 81.31 72.41
LC (q; 2) 0.00 0.00 0.00 3.57 18.46 26.06
LC (q; 3) 0.00 0.00 0.00 0.00 0.23 1.53
LR(q; 1) 100.00 100.00 100.00 96.43 81.31 72.41
LR(q; 2) 100.00 100.00 100.00 100.00 99.77 98.47
LR(q; 3) 100.00 100.00 100.00 100.00 100.00 100.00
The liquidity risk profile depends on the size of the redemption portfolio. For instance,
we can consider individual investment funds, whose total net assets are lower than $1 bn.
On the contrary, if we consider the largest asset managers, their aggregate exposure to the
stocks of the Eurostoxx 50 index is greater than $1 bn. In Figure 13 on page 38, we report
the liquidation ratio4when the total net assets are respectively equal to 1, 5, 10 and 20 bn.
If we consider a redemption rate of 10%, we obtain the results given in Figure 1for different
time horizons (one day, two days and one week). For instance, we obtain LR(q; 1) = 37.43%,
LR(q; 2) = 66.91% and LR (q; 1) = 99.47% when the total net assets are equal to e20 bn.
From the liquidation ratio, we can compute the liquidation time (Roncalli et al.,2021b,
page 18):
LT (q;p) = LR−1(q;p)
={inf h:LR(q;h)≥p}(8)
The liquidation period h+is equal to LT (q; 1). It measures the number of days required to
liquidate 100% of the redemption portfolio. In practice, the redemption portfolio can have
some small illiquid exposures on small-cap stocks. Therefore, it is better to define h+as the
99% quantile of the liquidation ratio:
h+=LR−1(q; 99%) (9)
In Figure 2, we report the computation of the liquidation ratio for the previous example.
4For that, we scale the original portfolio (ω1,...,ωn) by a factor of mwhere mis respectively equal to
1, 5, 10 and 20.
5
Liquidity Stress Testing in Asset Management
Figure 1: Liquidation ratio LR(q;h) in % when the redemption rate is equal to 10%
0 10 20 30 40 50 60 70
0
20
40
60
80
100
Figure 2: Liquidation time h+=LR−1(q; 99%) in number of trading days
5 10 25 50 75 90
0
10
20
30
40
6
Liquidity Stress Testing in Asset Management
2.1.2 Time to liquidation approach
We now turn to the computation of the redemption coverage ratio. Following Roncalli
(2021c, Equation (12), page 6), we have:
RCR (h) = LR(q;h)·V(q)
R·V(ω)(10)
where V(q) = Pn
i=1 qi·Piand V(ω) = Pn
i=1 ωi·Pi= TNA. The liquidity shortfall is the
amount of additional assets to be sold to satisfy the redemption. Its relative value (with
respect to the total net assets) is equal to (Roncalli,2021c, Equation (9), page 6):
LS (h) = R·max (0,1−RCR (h)) (11)
As noticed by Roncalli (2021c), the redemption coverage ratio is exactly equal to the liqui-
dation ratio when the redemption shock R=R·TNA is equal to the mark-to-market V(q)
of the redemption portfolio. The reason is that the redemption portfolio is defined using the
vertical slicing and its value is exactly equal to the redemption portfolio (Roncalli,2021c,
page 9). Therefore, in the case of the naive vertical slicing approach, we always have:
q=R·ω=⇒RCR (h)≤1 (12)
It is obvious that the RCR cannot be computed with the naive vertical slicing approach.
It is better to consider the waterfall approach: q=ω. In this case, we use the following
formula (Roncalli,2021c, Equations (7) and (8), page 6):
RCR (h) =
h
X
k=1 Pn
i=1 ωi(k)·Pi
R·TNA =1
R
h
X
k=1 Pn
i=1 ωi(k)·Pi
Pn
i=1 ωi·Pi
(13)
In Table 3, we report the redemption coverage ratio for different redemption rate values
when the total net assets are equal to e1 bn. When the liquidation approach corresponds
to the naive vertical slicing, we obtain the same figures as the liquidation ratio (see Table
2on page 5). When we use the waterfall approach, the RCR is higher. Figure 3shows the
evolution of the RCR with respect to the TNA. In the case where the redemption rate is
equal to 10%, the RCR is below one when the TNA is larger than 7.6 bn for the one-day time
horizon, 15.1 bn for the two-day time horizon and 37.7 bn for the one-week time horizon.
Table 3: Redemption coverage ratio (TNA = e1 bn)
Redemption rate R5% 10% 25% 50% 75% 90%
RCR (1) 1.00 1.00 1.00 0.96 0.81 0.72
Vertical slicing RCR (2) 1.00 1.00 1.00 1.00 1.00 0.98
RCR (3) 1.00 1.00 1.00 1.00 1.00 1.00
RCR (1) 13.38 6.69 2.68 1.34 0.89 0.74
Waterfall liquidation RCR (2) 19.29 9.64 3.86 1.93 1.29 1.07
RCR (3) 20.00 10.00 4.00 2.00 1.33 1.11
2.1.3 Impact of the stress test scenario
To compute the redemption coverage ratio in a stress period, we can shock the redemption
rate Rand the daily trading volume vi. For the first parameter, we use a larger value
7
Liquidity Stress Testing in Asset Management
Figure 3: Redemption coverage ratio when the redemption rate is equal to 10% (waterfall
liquidation)
0 10 20 30 40 50 60 70
0
0.5
1
1.5
2
2.5
3
Rstress for the redemption shock. For the second parameter, we recall that the liquidation
policy is defined as q+
i=x+
i·vi. Following Roncalli et al. (2021b, page 56), we introduce a
multiplicative parameter mv≤1 so that the liquidation policy in a stress period becomes:
q+
i=mv·x+
i·vi(14)
We consider the previous equity portfolio. We assume that the redemption rate Ris
equal to 5% in a normal period5. Results are given in Table 4. For the stress testing
exercise, we consider a higher redemption rate value (Rstress = 20%) and different asset
liquidity scenarios. Results are given in Table 5. For instance, if we assume that the asset
liquidity is reduced by a factor of 2 (mv= 50%), the value of RCRstress (1) is equal to 0.38
for a one-day time horizon and a TNA of e5 bn, while it was equal to 3.02 in a normal
period.
Table 4: Redemption coverage ratio in a normal period (R= 5%, waterfall liquidation)
TNA e1 bn e5 bn e10 bn e20 bn
h= 1 13.38 3.02 1.51 0.75
h= 2 19.29 6.04 3.02 1.51
h= 5 20.00 13.38 7.49 3.77
Remark 1 We can break down the impact of the stress test scenario by distinguishing the
effect of the redemption shock and the impact of the asset liquidity.
5This figure is far overestimated when it concerns normal market periods.
8
Liquidity Stress Testing in Asset Management
Table 5: Stress testing of the redemption coverage ratio (Rstress = 20%, waterfall liquidation)
mv= 1.00 mv= 0.75
TNA e1 bn e5 bn e10 bn e20 bn e1 bn e5 bn e10 bn e20 bn
h= 1 3.35 0.75 0.38 0.19 2.67 0.57 0.28 0.14
h= 2 4.82 1.51 0.75 0.38 4.33 1.13 0.57 0.28
h= 5 5.00 3.35 1.87 0.94 5.00 2.67 1.41 0.71
mv= 0.50 mv= 0.10
TNA e1 bn e5 bn e10 bn e20 bn e1 bn e5 bn e10 bn e20 bn
h= 1 1.87 0.38 0.19 0.09 0.38 0.08 0.04 0.02
h= 2 3.35 0.75 0.38 0.19 0.75 0.15 0.08 0.04
h= 5 4.97 1.87 0.94 0.47 1.87 0.38 0.19 0.09
2.1.4 The case of small-cap portfolios
The previous example may be misleading because we obtain high redemption coverage ratio
figures as we are considering large-cap liquid stocks. In fact, liquidity stress testing makes
more sense when the portfolio contains small- and mid-cap stocks. In Table 14 on page 27,
we consider a second portfolio, which is equally weighted on 20 stocks. Since some stocks
present a low free-float market capitalization, liquidating this portfolio is more challenging.
For instance, 144 trading days are required to liquidate 99% of a e1 bn exposure on this
portfolio6. Therefore, it is unsurprising that we obtained the redemption coverage ratio
values given in Tables 6and 7. Using the waterfall approach, the redemption coverage ratio
RCRstress (1) is below one for a e1 bn exposure even though the multiplicative factor mvis
equal to 1.
Table 6: Redemption coverage ratio in a normal period (small-cap portfolio, R= 5%,
waterfall liquidation)
TNA e1 bn e2 bn e3 bn e4 bn
h= 1 1.28 0.64 0.43 0.32
h= 2 2.56 1.28 0.85 0.64
h= 5 5.89 3.20 2.13 1.60
Table 7: Stress testing of the redemption coverage ratio (small-cap portfolio, Rstress = 20%,
waterfall liquidation)
mv= 1.00 mv= 0.75
TNA e1 bn e2 bn e3 bn e4 bn e1 bn e2 bn e3 bn e4 bn
h= 1 0.32 0.06 0.03 0.02 0.24 0.05 0.02 0.01
h= 2 0.64 0.13 0.06 0.03 0.48 0.10 0.05 0.02
h= 5 1.47 0.32 0.16 0.08 1.17 0.24 0.12 0.06
mv= 0.50 mv= 0.10
TNA e1 bn e2 bn e3 bn e4 bn e1 bn e2 bn e3 bn e4 bn
h= 1 0.16 0.03 0.02 0.01 0.03 0.01 0.00 0.00
h= 2 0.32 0.06 0.03 0.02 0.06 0.01 0.01 0.00
h= 5 0.80 0.16 0.08 0.04 0.16 0.03 0.02 0.01
6Figures 14 and 15 on page 38 show other liquidation statistics.
9
Liquidity Stress Testing in Asset Management
2.1.5 HQLA approach
The high-quality liquid assets (HQLA) method is explained in detail in Roncalli (2021c,
Section 2.1.2, pages 13-18). Let CCFkbe the cash conversion factor (CCF) of the kth
HQLA class. The redemption coverage ratio is defined as:
RCR (h) = Pm
k=1 wk·CCFk(h)
R(15)
where wkis the weight of the kth HQLA class. In the case of the Basel III framework,
CCFk(h) is fixed and does not depend on the time horizon h. In the case of the risk
sensitive framework, Roncalli (2021c) proposed the following formula:
CCFk,j (h) = LFk(h)·1−DFkh
2·(1 −SFk(TNAj,Hj)) (16)
where LFk(h)∈[0,1] is the liquidity factor, DFk(τh)∈[0,MDDk] is the drawdown factor
and SFk∈[0,1] is the specific risk factor associated to the fund j. Following Roncalli
(2021c), we specify these functions as follows:
LFk(h) = min (1.0, λk·h)
DFk(h) = min MDDk, ηk·√h
SFk(TNAj,Hj) = min
ξsize
kTNAj
TNA?−1+
+ξconcentration
k rHj
H?−1!+
,SF+
(17)
where λkis the selling intensity, MDDkis the maximum drawdown, ηkis the loss intensity,
TNAjis the total net assets and Hjis the Herfindahl index of the fund.
In the Basel framework, the value of CCFk(h) is set to 50%. Concerning the risk
sensitive framework, we assume that λk= 2%, ηk= 5%, MDDk= 50%, ξsize
k= 10%,
ξconcentration
k= 25%, TNA?= 1 bn, H?= 2% and SF+= 0.80. Results are given7in Figures
4and 5when the stressed redemption rate Rstress is equal to 20%. We also compare the
HQLA approach with the time-to-liquidation (TTL) approach8.
2.2 Reverse stress testing
According to Roncalli (2021c), reverse stress testing consists in finding the liquidity scenario
such that RCR(h) = RCR−where RCR−is the minimum acceptable level of the redemption
coverage ratio9.
2.2.1 Liability RST scenario
From a liability perspective, reverse stress testing consists in finding the redemption shock
above which the redemption coverage ratio is lower than the minimum acceptable level:
RCR (h)≤RCR−⇔R≥RRST (18)
RRST is computed by solving the non-linear equation:
RRST =R∈[0,1] : RCR (h) = RCR−(19)
7For the small-cap portfolio, we assume that the selling intensity λkis reduced by 25%, the loss intensity
ηkis increased by 20% and the threshold TNA?is divided by a factor of two.
8The stress multiplier mvis set to 50%.
9A standard value of RCR−is 50%.
10
Liquidity Stress Testing in Asset Management
Figure 4: Redemption coverage ratio of the large-cap portfolio (TNA = e1 bn, Rstress = 20%
and mv= 50%)
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
0
1
2
3
4
5
Figure 5: Redemption coverage ratio of the small-cap portfolio (TNA = e1 bn, Rstress = 20%
and mv= 50%)
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50
0
1
2
3
4
11
Liquidity Stress Testing in Asset Management
In the case where the liquidation portfolio qdoes not depend on the redemption rate, we
obtain an analytical expression:
Ph
k=1 Pn
i=1 qi(k)·Pi
RRST ·TNA = RCR−⇔RRST =Ph
k=1 Pn
i=1 qi(k)·Pi
RCR−·TNA (20)
For instance, this solution is valid for the waterfall approach since we have q=ω. However,
the analytical solution cannot be used for the pro-rata liquidation: q=R·ω. In this case,
we must solve the non-linear equation (19).
Table 8gives the liability RST scenario RRST when the minimum acceptable redemption
coverage ratio is set to 50% and we consider the pro-rata liquidation policy. We notice that
RRST may be greater than one, implying that the size of the fund is too small to experience
a liquidity stress test scenario. In this case, we report the total net assets TNARST =
RRST ·TNA such that the non-linear equation RCR (h) = RCR−is satisfied for a 100%
redemption rate. If we consider the large-cap portfolio with mv= 0.50, the reverse stress
testing scenario for the one-day time horizon is RRST = 72.1%. If mv= 1.00, the value of
Ris always greater than one. Therefore, the reverse stress testing scenario for the one-day
time horizon is given by the variable TNARST instead of the metric RRST. In our case,
we obtain TNARST =e1.441 bn. Therefore, by assuming a 100% redemption shock, the
size of the fund must be greater than e1.441 bn to breach the stress test scenario. On the
contrary, if we consider the small-cap portfolio, reverse stress testing implies low values of
the redemption rate RRST. For example, if we observe a 50% reduction in the asset liquidity,
a redemption scenario of 4.9% is sufficient to observe a redemption coverage ratio lower than
50% when the time horizon his set to one trading day.
Table 8: Liability reverse stress testing scenario RRST in % (TNA = e1 bn, pro-rata
liquidation, RCR−= 50%)
Large-cap portfolio Small-cap portfolio
mv1.00 0.75 0.50 0.10 1.00 0.75 0.50 0.10
h= 1 144.1 108.1 72.1 14.5 9.7 7.3 4.9 1.0
h= 2 288.2 216.2 144.1 28.9 19.3 14.5 9.7 2.0
h= 3 432.3 324.2 216.2 43.3 28.9 21.7 14.5 2.9
h= 4 576.3 432.3 288.2 57.7 38.5 28.9 19.3 3.9
h= 5 720.4 540.3 360.2 72.1 48.1 36.1 24.1 4.9
2.2.2 Asset RST scenario
From an asset perspective, reverse stress testing consists in finding the asset liquidity shock
mvbelow which the redemption coverage ratio is lower than the minimum acceptable level:
RCR (h)≤RCR−⇔mv≤mRST
v(21)
Since the liquidation policy q+
i=mv·x+
i·videpends on the value mv, there is no analytical
solution. The numerical solution corresponds then to the root of the non-linear equation
mv∈[0,1] : RCR (h) = RCR−.
For the large-cap portfolio, the values taken by mRST
vare very small. This indicates that
the asset liquidity must be dramatically reduced to observe a stress test scenario, which is
not realistic. For instance, in the case where the redemption shock is equal to 10%, the
current liquidity must be reduced by a factor greater than 10 (mRST
vis equal to 0.07 for
h= 1). In the case of the small-cap portfolio, the current liquidity is not enough to liquidate
10% of the total net assets in one day since we have mRST
v= 1.04.
12
Liquidity Stress Testing in Asset Management
Table 9: Asset reverse stress testing scenario mRST
v(TNA = e1 bn, pro-rata liquidation,
RCR−= 50%)
Large-cap portfolio Small-cap portfolio
R5% 10% 20% 50% 5% 10% 20% 50%
h= 1 0.04 0.07 0.14 0.35 0.52 1.04 2.08 5.20
h= 2 0.02 0.04 0.07 0.17 0.26 0.52 1.04 2.60
h= 3 0.01 0.02 0.05 0.12 0.17 0.35 0.69 1.73
h= 4 0.01 0.02 0.04 0.09 0.13 0.26 0.52 1.30
h= 5 0.01 0.01 0.03 0.07 0.11 0.21 0.42 1.04
2.3 Transaction cost analysis
2.3.1 Analytics of the transaction cost function
The cost function corresponds to the square-root-linear model described in Roncalli et al.
(2021b, Section 2.4.2, page 11). Let xi(h) be the participation rate at the trading day h. It
is equal to:
xi(h) = qi(h)
vi
(22)
where viis the daily volume. The unit transaction cost function is equal to:
c
c
ci(xi(h)) =
1.25 si+ 0.40 σipxi(h) if xi(h)≤˜xi
1.25 si+0.40
√˜xi
σixiif ˜xi≤xi(h)≤x+
i
+∞if xi(h)> x+
i
(23)
where siis the bid-ask spread, σiis the daily volatility, x+
i= 10% and ˜xi=2
3x+
i. We can
break down this unit transaction cost as follows:
c
c
ci(xi(h)) = c
c
cs
i+c
c
cπ
π
π
i(xi(h)) (24)
where c
c
cs
iis the spread component and c
c
cπ
π
π
i(xi(h)) is the price impact component. Let Qi=
qi·Piand Qi(h) = qi(h)·Pibe the nominal value of qiand qi(h). The transaction cost of
the trade associated to qi(h) is therefore equal to:
T Ci(qi(h)) = Qi(h)·c
c
ci(xi(h)) (25)
Again, we have the following decomposition:
T Ci(qi(h)) = T Cs
i(qi(h)) + T Cπ
π
π
i(qi(h))
T Cs
i(qi(h)) = Qi(h)·c
c
cs
i
T Cπ
π
π
i(qi(h)) = T Ci(qi(h)) − T Cs
i
(26)
We can now compute the total and unit costs of the redemption portfolio:
T C (q) =
h+
X
h=1
n
X
i=1 T Ci(qi(h)) (27)
and:
c
c
c(q) = T C (q)
Pn
i=1 qi·Pi
(28)
13
Liquidity Stress Testing in Asset Management
If we are interested in the contribution of the stock ior the trading day h, we have10:
T C (q;h) =
n
X
i=1 T Ci(qi(h)) (31)
and11:
T Ci(qi) =
h+
X
h=1 T Ci(qi(h)) (34)
We consider the redemption portfolio described in Table 1on page 4. Using the daily
volume, the annualized volatility and the bid-ask quotes given in Table 13 on page 26, we
compute the daily participation rate12 xi(h), the daily volatility13 σiand the (half) bid-ask
spread14 si. Then, we define the spread and price impact component c
c
cs
iand c
c
cπ
π
π
i(xi(h)) and
we deduce the unit transaction cost c
c
ci(xi(h)). Results are given in Table 16 on page 30.
For example, if we consider the first stock (i= 1), we have x1(1) = 9.18%, σ1= 1.59%,
s1= 0.89 bps, c
c
cs
1= 1.11 bps, c
c
cπ
π
π
1(x1(1)) = 22.67 bps and c
c
c1(x1(1)) = 23.78 bps. When
his equal to 2 or 3, we have c
c
c1(x1(h)) = 0 bps because x1(h) = 0. Then, we compute
Qi(h) = qi(h)·Piand deduce T Ci(qi(h)) and T Ci(qi) using Equations (25) and (34).
Results are reported in Table 17 on page 31. The transaction cost to liquidate q1is equal
to e31 936. Finally, liquidating the redemption portfolio implies a total cost T C (q) of
e1 738 156. This represents a unit transaction cost c
c
c(q) of 21.73 bps. Table 18 on page 32
also provides the break-down between the spread component T Cs
i(qi) and the price impact
component T Cπ
π
π
i(qi).
Remark 2 From a trading viewpoint, computing the unit transaction cost c
c
c(q)makes a lot
of sense because we compare the total transaction cost to the value of the redemption portfolio.
From a liquidity stress testing viewpoint, it is better to normalize the total transaction cost
by the total net assets:
˜
c
c
c(q) = T C (q)
TNA (37)
10We have the following decomposition:
T Cs(q;h) =
n
X
i=1
Qi(h)·c
c
cs
i(29)
and:
T Cπ
π
π(q;h) = T C (q;h)− T Cs(q;h) (30)
11We have the following decomposition:
T Cs
i(qi) = Qi·c
c
cs
i(32)
and:
T Cπ
π
π
i(qi) = T Ci(qi)− T Cs
i(qi) (33)
12We use Equation (22).
13We have:
σi=σyearly
i
√260 (35)
14We have:
si=Pask
i−Pbid
i
Pask
i+Pbid
i
(36)
14
Liquidity Stress Testing in Asset Management
Indeed, the computation of c
c
c(q)measures the real impact of the redemption portfolio by
combining the redemption shock and the liquidation cost:
˜
c
c
c(q) = T C (q)
Pn
i=1 ωi·Pi
=R·c
c
c(q) (38)
Moreover, without the implementation of swing pricing, this is the cost borne by the final
investor. In the example above, we obtain ˜
c
c
c(q) = 17.38 bps and c
c
c(q) = 21.73 bps.
In Figure 6, we report the transaction cost functions T C (q), c
c
c(q) and ˜
c
c
c(q). For each
cost function, we indicate the total value and also the breakdown between the spread and
the price impact. In the last panel, we show the proportion of c
c
c(q) due to the price impact.
In Figure 7, we illustrate the breakdown per asset when the redemption rate is equal to 5%.
Figure 6: Transaction cost of the large-cap portfolio (TNA = e1 bn)
0 20 40 60 80 100
0
0.5
1
1.5
2
0 20 40 60 80 100
0
5
10
15
20
0 20 40 60 80 100
0
5
10
15
20
0 20 40 60 80 100
60
70
80
90
100
2.3.2 The case of small-cap stocks
For small-cap stocks, the only difference concerns the sensitivity coefficients of the unit
transaction cost function. Following Roncalli et al. (2021b, Equation (44), page 40), we
have:
c
c
ci(xi(h)) =
1.40 si+ 0.50 σipxi(h) if xi(h)≤˜xi
1.40 si+0.50
√˜xi
σixiif ˜xi≤xi(h)≤x+
i
+∞if xi(h)> x+
i
(39)
We consider the small-cap portfolio. Figure 8shows the breakdown per trading day when
we assume that the redemption rate is equal to 5%. Finally, we obtain T C (q) = e147 560.
15
Liquidity Stress Testing in Asset Management
Figure 7: Breakdown per asset in e(large-cap portfolio, TNA = e1 bn, R= 5%)
0 5 10 15 20 25 30 35 40 45 50
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Figure 8: Breakdown per trading day in e(small-cap portfolio, TNA = e1 bn, R= 5%)
12345678910
0
1
2
3
4
5
6
7
8
9
10 104
16
Liquidity Stress Testing in Asset Management
2.3.3 Stress testing
The transaction cost function remains the same for the stress period, but the three param-
eters si,σiand vichange because we apply shocks to them. Since the stress test scenario is
defined by the triplet (∆s,∆σ, mv), it follows that Equation (23) becomes:
c
c
ci(xi(h)) =
1.25 (si+ ∆s)+0.40 σi+∆σ
√260√xiif xi(h)≤˜xi
1.25 (si+ ∆s) + 0.40
√˜xiσi+∆σ
√260xiif ˜xi≤xi(h)≤x+
i
+∞if xi(h)> x+
i
(40)
where:
xi(h) = qi(h)
mv·vi
(41)
Therefore, a stress test scenario has three impacts: (1) for a given value of qi(h), the
participation rate xi(h) is larger; (2) the transaction cost c
c
ci(xi(h)) is higher because the
participation rate, the bid-ask spread and the volatility are higher; (3) moreover, it takes
more time to liquidate the redemption portfolio since we have q+
i=mv·x+
i·vi, meaning
that the daily trading limits expressed in number of shares are reduced.
Figure 9: Breakdown per trading day in e(large-cap portfolio, TNA = e1 bn, R= 80%)
12345
0
0.5
1
1.5
2106
We consider the example described in Section 2.3.1 on page 13. Previously, we found that
the total transaction cost T C (q) was equal to 1 738 156 euros. representing a redemption
transaction cost c
c
c(q) of 21.73 bps and a portfolio transaction cost ˜
c
c
c(q) of 17.38 bps. Let us
assume the following stress test scenario: ∆s= 8 bps, ∆σ= 20% and mv= 0.50. We now
obtain15 T Cstress (q) = 4 124 811 euros, c
c
c(q) = 51.56 bps and ˜
c
c
c(q) = 41.25 bps. In Figure 9,
we compare the breakdown per trading day for the normal and stress cases.
15See Tables 19,20 and 21 on page 33 for computational details.
17
Liquidity Stress Testing in Asset Management
3 The case of bond portfolios
The case of bond portfolios is similar to the case of equity portfolios. This is why we only
focus on the most important differences when doing a liquidity stress testing exercise with
bond portfolios.
3.1 Computing the liquidation portfolio
We define the redemption portfolio by the vector of liquidated bonds q= (q1, . . . , qn).
Nevertheless, in the fixed-income universe, we may prefer to consider nominal values Qi=
qi·Piinstead of real values qi. Therefore, the redemption portfolio is also defined by
Q= (Q1, . . . , Qn). Let Qi(h) be the exposure in the ith bond liquidated after htrading
days. We have:
Qi(h) = min
Qi−
h−1
X
k=0
Qi(k)!+
, Q+
i
(42)
where Qi(0) = 0 and Q+
iis the maximum trading limit that can be sold during a trading
day for the bond i. Equation (42) is equivalent to Equation (1) on page 3because we have:
qi(h) = Qi(h)
Pi
(43)
The formulas for LR(q;h), RCR (h) and LS (h) remain the same as previously.
We consider the portfolio described on page 28. It is made up of 11 sovereign bonds and
36 corporate bonds. In Table 15, we report the values of Q+
i. We assume that Q+
iis equal to
$50 mn for US sovereign bonds, $6 mn for senior corporate bonds and $3 mn for non-senior
corporate bonds (e.g. subordinated debt). If we consider a redemption rate equal to 30%,
we obtain the liquidation values Qi(h) given in Table 10. We have LR(q; 1) = 0.9566,
LR(q; 2) = 0.9958 and LR(q; 3) = 1. The redemption portfolio can almost be liquidated
during the first day, since only the exposures on the four subordinated bonds require more
than one day to be sold.
3.2 Computing the redemption coverage ratio
We scale up the portfolio in order to obtain TNA = $10 bn or TNA = $20 bn. Results
are given in Table 11. When the total net assets are equal to $10 bn and the redemption
rate is set to 30%, the redemption coverage ratio RCR(1) is equal to 0.251 if we consider
a pro-rata liquidation or a waterfall liquidation. Therefore, we notice that the liquidation
policy may have no impact on the RCR. The reason lies in the fact that Qi(h) is equal to
Q+
iwhen the time horizon his low. This means that the trading limits are reached for all
the bonds. In this case, we cannot use a bond ito liquidate the exposure on the bond j.
Stress testing is implemented by considering a shock on the trading limits:
Qi(h) = min
Qi−
h−1
X
k=0
Qi(k)!+
, mQ+·Q+
i
(44)
where mQ+≤1 is the stress parameter of the trading limit Q+
i. For instance, if mvis equal
to 50%, we obtain the right panel in Table 11. We notice the high impact of the stress
parameter mQ+on the redemption coverage ratio16.
16See Figures 17 and 18 on page 40 for a more visual illustration.
18
Liquidity Stress Testing in Asset Management
Table 10: Liquidation of the redemption portfolio (bond portfolio, TNA = $1 bn, R= 30%,
vertical slicing)
i QiQ+
iQi(1) Qi(2) Qi(3) P3
h=1 Qi(h)
1 16 255 353 50 000 000 16 255 353 0 0 16 255 353
2 12 718 447 50 000 000 12 718 447 0 0 12 718 447
3 12 017 318 50 000 000 12 017 318 0 0 12 017 318
4 13 233 346 50 000 000 13 233 346 0 0 13 233 346
5 12 063 483 50 000 000 12 063 483 0 0 12 063 483
6 11 544 025 50 000 000 11 544 025 0 0 11 544 025
7 21 235 655 50 000 000 21 235 655 0 0 21 235 655
8 20 407 521 50 000 000 20 407 521 0 0 20 407 521
9 20 072 491 50 000 000 20 072 491 0 0 20 072 491
10 13 683 284 50 000 000 13 683 284 0 0 13 683 284
11 26 768 829 50 000 000 26 768 829 0 0 26 768 829
12 3 802 896 6 000 000 3 802 896 0 0 3 802 896
13 3 282 954 6 000 000 3 282 954 0 0 3 282 954
14 1 476 301 6 000 000 1 476 301 0 0 1 476 301
15 2 064 731 6 000 000 2 064 731 0 0 2 064 731
16 2 954 414 6 000 000 2 954 414 0 0 2 954 414
17 2 359 638 6 000 000 2 359 638 0 0 2 359 638
18 1 887 457 6 000 000 1 887 457 0 0 1 887 457
19 3 538 056 6 000 000 3 538 056 0 0 3 538 056
20 6 906 942 3 000 000 3 000 000 3 000 000 906 942 6 906 942
21 6 153 350 3 000 000 3 000 000 3 000 000 153 350 6 153 350
22 2 639 184 6 000 000 2 639 184 0 0 2 639 184
23 4 773 841 6 000 000 4 773 841 0 0 4 773 841
24 3 007 583 6 000 000 3 007 583 0 0 3 007 583
25 5 735 256 3 000 000 3 000 000 2 735 256 0 5 735 256
26 6 210 205 3 000 000 3 000 000 3 000 000 210 205 6 210 205
27 3 561 300 6 000 000 3 561 300 0 0 3 561 300
28 2 978 950 6 000 000 2 978 950 0 0 2 978 950
29 4 166 994 6 000 000 4 166 994 0 0 4 166 994
30 2 979 133 6 000 000 2 979 133 0 0 2 979 133
31 2 920 499 6 000 000 2 920 499 0 0 2 920 499
32 2 358 223 6 000 000 2 358 223 0 0 2 358 223
33 2 652 842 6 000 000 2 652 842 0 0 2 652 842
34 4 958 507 6 000 000 4 958 507 0 0 4 958 507
35 3 535 819 6 000 000 3 535 819 0 0 3 535 819
36 2 650 657 6 000 000 2 650 657 0 0 2 650 657
37 2 662 310 6 000 000 2 662 310 0 0 2 662 310
38 3 140 845 6 000 000 3 140 845 0 0 3 140 845
39 4 155 263 6 000 000 4 155 263 0 0 4 155 263
40 2 803 320 6 000 000 2 803 320 0 0 2 803 320
41 2 293 116 6 000 000 2 293 116 0 0 2 293 116
42 1 946 863 6 000 000 1 946 863 0 0 1 946 863
43 2 656 800 6 000 000 2 656 800 0 0 2 656 800
44 2 295 612 6 000 000 2 295 612 0 0 2 295 612
45 3 267 302 6 000 000 3 267 302 0 0 3 267 302
46 2 270 545 6 000 000 2 270 545 0 0 2 270 545
47 2 952 538 6 000 000 2 952 538 0 0 2 952 538
19
Liquidity Stress Testing in Asset Management
Table 11: Redemption coverage ratio RCR(h) (bond portfolio, R= 30%)
Normal period Stress period
Pro-rata Waterfall Pro-rata Waterfall
TNA 10 20 10 20 10 20 10 20
h
1 0.251 0.126 0.251 0.126 0.126 0.063 0.126 0.063
2 0.503 0.251 0.503 0.251 0.251 0.126 0.251 0.126
3 0.704 0.377 0.754 0.377 0.377 0.188 0.377 0.188
4 0.835 0.503 1.005 0.503 0.503 0.251 0.503 0.251
5 0.900 0.622 1.257 0.628 0.622 0.314 0.628 0.314
6 0.928 0.704 1.508 0.754 0.704 0.377 0.754 0.377
7 0.940 0.773 1.759 0.880 0.773 0.440 0.880 0.440
8 0.948 0.835 2.006 1.005 0.835 0.503 1.005 0.503
9 0.953 0.873 2.195 1.131 0.873 0.565 1.131 0.565
10 0.957 0.900 2.346 1.257 0.900 0.622 1.257 0.628
Figure 10: Multiplier stress factor mRCR (h) (mQ+= 50%)
0 5 10 15 20
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20
0.5
0.6
0.7
0.8
0 5 10 15 20
0.5
0.6
0.7
0.8
20
Liquidity Stress Testing in Asset Management
Remark 3 The stress parameter mQ+is exactly equal to the previous stress parameter mv.
Indeed, we have q+
i=x+
i·viand Q+
i=q+
i·Pi. The stressed values are defined as q+,stress
i=
mv·q+
iand Q+,stress
i=mQ+·Q+
i. We deduce that Q+,stress
i=q+
i·Pi=mv·q+
i·Pi=mv·Q+
i
and mQ+=mv.
We note RCRnormal (h) the redemption coverage ratio when mQ+(or mv) is equal to
one. The stressed value RCRstress (h) is computed with mQ+<1. We define the multiplier
stress factor of the redemption coverage ratio as:
mRCR (h) = RCRstress (h)
RCRnormal (h)(45)
By construction, we have RCRstress (∞) = RCRnormal (∞), implying that:
mQ+≤mRCR (h)≤1 (46)
In Figure 10, we report mRCR (h) when we consider the pro-rata liquidation and the waterfall
liquidation. We notice that the stress test scenario has a bigger impact on the pro-rata
liquidation than on the waterfall liquidation. In particular, in this example, mRCR (h) does
not depend on the redemption shock when we use the waterfall liquidation.
3.3 Computing the transaction cost
In the case of bonds, the participation rate is defined with respect to the outstanding amount
Miand not the daily volume vi:
yi(h) = Qi(h)
Mi
(47)
Following Roncalli et al. (2021b), we use the following unit transaction cost for the sovereign
bonds:
c
c
ci(yi(h)) = 1.25 si+ 3.00 σiyi(h)0.25 if yi(h)≤y+
i(48)
where:
y+
i=Q+
i
Mi
(49)
In the case of corporate bonds, the formula becomes:
c
c
ci(yi(h)) = 1.50 si+ 0.125 DTSiyi(h)0.25 if yi(h)≤y+
i(50)
where DTSiis the duration-times-spread measure of the bond i. Again, we can introduce a
second transaction cost regime, where the price impact is linear. In this case, we have:
c
c
ci(yi(h)) =
1.25 si+ 3.00 σiyi(h)0.25 if yi(h)≤˜yi
1.25 si+3.00
˜y0.75
i
σiyi(h) if ˜yi≤yi(h)≤y+
i
(51)
and:
c
c
ci(yi(h)) =
1.50 si+ 0.125 DTSiyi(h)0.25 if yi(h)≤˜yi
1.50 si+0.125
˜y0.75
i
DTSiyi(h) if ˜yi≤yi(h)≤y+
i
(52)
where ˜yi=2
3y+
i.
21
Liquidity Stress Testing in Asset Management
Using the data given in Table 15 on page 28, we compute the unit transaction cost with
respect to Qi(h) in Figure 11. The first panel uses the formula of sovereign bonds while the
three other panels use the formula of corporate bonds. i= 1 corresponds to the UST bond.
The spread siis equal to 1.33 bps and the yearly volatility is equal to 16 bps. This explains
the low transaction cost. i= 45 corresponds to the Petronas bond. The transaction cost is
very high because the bid-ask spread siis equal to 57.15 bps and the DTS is equal to 2635
bps.
Figure 11: Unit transaction cost function
0 10 20 30 40 50
1.6
1.8
2
2.2
0246
5
5.5
6
6.5
7
0 1 2 3
15
20
25
30
0246
100
120
140
160
We consider the bond portfolio, whose total net assets are equal to $10 bn. By assuming
a 30% redemption rate and a vertical slicing approach, we obtain T C (q) = 10.68 million
dollars. This implies a liquidation cost c
c
c(q) of 35.60 bps and an investment cost ˜
c
c
c(q) of
10.68 bps. The price impact cost represents 68.90% of the total cost17.
Let us consider the following stress test scenario: ∆s= 3 bps, ∆σyearly = 2%, ∆DTS = 100
bps and mQ+= 0.50. We now obtain T Cstress (q) = 12.29 million dollars. In Table 12, we
compare the breakdown between the spread and price impact components. We notice that
the ratio of the transaction cost that is explained by the price impact is greater for the
normal scenario than for the stress test scenario. At first sight, this result may be curious,
because we expect that price impacts increase when we stress the liquidity condition. The
issue comes from the definition of the participation rate. In the case of stocks, we have
xi(h) = qi(h)
vi
. When we stress the volume viby applying the stress factor mv, we have
two effects: an impact on qi(h) and an impact on vi. Finally, the stressed value of xi(h) is
greater or equal to the normal value of xi(h). This is not the case for bonds. Indeed, we
have:
yi(h) = Qi(h)
Mi
(53)
17More figures are given in Tables 22 and 23 on page 36.
22
Liquidity Stress Testing in Asset Management
Figure 12: Ratio of transaction cost explained by the price impact (bond portfolio, TNA =
$10 bn, R= 30%, vertical slicing)
00.10.20.30.40.50.60.70.80.91
0.3
0.4
0.5
0.6
0.7
When we stress the liquidity policy by applying the multiplicative factor mQ+, we have
an impact on the liquidation shares Qi(h), but no effect on the outstanding amount Mi.
Therefore, the stressed value of yi(h) is less than the normal value of yi(h). Finally, the
part of the transaction cost due to the price impact is reduced when we decrease mQ+(see
Figure 12). In order to be consistent with the equity framework, we propose to scale the
participation rate:
ystress∗
i(h) = 1
mQ+Qi(h)
Mi(54)
This approach is called “stress∗” in order to distinguish it from the previous one. In Table
12 we report the transaction cost of the redemption portfolio when we assume that R= 30%
and the fund size is equal to $10 bn. In the normal case, the investment cost is equal to
10.68 bps. However, this figure is not representative of a normal cost. Indeed, we use a too
high redemption value (R= 30%). If we assume that R= 30%, we obtain ˜
c
c
c(q)=1.53 bps.
This investment cost becomes 12.29 bps when we consider the stress test scenario. If we
implement the correction given by Equation (54), we finally obtain ˜
c
c
c(q) = 13.75 bps.
Table 12: Spread and price impact components in bps (bond portfolio, TNA = $10 bn,
R= 30%, vertical slicing)
Scenario c
c
cs(q)c
c
cπ
π
π(q)c
c
c(q)˜
c
c
cs(q)˜
c
c
cπ
π
π(q)˜
c
c
c(q)
Normal (R= 5%) 11.07 19.51 30.58 0.55 0.98 1.53
Normal 11.07 24.53 35.60 3.32 7.36 10.68
Stress 15.12 25.84 40.96 4.54 7.75 12.29
Stress∗15.12 30.73 45.85 4.54 9.22 13.75
23
Liquidity Stress Testing in Asset Management
4 Conclusion
This working paper complements the publication of the three previous research papers ded-
icated to the liquidity stress testing in asset management (Roncalli et al.,2021a,b;Roncalli,
2021c). Using three portfolios, we have explained how to compute the redemption coverage
ratio, implement reverse stress testing and estimate the liquidation cost of the redemption
portfolio. The first portfolio is an index portfolio corresponding to the Eurostoxx 50 index.
The second portfolio is an active equity portfolio, which is made up of 20 small and mid-cap
stocks. Finally, the third portfolio corresponds to 47 sovereign and corporate bonds. These
examples illustrate the main asset classes that are concerned by liquidity stress testing:
large-cap stocks, small-cap stocks, sovereign bonds, corporate bonds. We have excluded
derivatives from this research project because of the lack of research on this topic. Nev-
ertheless, these examples must enable asset managers to understand the common basics of
liquidity stress testing in asset management.
References
Bouveret, A. (2017), Liquidity Stress Tests for Investment Funds: A Practical Guide, IMF
Working Paper, 17/226.
European Securities and Markets Authority (2019a), Guidelines on Liquidity Stress Testing
in UCITS and AIFs, Final Report,ESMA34-39-882, September.
European Securities and Markets Authority (2019b), Stress Simulation for Investment Funds
(Stresi), Economic Report,ESMA50-164-2458, September.
European Securities and Markets Authority (2020a), Guidelines on Liquidity Stress Testing
in UCITS and AIFs, ESMA34-39-897, July.
European Securities and Markets Authority (2020b), Recommendation of the European
Systemic Risk Board (ESRB) on Liquidity Risk in Investment Funds, Report,ESMA34-
39-1119, November.
International Monetary Fund (2017), Luxembourg Financial Sector Assessment Program
(FSAP): Technical Note & Risk Analysis, IMF Staff Country Reports, 17/261, August.
International Monetary Fund (2020), United States Financial Sector Assessment Program
(FSAP): Technical Note, Risk Analysis and Stress Testing the Financial Sector, IMF Staff
Country Reports, 20/247, August.
Roncalli, T., Karray-Meziou, F., Pan, F., and Regnault, M. (2021a), Liquidity Stress
Testing in Asset Management — Part 1. Modeling the Liability Liquidity Risk, Amundi
Working Paper.
Roncalli, T., Cherief, A., Karray-Meziou, F., and Regnault, M. (2021b), Liquidity
Stress Testing in Asset Management — Part 2. Modeling the Asset Liquidity Risk, Amundi
Working Paper.
Roncalli, T. (2021c), Liquidity Stress Testing in Asset Management — Part 3. Managing
the Asset-Liability Liquidity Risk, Amundi Working Paper.
24
Liquidity Stress Testing in Asset Management
Appendix
A Data
We use three portfolios18 :
1. Table 13 presents a first equity portfolio based on the Eurostoxx 50 index. It is made
up of 50 stocks with the following correspondence: (1) Adidas, (2) Adyen, (3) Air
Liquide, (4) Airbus, (5) Allianz, (6) Anheuser-Busch, (7) ASML, (8) AXA, (9) Banco
Santander, (10) BASF, (11) Bayer, (12) BMW, (13) BBVA, (14) BNP Paribas, (15)
CRH, (16) Daimler, (17) Danone, (18) Deutsche Boerse, (19) Deutsche Post, (20)
Deutsche Telekom, (21) Enel, (22) Eni, (23) EssilorLuxottica, (24) Flutter Entertain-
ment, (25) Iberdrola, (26) Inditex, (27) Infineon Technolog, (28) ING, (29) Intesa
Sanpaolo, (30) Kering, (31) Kon Ahold Delhaize, (32) Kone, (33) Koninklijke Philips,
(34) L’Oreal, (35) Linde, (36) LVMH, (37) Muenchener Rueckve, (38) Pernod Ricard,
(39) Prosus, (40) Safran, (41) Sanofi, (42) SAP, (43) Schneider Electric, (44) Siemens,
(45) Stellantis, (46) TotalEnergies, (47) Universal Music, (48) Vinci, (49) Volkswagen,
(50) Vonovia.
2. In Table 14, we consider a second equity portfolio with more small and mid-cap stocks.
We have the following correspondence: (1) Heineken, (2) Zalando, (3) ArcelorMittal,
(4) Kerry Group, (5) Porsche, (6) Puma, (7) Hannover Rueck, (8) Solvay, (9) Euronext,
(10) Sodexo, (11) Gecina, (12) Ubisoft Entertainment, (13) Rational, (14) Deutsche
Lufthansa, (15) Ackermans & van Haaren, (16) Aeroports de Paris, (17) Telecom
Italia, (18) Christian Dior, (19) Sopra Steria Group, (20) Dassault Aviation.
3. The last portfolio is made up of 11 US bonds and 36 USD-denominated corporate
bonds. Table 15 gives the different characteristics for the different bonds: (1) UST (T
1 5/8 11/15/22), (2) UST (T 0 1/4 05/15/24), (3) UST (T 0 3/8 08/15/24), (4) UST
(T 1 5/8 08/15/29), (5) UST (T 2 3/8 05/15/29), (6) UST (T 3 1/8 11/15/28), (7) UST
(T 1 5/8 05/15/31), (8) UST (T 1 1/4 08/15/31), (9) UST (T 1 1/8 02/15/31), (10)
UST (T 2 3/8 05/15/51), (11) UST (T 2 08/15/51), (12) Apple (AAPL 3 02/09/24),
(13) DNB Bank (DNBNO 0.856 09/30/25), (14) Equinor (EQNR 2 7/8 04/06/25), (15)
Bank of America (BAC 3.093 10/01/25), (16) Comcast (CMCSA 3.7 04/15/24), (17)
Cnooc Finance (CNOOC 3 05/09/23), (18) EMD Finance (MRKGR 3 1/4 03/19/25),
(19) Boeing (BA 4.508 05/01/23), (20) Bank of America (BCHINA 5 11/13/24), (21)
BP Capital Markets (BPLN 4 3/8 PERP), (22) Apple (AAPL 3.35 02/09/27), (23)
Nestle (NESNVX 3 5/8 09/24/28), (24) Alibaba (BABA 3.4 12/06/27), (25) Bank of
America (BAC 4.183 11/25/27), (26) Dai-Ichi Life Insurance (DAIL 4 PERP), (27)
NTT Finance (NTT 1.162 04/03/26), (28) Anheuser-Busch (ABIBB 4 04/13/28), (29)
Bat Capital (BATSLN 3.557 08/15/27), (30) Bayer (BAYNGR 4 1/4 12/15/25), (31)
Nissan (NSANY 4.345 09/17/27), (32) Amazon (AMZN 1 1/2 06/03/30), (33) Alpha-
bet (GOOGL 1.1 08/15/30), (34) Shell (RDSALN 2 3/4 04/06/30), (35) Bank of Amer-
ica (BAC 2.592 04/29/31), (36) BNP Paribas (BNP 2.871 04/19/32), (37) Petronas
(PETMK 3 1/2 04/21/30), (38) British Telecom (BRITEL 9 5/8 12/15/30), (39)
Deutsche Telekom (DT 8 1/4 06/15/30), (40) Orange (ORAFP 8 1/2 03/01/31), (41)
Shell (RDSALN 3 1/4 04/06/50), (42) Abbott Laboratories (ABT 4 3/4 11/30/36),
(43) Saudi Arabian Oil (ARAMCO 3 1/4 11/24/50), (44) Bank of America (BAC
4.443 01/20/48), (45) Petronas (PETMK 4.55 04/21/50), (46) Credit Suisse (CS 4
7/8 05/15/45) and (47) Telefonica (TELEFO 5.213 03/08/47).
18The raw data are available at https://www.researchgate.net/publication/356849853.
25
Liquidity Stress Testing in Asset Management
Table 13: Large-cap equity portfolio (Eurostoxx 50 index), TNA = e1 bn)
i ωiPiPbid
iPask
iσyearly
ivi
1 59 106 284.050 281.750 281.800 25.69 514 842
2 8 883 2630.500 2567.500 2568.500 31.14 56 255
3 150 027 143.600 142.880 142.920 13.75 629 509
4 184 310 112.000 111.640 111.660 26.42 1 316 600
5 130 520 200.950 200.150 200.200 21.75 750 684
6 268 123 54.460 53.280 53.290 27.08 1 736 372
7 131 520 698.400 690.100 690.300 31.82 754 901
8 651 421 24.415 24.760 24.765 18.72 4 358 304
9 3 192 430 5.641 5.988 5.990 33.74 54 130 721
10 5 544 072 3.2805 3.2465 3.2480 29.85 72 371 040
11 242 317 62.550 62.190 62.200 20.69 2 473 040
12 181 800 48.735 48.805 48.820 21.39 2 444 130
13 136 334 87.340 86.610 86.630 26.13 1 183 053
14 365 067 57.460 58.250 58.260 26.61 2 390 614
15 251 719 41.360 41.060 41.070 20.83 1 434 050
16 265 762 83.850 84.870 84.900 26.15 2 672 846
17 206 041 56.040 55.810 55.830 18.72 1 579 517
18 60 148 144.200 142.700 142.750 17.01 339 768
19 311 879 54.040 53.160 53.170 19.38 2 536 147
20 1 026 503 16.032 15.942 15.944 19.77 9 225 311
21 2 459 244 7.270 7.213 7.215 23.17 30 518 046
22 795 234 12.164 12.408 12.412 21.32 19 419 467
23 95 262 172.860 177.120 177.160 20.43 491 647
24 55 520 165.600 163.600 163.650 33.71 212 501
25 1 840 196 10.250 10.225 10.230 24.90 14 316 692
26 351 837 31.050 30.910 30.930 25.62 7 543 014
27 413 417 40.100 39.700 39.710 29.69 3 643 730
28 1 235 905 13.082 13.122 13.126 25.02 15 954 487
29 5 774 696 2.438 2.441 2.442 22.67 106 942 206
30 23 113 650.900 646.100 646.200 31.69 204 628
31 161 807 57.460 57.960 57.980 19.61 786 412
32 261 645 28.085 27.965 27.970 15.39 2 285 287
33 290 422 40.665 40.250 40.260 21.56 2 489 971
34 76 637 393.450 388.850 388.950 20.44 372 415
35 162 956 271.800 271.300 271.400 17.81 578 973
36 83 427 671.700 666.100 666.200 28.37 364 566
37 44 351 254.400 255.300 255.400 21.80 256 421
38 64 954 201.100 197.000 197.050 17.37 363 103
39 282 801 76.530 75.000 75.020 46.16 2 135 693
40 120 062 114.080 116.920 116.940 30.87 794 444
41 362 506 85.990 85.770 85.780 14.72 1 551 395
42 345 779 126.300 123.060 123.080 21.01 1 859 400
43 180 140 148.900 145.600 145.620 22.27 818 822
44 237 952 139.840 136.940 136.980 25.33 1 136 151
45 660 350 17.286 17.078 17.084 28.79 10 497 975
46 835 885 43.295 43.610 43.620 21.90 6 596 020
47 247 964 25.120 24.970 24.980 12.73 1 943 066
48 189 196 91.670 91.590 91.620 20.21 912 539
49 57 954 194.780 193.200 193.260 29.39 1 071 749
50 163 680 54.060 52.680 52.700 19.20 976 446
The portfolio holding ωiand the daily volume viare measured in number of shares, the yearly volatility
σyearly
iis expressed in %, whereas the prices (Pi,Pbid
iand Pask
i) are in euros.
26
Liquidity Stress Testing in Asset Management
Table 14: Small-cap equity portfolio (Eurostoxx index), TNA = e1 bn)
i ωiPiPbid
iPask
iσyearly
ivi
1 505 766 98.860 98.840 98.860 17.68 432 593
2 645 495 77.460 77.460 77.480 34.36 725 019
3 1 830 496 27.315 27.310 27.320 41.55 5 513 446
4 436 110 114.650 114.650 114.750 16.88 221 468
5 592 839 84.340 84.300 84.320 27.48 535 423
6 447 628 111.700 111.650 111.750 21.57 291 008
7 309 311 161.650 161.600 161.700 20.60 92 108
8 473 261 105.650 105.600 105.700 17.23 134 879
9 531 915 94.000 93.950 94.050 21.40 201 095
10 597 944 83.620 83.600 83.660 26.02 298 374
11 407 997 122.550 122.500 122.550 17.73 90 460
12 1 066 100 46.900 46.890 46.910 30.24 566 323
13 56 883 879.000 878.400 879.400 34.17 8 328
14 7 423 908 6.735 6.734 6.738 38.20 12 831 900
15 324 464 154.100 154.000 154.200 14.60 22 415
16 413 908 120.800 120.600 120.750 30.83 90 763
17 153 940 887 0.325 0.324 0.325 25.92 179 923 300
18 70 521 709.000 709.000 709.500 28.73 4 061
19 289 687 172.600 172.500 172.700 26.86 19 850
20 537 923 92.950 92.850 93.000 24.35 41 414
The portfolio holding ωiand the daily volume viare measured in number of shares, the yearly volatility
σyearly
iis expressed in %, whereas the prices (Pi,Pbid
iand Pask
i) are in euros.
27
Liquidity Stress Testing in Asset Management
Table 15: Bond portfolio (USD-denominated, TNA = $1 bn)
iIsin ωiPisiσyearly
iDTSiMiQ+
i
1US912828TY62 529 725 102.288 1.33 0.16 121 993 50
2US91282CCC38 427 182 99.243 1.56 0.99 88 769 50
3US91282CCT62 403 263 99.334 1.21 1.46 83 876 50
4US912828YB05 432 386 102.018 2.31 4.65 92 619 50
5US9128286T26 372 123 108.060 1.46 4.31 84 427 50
6US9128285M81 339 801 113.243 2.10 3.91 80 506 50
7US91282CCB54 695 025 101.846 1.55 5.90 148 501 50
8US91282CCS89 695 930 97.747 1.60 6.25 142 197 50
9US91282CBL46 689 215 97.079 1.61 5.88 140 063 50
10 US912810SX72 412 836 110.482 2.86 15.15 95 481 50
11 US912810SZ21 880 939 101.289 3.10 17.88 91 407 50
12 US037833CG39 120 000 105.636 3.43 0.92 43 1 750 6
13 US25601B2A27 110 140 99.357 6.52 2.44 78 1 250 6
14 US29446MAD48 46 500 105.828 9.09 2.38 101 1 250 6
15 US06051GGT04 65 100 105.721 9.87 1.19 137 1 750 6
16 US20030NCR08 92 000 107.044 4.07 1.06 53 2 500 6
17 US12625GAC87 75 300 104.455 6.65 0.88 119 2 000 6
18 US26867LAL45 59 000 106.636 6.17 1.57 124 1 600 6
19 US097023CS21 112 300 105.018 2.19 1.30 101 3 000 6
20 US061202AA55 205 000 112.308 9.37 2.08 265 3 000 3
21 US05565QDU94 192 000 106.829 3.56 4.29 559 2 500 3
22 US037833CJ77 80 000 109.966 7.56 2.44 137 2 250 6
23 US641062AF17 142 000 112.062 13.08 3.94 243 1 250 6
24 US01609WAT99 92 800 108.031 10.18 3.74 518 2 550 6
25 US06051GGC78 170 000 112.456 10.85 2.74 392 2 000 3
26 US23380YAD94 190 500 108.665 26.74 2.10 555 2 500 3
27 US62954WAC91 120 000 98.925 7.94 2.20 168 3 000 6
28 US035240AL43 88 000 112.839 13.64 3.54 300 2 500 6
29 US05526DBB01 130 000 106.846 15.23 2.79 566 3 500 6
30 US07274NAJ28 89 000 111.578 11.56 2.04 257 2 500 6
31 US654744AC50 89 000 109.382 7.22 3.08 753 2 500 6
32 US023135BS49 80 500 97.649 14.43 4.97 296 2 000 6
33 US02079KAD90 93 800 94.273 12.76 5.05 425 2 250 6
34 US822582CG52 156 000 105.951 11.78 5.83 413 1 750 6
35 US06051GJB68 116 000 101.604 13.98 5.32 729 3 000 6
36 US09659W2P81 86 400 102.263 15.38 5.57 926 2 250 6
37 US716743AP46 82 000 108.224 23.37 10.20 763 2 250 6
38 US111021AE12 68 000 153.963 16.15 4.43 1193 2 670 6
39 US25156PAC77 92 000 150.553 14.01 3.75 839 3 500 6
40 US35177PAL13 60 000 155.740 17.46 4.65 763 2 460 6
41 US822582CH36 70 000 109.196 26.31 15.97 1619 2 000 6
42 US002824BG43 49 900 130.051 37.90 7.36 817 1 650 6
43 US80414L2L80 90 000 98.400 59.02 11.07 2872 2 250 6
44 US06051GGG82 60 000 127.534 56.44 12.08 1560 2 000 6
45 US716743AR02 87 000 125.184 57.14 12.43 2336 2 750 6
46 US225433AF86 58 600 129.155 29.29 10.24 1962 1 925 6
47 US87938WAU71 77 700 126.664 27.90 10.78 2635 2 500 6
The portfolio holding ωiis measured in number of shares, the price Piis in US dollars, the yearly volatility
σyearly
iis expressed in %, the (half) bid-ask spread siand the duration-times-spread DTSiare in bps,
whereas the outstanding amount Miand the daily trading limit Q+
iare expressed in $ mn.
28
Liquidity Stress Testing in Asset Management
B Additional results
This appendix contains additional tables and figures.
29
Liquidity Stress Testing in Asset Management
Table 16: Computation of the unit transaction cost c
c
ci(xi(h)) (large-cap portfolio, TNA =
e1 bn, R= 80%, vertical slicing)
isic
c
cs
iσixi(h)c
c
cπ
π
π
i(xi(h)) c
c
ci(xi(h))
1 0.89 1.11 1.59 9.18 0.00 0.00 22.67 0.00 0.00 23.78 0.00 0.00
2 1.95 2.43 1.93 10.00 2.63 0.00 29.92 12.53 0.00 32.35 14.97 0.00
3 1.40 1.75 0.85 10.00 9.07 0.00 13.21 11.98 0.00 14.96 13.73 0.00
4 0.90 1.12 1.64 10.00 1.20 0.00 25.38 7.18 0.00 26.50 8.30 0.00
5 1.25 1.56 1.35 10.00 3.91 0.00 20.90 10.67 0.00 22.46 12.23 0.00
6 0.94 1.17 1.68 10.00 2.35 0.00 26.02 10.31 0.00 27.19 11.48 0.00
7 1.45 1.81 1.97 10.00 3.94 0.00 30.57 15.66 0.00 32.38 17.47 0.00
8 1.01 1.26 1.16 10.00 1.96 0.00 17.99 6.50 0.00 19.25 7.76 0.00
9 1.67 2.09 2.09 4.72 0.00 0.00 18.18 0.00 0.00 20.27 0.00 0.00
10 2.31 2.89 1.85 6.13 0.00 0.00 18.33 0.00 0.00 21.22 0.00 0.00
11 0.80 1.00 1.28 7.84 0.00 0.00 15.58 0.00 0.00 16.59 0.00 0.00
12 1.54 1.92 1.33 5.95 0.00 0.00 12.94 0.00 0.00 14.86 0.00 0.00
13 1.15 1.44 1.62 9.22 0.00 0.00 23.14 0.00 0.00 24.59 0.00 0.00
14 0.86 1.07 1.65 10.00 2.22 0.00 25.57 9.83 0.00 26.64 10.90 0.00
15 1.22 1.52 1.29 10.00 4.04 0.00 20.01 10.39 0.00 21.53 11.91 0.00
16 1.77 2.21 1.62 7.95 0.00 0.00 19.98 0.00 0.00 22.19 0.00 0.00
17 1.79 2.24 1.16 10.00 0.44 0.00 17.99 3.07 0.00 20.22 5.30 0.00
18 1.75 2.19 1.05 10.00 4.16 0.00 16.34 8.61 0.00 18.53 10.80 0.00
19 0.94 1.18 1.20 9.84 0.00 0.00 18.32 0.00 0.00 19.49 0.00 0.00
20 0.63 0.78 1.23 8.90 0.00 0.00 16.91 0.00 0.00 17.69 0.00 0.00
21 1.39 1.73 1.44 6.45 0.00 0.00 14.59 0.00 0.00 16.33 0.00 0.00
22 1.61 2.01 1.32 3.28 0.00 0.00 9.57 0.00 0.00 11.59 0.00 0.00
23 1.13 1.41 1.27 10.00 5.50 0.00 19.63 11.89 0.00 21.04 13.30 0.00
24 1.53 1.91 2.09 10.00 10.00 0.90 32.39 32.39 7.94 34.30 34.30 9.85
25 2.44 3.06 1.54 10.00 0.28 0.00 23.92 3.28 0.00 26.98 6.34 0.00
26 3.23 4.04 1.59 3.73 0.00 0.00 12.28 0.00 0.00 16.32 0.00 0.00
27 1.26 1.57 1.84 9.08 0.00 0.00 25.89 0.00 0.00 27.47 0.00 0.00
28 1.52 1.90 1.55 6.20 0.00 0.00 15.45 0.00 0.00 17.36 0.00 0.00
29 2.05 2.56 1.41 4.32 0.00 0.00 11.69 0.00 0.00 14.25 0.00 0.00
30 0.77 0.97 1.97 9.04 0.00 0.00 27.51 0.00 0.00 28.48 0.00 0.00
31 1.73 2.16 1.22 10.00 6.46 0.00 18.84 12.36 0.00 21.00 14.52 0.00
32 0.89 1.12 0.95 9.16 0.00 0.00 13.54 0.00 0.00 14.66 0.00 0.00
33 1.24 1.55 1.34 9.33 0.00 0.00 19.33 0.00 0.00 20.88 0.00 0.00
34 1.29 1.61 1.27 10.00 6.46 0.00 19.64 12.89 0.00 21.25 14.50 0.00
35 1.84 2.30 1.10 10.00 10.00 2.52 17.11 17.11 7.01 19.41 19.41 9.31
36 0.75 0.94 1.76 10.00 8.31 0.00 27.26 22.64 0.00 28.20 23.58 0.00
37 1.96 2.45 1.35 10.00 3.84 0.00 20.94 10.59 0.00 23.39 13.04 0.00
38 1.27 1.59 1.08 10.00 4.31 0.00 16.69 8.95 0.00 18.27 10.53 0.00
39 1.33 1.67 2.86 10.00 0.59 0.00 44.35 8.82 0.00 46.02 10.49 0.00
40 0.86 1.07 1.91 10.00 2.09 0.00 29.66 11.07 0.00 30.73 12.14 0.00
41 0.58 0.73 0.91 10.00 8.69 0.00 14.14 12.29 0.00 14.87 13.02 0.00
42 0.81 1.02 1.30 10.00 4.88 0.00 20.19 11.51 0.00 21.20 12.53 0.00
43 0.69 0.86 1.38 10.00 7.60 0.00 21.40 16.26 0.00 22.25 17.12 0.00
44 1.46 1.83 1.57 10.00 6.75 0.00 24.34 16.44 0.00 26.16 18.26 0.00
45 1.76 2.20 1.79 5.03 0.00 0.00 16.02 0.00 0.00 18.22 0.00 0.00
46 1.15 1.43 1.36 10.00 0.14 0.00 21.04 2.02 0.00 22.47 3.45 0.00
47 2.00 2.50 0.79 10.00 0.21 0.00 12.23 1.44 0.00 14.73 3.95 0.00
48 1.64 2.05 1.25 10.00 6.59 0.00 19.42 12.87 0.00 21.46 14.91 0.00
49 1.55 1.94 1.82 4.33 0.00 0.00 15.16 0.00 0.00 17.10 0.00 0.00
50 1.90 2.37 1.19 10.00 3.41 0.00 18.45 8.80 0.00 20.82 11.17 0.00
The daily volatility σiand the daily participation rate xi(h) are expressed in %, whereas the bid-ask
spread siand the unit transaction costs c
c
cs
i,c
c
cπ
π
π
i(xi(h)) and c
c
ci(xi(h)) are measured in bps.
30
Liquidity Stress Testing in Asset Management
Table 17: Computation of the total transaction cost T Ci(qi(h)) (large-cap portfolio, TNA =
e1 bn, R= 80%, vertical slicing)
i Qi(h)T Ci(qi(h)) T Ci(qi)
1 13 431 247.44 0.00 0.00 31 936.75 0.00 0.00 31 936.75
2 14 797 877.75 3 895 507.45 0.00 47 874.31 5 830.54 0.00 53 704.85
3 9 039 749.24 8 195 352.52 0.00 13 523.51 11 248.99 0.00 24 772.50
4 14 745 920.00 1 768 256.00 0.00 39 081.24 1 467.05 0.00 40 548.28
5 15 084 994.98 5 897 400.22 0.00 33 877.67 7 212.12 0.00 41 089.78
6 9 456 281.91 2 225 300.95 0.00 25 712.16 2 554.23 0.00 28 266.39
7 52 722 285.84 20 760 568.56 0.00 170 729.19 36 278.74 0.00 207 007.93
8 10 640 799.22 2 082 755.76 0.00 20 480.97 1 616.00 0.00 22 096.97
9 14 406 798.10 0.00 0.00 29 198.99 0.00 0.00 29 198.99
10 14 549 862.56 0.00 0.00 30 872.51 0.00 0.00 30 872.51
11 12 125 542.68 0.00 0.00 20 112.48 0.00 0.00 20 112.48
12 7 088 018.40 0.00 0.00 10 535.97 0.00 0.00 10 535.97
13 9 525 929.25 0.00 0.00 23 421.99 0.00 0.00 23 421.99
14 13 736 468.04 3 044 931.81 0.00 36 592.48 3 319.27 0.00 39 911.75
15 5 931 230.80 2 397 647.47 0.00 12 772.78 2 855.88 0.00 15 628.67
16 17 827 314.96 0.00 0.00 39 565.34 0.00 0.00 39 565.34
17 8 851 613.27 385 616.84 0.00 17 902.33 204.55 0.00 18 106.88
18 4 899 454.56 2 039 218.72 0.00 9 079.77 2 201.99 0.00 11 281.76
19 13 483 152.93 0.00 0.00 26 283.29 0.00 0.00 26 283.29
20 13 165 516.88 0.00 0.00 23 292.62 0.00 0.00 23 292.62
21 14 302 963.10 0.00 0.00 23 351.72 0.00 0.00 23 351.72
22 7 738 581.10 0.00 0.00 8 966.85 0.00 0.00 8 966.85
23 8 498 610.04 4 674 981.41 0.00 17 880.92 6 216.75 0.00 24 097.67
24 3 519 016.56 3 519 016.56 317 256.48 12 069.30 12 069.30 312.50 24 451.10
25 14 674 609.30 414 997.90 0.00 39 590.09 263.12 0.00 39 853.21
26 8 739 631.08 0.00 0.00 14 262.89 0.00 0.00 14 262.89
27 13 262 417.36 0.00 0.00 36 426.36 0.00 0.00 36 426.36
28 12 934 487.37 0.00 0.00 22 448.98 0.00 0.00 22 448.98
29 11 262 967.08 0.00 0.00 16 047.96 0.00 0.00 16 047.96
30 12 035 401.36 0.00 0.00 34 275.96 0.00 0.00 34 275.96
31 4 518 723.35 2 919 220.82 0.00 9 487.95 4 238.94 0.00 13 726.89
32 5 878 639.86 0.00 0.00 8 618.38 0.00 0.00 8 618.38
33 9 448 008.50 0.00 0.00 19 728.26 0.00 0.00 19 728.26
34 14 652 668.17 9 469 593.95 0.00 31 129.91 13 728.40 0.00 44 858.31
35 15 736 486.14 15 736 486.14 3 960 180.36 30 551.75 30 551.75 3 687.74 64 791.24
36 24 487 898.22 20 342 434.50 0.00 69 044.22 47 969.50 0.00 117 013.72
37 6 523 350.24 2 502 965.28 0.00 15 259.67 3 264.04 0.00 18 523.71
38 7 302 001.33 3 147 798.19 0.00 13 344.15 3 315.46 0.00 16 659.62
39 16 344 458.53 969 749.90 0.00 75 209.91 1 016.95 0.00 76 226.86
40 9 063 017.15 1 894 321.22 0.00 27 848.80 2 299.77 0.00 30 148.57
41 13 340 445.60 11 597 067.15 0.00 19 838.80 15 102.83 0.00 34 941.63
42 23 484 222.00 11 453 288.16 0.00 49 789.93 14 346.06 0.00 64 135.98
43 12 192 259.58 9 266 017.22 0.00 27 133.61 15 862.95 0.00 42 996.55
44 15 887 935.58 10 732 230.56 0.00 41 565.42 19 601.77 0.00 61 167.20
45 9 131 848.08 0.00 0.00 16 635.12 0.00 0.00 16 635.12
46 28 557 468.59 394 244.27 0.00 64 179.58 136.08 0.00 64 315.65
47 4 880 981.79 102 102.75 0.00 7 191.19 40.30 0.00 7 231.49
48 8 365 245.01 5 509 632.84 0.00 17 955.14 8 216.72 0.00 26 171.86
49 9 030 624.10 0.00 0.00 15 446.58 0.00 0.00 15 446.58
50 5 278 667.08 1 800 165.56 0.00 10 989.73 2 010.43 0.00 13 000.16
Total 626 583 692.07 169 138 870.69 4 277 436.84 1 459 115.46 275 040.48 4 000.24 1 738 156.17
All the metrics are expressed in e.
31
Liquidity Stress Testing in Asset Management
Table 18: Break-down of the total transaction cost T C (q) (large-cap portfolio, TNA = e1
bn, R= 80%, vertical slicing)
iT Ci(qi)T Cs
i(qi)T Cπ
π
π
i(qi)
1 31 936.75 1 489.58 30 447.17
2 53 704.85 4 549.60 49 155.26
3 24 772.50 3 015.24 21 757.26
4 40 548.28 1 848.88 38 699.40
5 41 089.78 3 275.63 37 814.15
6 28 266.39 1 370.18 26 896.21
7 207 007.93 13 308.25 193 699.67
8 22 096.97 1 605.70 20 491.27
9 29 198.99 3 006.93 26 192.06
10 30 872.51 4 200.63 26 671.88
11 20 112.48 1 218.50 18 893.98
12 10 535.97 1 361.34 9 174.63
13 23 421.99 1 374.67 22 047.31
14 39 911.75 1 800.42 38 111.33
15 15 628.67 1 267.64 14 361.03
16 39 565.34 3 937.82 35 627.51
17 18 106.88 2 068.53 16 038.35
18 11 281.76 1 519.24 9 762.52
19 26 283.29 1 585.06 24 698.23
20 23 292.62 1 032.23 22 260.39
21 23 351.72 2 478.33 20 873.38
22 8 966.85 1 558.94 7 407.91
23 24 097.67 1 859.21 22 238.47
24 24 451.10 1 404.75 23 046.35
25 39 853.21 4 610.61 35 242.60
26 14 262.89 3 533.16 10 729.73
27 36 426.36 2 087.65 34 338.71
28 22 448.98 2 463.90 19 985.08
29 16 047.96 2 883.21 13 164.75
30 34 275.96 1 164.15 33 111.82
31 13 726.89 1 603.83 12 123.05
32 8 618.38 656.86 7 961.52
33 19 728.26 1 466.90 18 261.36
34 44 858.31 3 876.68 40 981.63
35 64 791.24 8 161.31 56 629.92
36 117 013.72 4 206.10 112 807.62
37 18 523.71 2 209.30 16 314.41
38 16 659.62 1 657.44 15 002.18
39 76 226.86 2 885.32 73 341.54
40 30 148.57 1 171.36 28 977.21
41 34 941.63 1 817.07 33 124.56
42 64 135.98 3 548.54 60 587.44
43 42 996.55 1 842.10 41 154.45
44 61 167.20 4 859.11 56 308.09
45 16 635.12 2 004.83 14 630.29
46 64 315.65 4 148.76 60 166.89
47 7 231.49 1 247.02 5 984.47
48 26 171.86 2 839.95 23 331.90
49 15 446.58 1 752.57 13 694.02
50 13 000.16 1 679.36 11 320.80
Total 1 738 156.17 132 514.40 1 605 641.78
All the metrics are expressed in e.
32
Liquidity Stress Testing in Asset Management
Table 19: Stress testing — Liquidation portfolio qi(h) (large-cap portfolio, TNA = e1 bn,
R= 80%, vertical slicing)
i qiq+
iqi(1) qi(2) qi(3) qi(4) qi(5)
1 47 284.80 25 742.10 25 742.10 21 542.70 0.00 0.00 0.00
2 7 106.40 2 812.75 2 812.75 2 812.75 1 480.90 0.00 0.00
3 120 021.60 31 475.45 31 475.45 31 475.45 31 475.45 25 595.25 0.00
4 147 448.00 65 830.00 65 830.00 65 830.00 15 788.00 0.00 0.00
5 104 416.00 37 534.20 37 534.20 37 534.20 29 347.60 0.00 0.00
6 214 498.40 86 818.60 86 818.60 86 818.60 40 861.20 0.00 0.00
7 105 216.00 37 745.05 37 745.05 37 745.05 29 725.90 0.00 0.00
8 521 136.80 217 915.20 217 915.20 217 915.20 85 306.40 0.00 0.00
9 2 553 944.00 2 706 536.05 2 553 944.00 0.00 0.00 0.00 0.00
10 4 435 257.60 3 618 552.00 3 618 552.00 816 705.60 0.00 0.00 0.00
11 193 853.60 123 652.00 123 652.00 70 201.60 0.00 0.00 0.00
12 145 440.00 122 206.50 122 206.50 23 233.50 0.00 0.00 0.00
13 109 067.20 59 152.65 59 152.65 49 914.55 0.00 0.00 0.00
14 292 053.60 119 530.70 119 530.70 119 530.70 52 992.20 0.00 0.00
15 201 375.20 71 702.50 71 702.50 71 702.50 57 970.20 0.00 0.00
16 212 609.60 133 642.30 133 642.30 78 967.30 0.00 0.00 0.00
17 164 832.80 78 975.85 78 975.85 78 975.85 6 881.10 0.00 0.00
18 48 118.40 16 988.40 16 988.40 16 988.40 14 141.60 0.00 0.00
19 249 503.20 126 807.35 126 807.35 122 695.85 0.00 0.00 0.00
20 821 202.40 461 265.55 461 265.55 359 936.85 0.00 0.00 0.00
21 1 967 395.20 1 525 902.30 1 525 902.30 441 492.90 0.00 0.00 0.00
22 636 187.20 970 973.35 636 187.20 0.00 0.00 0.00 0.00
23 76 209.60 24 582.35 24 582.35 24 582.35 24 582.35 2 462.55 0.00
24 44 416.00 10 625.05 10 625.05 10 625.05 10 625.05 10 625.05 1 915.80
25 1 472 156.80 715 834.60 715 834.60 715 834.60 40 487.60 0.00 0.00
26 281 469.60 377 150.70 281 469.60 0.00 0.00 0.00 0.00
27 330 733.60 182 186.50 182 186.50 148 547.10 0.00 0.00 0.00
28 988 724.00 797 724.35 797 724.35 190 999.65 0.00 0.00 0.00
29 4 619 756.80 5 347 110.30 4 619 756.80 0.00 0.00 0.00 0.00
30 18 490.40 10 231.40 10 231.40 8 259.00 0.00 0.00 0.00
31 129 445.60 39 320.60 39 320.60 39 320.60 39 320.60 11 483.80 0.00
32 209 316.00 114 264.35 114 264.35 95 051.65 0.00 0.00 0.00
33 232 337.60 124 498.55 124 498.55 107 839.05 0.00 0.00 0.00
34 61 309.60 18 620.75 18 620.75 18 620.75 18 620.75 5 447.35 0.00
35 130 364.80 28 948.65 28 948.65 28 948.65 28 948.65 28 948.65 14 570.20
36 66 741.60 18 228.30 18 228.30 18 228.30 18 228.30 12 056.70 0.00
37 35 480.80 12 821.05 12 821.05 12 821.05 9 838.70 0.00 0.00
38 51 963.20 18 155.15 18 155.15 18 155.15 15 652.90 0.00 0.00
39 226 240.80 106 784.65 106 784.65 106 784.65 12 671.50 0.00 0.00
40 96 049.60 39 722.20 39 722.20 39 722.20 16 605.20 0.00 0.00
41 290 004.80 77 569.75 77 569.75 77 569.75 77 569.75 57 295.55 0.00
42 276 623.20 92 970.00 92 970.00 92 970.00 90 683.20 0.00 0.00
43 144 112.00 40 941.10 40 941.10 40 941.10 40 941.10 21 288.70 0.00
44 190 361.60 56 807.55 56 807.55 56 807.55 56 807.55 19 938.95 0.00
45 528 280.00 524 898.75 524 898.75 3 381.25 0.00 0.00 0.00
46 668 708.00 329 801.00 329 801.00 329 801.00 9 106.00 0.00 0.00
47 198 371.20 97 153.30 97 153.30 97 153.30 4 064.60 0.00 0.00
48 151 356.80 45 626.95 45 626.95 45 626.95 45 626.95 14 475.95 0.00
49 46 363.20 53 587.45 46 363.20 0.00 0.00 0.00 0.00
50 130 944.00 48 822.30 48 822.30 48 822.30 33 299.40 0.00 0.00
33
Liquidity Stress Testing in Asset Management
Table 20: Stress testing — Participation rate xi(h) and unit transaction cost c
c
ci(xi(h))
(large-cap portfolio, TNA = e1 bn, R= 80%, vertical slicing)
isiσixi(h)c
c
ci(xi(h))
1 8.89 2.83 10.00 8.37 0.00 0.00 0.00 55.01 47.85 0.00 0.00 0.00
2 9.95 3.17 10.00 10.00 5.26 0.00 0.00 61.57 61.57 41.54 0.00 0.00
3 9.40 2.09 10.00 10.00 10.00 8.13 0.00 44.18 44.18 44.18 38.12 0.00
4 8.90 2.88 10.00 10.00 2.40 0.00 0.00 55.72 55.72 28.95 0.00 0.00
5 9.25 2.59 10.00 10.00 7.82 0.00 0.00 51.67 51.67 42.92 0.00 0.00
6 8.94 2.92 10.00 10.00 4.71 0.00 0.00 56.41 56.41 36.51 0.00 0.00
7 9.45 3.21 10.00 10.00 7.88 0.00 0.00 61.60 61.60 51.02 0.00 0.00
8 9.01 2.40 10.00 10.00 3.91 0.00 0.00 48.46 48.46 30.27 0.00 0.00
9 9.67 3.33 9.44 0.00 0.00 0.00 0.00 60.81 0.00 0.00 0.00 0.00
10 10.31 3.09 10.00 2.26 0.00 0.00 0.00 60.78 31.47 0.00 0.00 0.00
11 8.80 2.52 10.00 5.68 0.00 0.00 0.00 50.10 35.06 0.00 0.00 0.00
12 9.54 2.57 10.00 1.90 0.00 0.00 0.00 51.69 26.08 0.00 0.00 0.00
13 9.15 2.86 10.00 8.44 0.00 0.00 0.00 55.76 48.84 0.00 0.00 0.00
14 8.86 2.89 10.00 10.00 4.43 0.00 0.00 55.85 55.85 35.42 0.00 0.00
15 9.22 2.53 10.00 10.00 8.08 0.00 0.00 50.75 50.75 43.24 0.00 0.00
16 9.77 2.86 10.00 5.91 0.00 0.00 0.00 56.55 40.04 0.00 0.00 0.00
17 9.79 2.40 10.00 10.00 0.87 0.00 0.00 49.44 49.44 21.21 0.00 0.00
18 9.75 2.30 10.00 10.00 8.32 0.00 0.00 47.75 47.75 41.79 0.00 0.00
19 8.94 2.44 10.00 9.68 0.00 0.00 0.00 49.01 47.78 0.00 0.00 0.00
20 8.63 2.47 10.00 7.80 0.00 0.00 0.00 48.99 40.60 0.00 0.00 0.00
21 9.39 2.68 10.00 2.89 0.00 0.00 0.00 53.21 29.95 0.00 0.00 0.00
22 9.61 2.56 6.55 0.00 0.00 0.00 0.00 38.25 0.00 0.00 0.00 0.00
23 9.13 2.51 10.00 10.00 10.00 1.00 0.00 50.26 50.26 50.26 21.45 0.00
24 9.53 3.33 10.00 10.00 10.00 10.00 1.80 63.51 63.51 63.51 63.51 29.80
25 10.44 2.78 10.00 10.00 0.57 0.00 0.00 56.19 56.19 21.43 0.00 0.00
26 11.23 2.83 7.46 0.00 0.00 0.00 0.00 46.75 0.00 0.00 0.00 0.00
27 9.26 3.08 10.00 8.15 0.00 0.00 0.00 59.31 50.50 0.00 0.00 0.00
28 9.52 2.79 10.00 2.39 0.00 0.00 0.00 55.16 29.19 0.00 0.00 0.00
29 10.05 2.65 8.64 0.00 0.00 0.00 0.00 47.98 0.00 0.00 0.00 0.00
30 8.77 3.21 10.00 8.07 0.00 0.00 0.00 60.63 51.06 0.00 0.00 0.00
31 9.73 2.46 10.00 10.00 10.00 2.92 0.00 50.21 50.21 50.21 28.95 0.00
32 8.89 2.19 10.00 8.32 0.00 0.00 0.00 45.12 39.40 0.00 0.00 0.00
33 9.24 2.58 10.00 8.66 0.00 0.00 0.00 51.48 46.14 0.00 0.00 0.00
34 9.29 2.51 10.00 10.00 10.00 2.93 0.00 50.46 50.46 50.46 28.77 0.00
35 9.84 2.34 10.00 10.00 10.00 10.00 5.03 48.63 48.63 48.63 48.63 33.35
36 8.75 3.00 10.00 10.00 10.00 6.61 0.00 57.41 57.41 57.41 41.80 0.00
37 9.96 2.59 10.00 10.00 7.67 0.00 0.00 52.61 52.61 43.27 0.00 0.00
38 9.27 2.32 10.00 10.00 8.62 0.00 0.00 47.49 47.49 42.54 0.00 0.00
39 9.33 4.10 10.00 10.00 1.19 0.00 0.00 75.23 75.23 29.54 0.00 0.00
40 8.86 3.15 10.00 10.00 4.18 0.00 0.00 59.94 59.94 36.87 0.00 0.00
41 8.58 2.15 10.00 10.00 10.00 7.39 0.00 44.09 44.09 44.09 35.37 0.00
42 8.81 2.54 10.00 10.00 9.75 0.00 0.00 50.42 50.42 49.45 0.00 0.00
43 8.69 2.62 10.00 10.00 10.00 5.20 0.00 51.47 51.47 51.47 34.77 0.00
44 9.46 2.81 10.00 10.00 10.00 3.51 0.00 55.38 55.38 55.38 32.89 0.00
45 9.76 3.03 10.00 0.06 0.00 0.00 0.00 59.07 15.27 0.00 0.00 0.00
46 9.15 2.60 10.00 10.00 0.28 0.00 0.00 51.69 51.69 16.89 0.00 0.00
47 10.00 2.03 10.00 10.00 0.42 0.00 0.00 43.95 43.95 17.75 0.00 0.00
48 9.64 2.49 10.00 10.00 10.00 3.17 0.00 50.68 50.68 50.68 29.81 0.00
49 9.55 3.06 8.65 0.00 0.00 0.00 0.00 53.00 0.00 0.00 0.00 0.00
50 9.90 2.43 10.00 10.00 6.82 0.00 0.00 50.03 50.03 38.06 0.00 0.00
The daily volatility σiand the daily participation rate xi(h) are expressed in %, whereas the bid-ask
spread siand the unit transaction costs c
c
ci(xi(h)) are measured in bps.
34
Liquidity Stress Testing in Asset Management
Table 21: Stress testing — Break-down of the total transaction cost T C (q) (large-cap port-
folio, TNA = e1 bn, R= 80%, vertical slicing)
iT Ci(qi)T Cs
i(qi)T Cπ
π
π
i(qi)
1 69 498.63 14 920.83 54 577.80
2 107 290.00 23 242.98 84 047.02
3 73 910.28 20 250.34 53 659.94
4 87 281.60 18 363.05 68 918.54
5 103 263.27 24 258.03 79 005.24
6 61 463.65 13 051.76 48 411.89
7 430 680.96 86 791.11 343 889.85
8 57 872.23 14 329.25 43 542.97
9 87 604.76 17 413.73 70 191.03
10 80 581.70 18 750.49 61 831.21
11 54 141.87 13 344.04 40 797.83
12 33 736.06 8 449.35 25 286.71
13 50 102.25 10 900.60 39 201.65
14 87 508.74 18 581.82 68 926.92
15 40 467.87 9 596.51 30 871.36
16 89 878.23 21 765.14 68 113.09
17 44 580.36 11 305.76 33 274.60
18 31 915.40 8 457.91 23 457.49
19 65 268.45 15 068.21 50 200.23
20 59 659.28 14 197.75 45 461.53
21 68 639.06 16 781.30 51 857.76
22 29 601.62 9 297.52 20 304.10
23 64 977.96 15 032.80 49 945.16
24 45 645.94 8 760.04 36 885.90
25 83 351.95 19 700.22 63 651.73
26 40 860.80 12 272.79 28 588.01
27 73 414.82 15 350.07 58 064.75
28 64 855.26 15 398.39 49 456.88
29 54 038.96 14 146.18 39 892.79
30 67 823.25 13 199.55 54 623.71
31 35 944.56 9 041.78 26 902.78
32 24 997.62 6 535.50 18 462.12
33 46 297.30 10 914.91 35 382.39
34 117 072.56 27 998.94 89 073.61
35 166 258.54 43 594.46 122 664.07
36 244 729.74 49 036.44 195 693.30
37 45 147.21 11 235.62 33 911.59
38 48 068.44 12 107.24 35 961.21
39 125 825.97 20 199.53 105 626.44
40 61 311.15 12 128.69 49 182.46
41 105 645.33 26 754.59 78 890.75
42 175 033.80 38 486.05 136 547.75
43 105 152.21 23 300.38 81 851.83
44 141 145.26 31 479.28 109 665.98
45 53 687.02 11 136.67 42 550.35
46 148 277.36 33 100.47 115 176.89
47 21 632.44 6 230.10 15 402.34
48 67 548.14 16 714.83 50 833.30
49 47 858.61 10 783.19 37 075.42
50 33 262.97 8 758.19 24 504.78
Total 4 124 811.45 932 514.40 3 192 297.05
All the metrics are expressed in e.
35
Liquidity Stress Testing in Asset Management
Table 22: Break-down of the total transaction cost T C (q) (bond portfolio, TNA = $10 bn,
R= 30%, vertical slicing)
iT Ci(qi)T Cs
i(qi)T Cπ
π
π
i(qi)
1 36 012.29 27 024.52 8 987.77
2 69 885.29 24 800.97 45 084.32
3 82 529.58 18 176.19 64 353.39
4 255 188.46 38 211.29 216 977.17
5 212 256.83 22 015.86 190 240.97
6 199 177.04 30 303.07 168 873.97
7 457 182.39 41 144.08 416 038.31
8 475 933.42 40 815.04 435 118.38
9 448 409.64 40 395.89 408 013.75
10 783 664.66 48 917.74 734 746.92
11 1 897 014.61 103 729.21 1 793 285.40
12 26 113.54 19 565.90 6 547.64
13 43 144.34 32 107.29 11 037.05
14 26 290.67 20 129.36 6 161.31
15 41 572.40 30 568.34 11 004.05
16 23 824.88 18 036.70 5 788.18
17 34 493.76 23 537.39 10 956.36
18 27 033.64 17 468.42 9 565.22
19 24 224.80 11 622.52 12 602.29
20 152 183.10 97 077.07 55 106.03
21 140 308.97 32 858.89 107 450.08
22 43 327.04 29 928.35 13 398.69
23 145 124.09 93 662.76 51 461.33
24 103 949.26 45 925.79 58 023.47
25 168 041.37 93 341.29 74 700.07
26 356 675.94 249 091.31 107 584.63
27 63 633.66 42 415.08 21 218.57
28 94 233.98 60 949.31 33 284.67
29 175 895.36 95 194.98 80 700.38
30 80 175.43 51 658.16 28 517.28
31 112 149.16 31 629.01 80 520.15
32 78 265.60 51 043.74 27 221.86
33 92 505.62 50 775.40 41 730.22
34 170 309.73 87 616.82 82 692.91
35 165 002.62 74 146.13 90 856.49
36 152 015.61 61 150.66 90 864.95
37 168 452.93 93 327.29 75 125.64
38 211 293.24 76 086.97 135 206.27
39 206 312.96 87 322.85 118 990.11
40 150 154.94 73 418.95 76 735.99
41 232 066.72 90 497.82 141 568.90
42 174 454.08 110 679.19 63 774.90
43 517 530.40 235 206.50 282 323.90
44 331 010.74 194 346.51 136 664.23
45 550 434.34 280 040.49 270 393.85
46 270 117.83 99 756.39 170 361.44
47 410 992.48 123 563.71 287 428.78
Total 10 680 569.46 3 321 281.21 7 359 288.25
All the metrics are expressed in $.
36
Liquidity Stress Testing in Asset Management
Table 23: Break-down of the total transaction cost T C (q) (bond portfolio, TNA = $10 bn,
R= 30%, vertical slicing)
hT Ci(q;h)T Cs
i(q;h)T Cπ
π
π
i(q;h)
1 2 474 425.38 662 994.50 1 811 430.88
2 2 474 425.38 662 994.50 1 811 430.88
3 2 088 332.97 625 380.98 1 462 951.99
4 1 671 552.80 521 194.84 1 150 357.97
5 961 444.93 309 791.04 651 653.89
6 298 303.99 130 259.49 168 044.50
7 129 808.94 70 773.73 59 035.21
8 77 556.19 44 594.76 32 961.43
9 43 809.26 25 534.82 18 274.44
10 39 588.86 22 734.00 16 854.86
11 39 588.86 22 734.00 16 854.86
12 39 588.86 22 734.00 16 854.86
13 39 588.86 22 734.00 16 854.86
14 39 588.86 22 734.00 16 854.86
15 39 588.86 22 734.00 16 854.86
16 39 588.86 22 734.00 16 854.86
17 39 588.86 22 734.00 16 854.86
18 39 588.86 22 734.00 16 854.86
19 39 588.86 22 734.00 16 854.86
20 31 558.10 18 425.29 13 132.81
21 20 125.95 13 466.70 6 659.24
22 6 611.72 4 216.50 2 395.22
23 6 611.72 4 216.50 2 395.22
24 113.52 97.57 15.95
25 0.00 0.00 0.00
Total 10 680 569.46 3 321 281.21 7 359 288.25
All the metrics are expressed in $.
37
Liquidity Stress Testing in Asset Management
Figure 13: Liquidation ratio LR(q;h) in % (equity portfolio, vertical slicing)
123
0
20
40
60
80
100
2 4 6 8 10
0
20
40
60
80
100
5 10 15 20
0
20
40
60
80
100
10 20 30 40 50
0
20
40
60
80
100
Figure 14: Liquidation ratio LR(q;h) in % (small-cap portfolio, vertical slicing)
20 40 60 80 100
0
20
40
60
80
100
20 40 60 80 100
0
20
40
60
80
100
20 40 60 80 100
0
20
40
60
80
100
20 40 60 80 100
0
20
40
60
80
100
38
Liquidity Stress Testing in Asset Management
Figure 15: Liquidation time h+=LR−1(q; 99%) in number of trading days (small-cap
portfolio, vertical slicing)
5 10 25 50 75 90
0
100
200
300
400
500
Figure 16: Transaction cost of the large-cap portfolio (TNA = e5 bn)
0 20 40 60 80 100
0
5
10
15
0 20 40 60 80 100
0
10
20
30
0 20 40 60 80 100
0
10
20
30
0 20 40 60 80 100
75
80
85
90
95
39
Liquidity Stress Testing in Asset Management
Figure 17: Normal redemption coverage ratio (bond portfolio, R= 30%, mQ+= 1.0)
012345678910
0
0.5
1
1.5
2
Figure 18: Stressed redemption coverage ratio (bond portfolio, R= 30%, mQ+= 0.5)
012345678910
0
0.5
1
1.5
2
40
