# V488 Per revisited: no strong mid-infrared emission features and no evidence
for stellar/sub-stellar companions

Swetha Sankar email<EMAIL_ADDRESS>
Department of Physics and Astronomy, University of California, Los Angeles, CA
90095-1562, USA  Carl Melis email<EMAIL_ADDRESS>
Center for Astrophysics and Space Sciences, University of California, San
Diego, CA 92093-0424, USA  Beth L. Klein Department of Physics and Astronomy,
University of California, Los Angeles, CA 90095-1562, USA  B. J. Fulton NASA
Exoplanet Science Institute / Caltech-IPAC, Pasadena, CA 91125, USA  B.
Zuckerman Department of Physics and Astronomy, University of California, Los
Angeles, CA 90095-1562, USA  Inseok Song Department of Physics and Astronomy,
University of Georgia, Athens, GA 30602-2451, USA  Andrew W. Howard California
Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

We present characterization of the planetary system architecture for V488 Per,
the dustiest main sequence star known with a fractional infrared luminosity of
$\approx$16%. Far-infrared imaging photometry confirms the existence of an
outer planetary system dust population with blackbody-fit temperature of
$\approx$130 K. Mid-infrared spectroscopy probing the previously-identified
$\approx$800 K inner planetary system dust population does not detect any
obvious solid-state emission features, suggesting either large grain sizes
that mute such emission and/or grain compositions dominated by species like
amorphous carbon and metallic iron which do not produce such features. In the
latter case, the presence of significant quantities of iron-rich material
could be indicative of the active formation of a Mercury-like planet around
V488 Per. In any event, the absence of solid-state emission features is very
unusual among main sequence stars with copious amounts of warm orbiting dust
particles; we know of no other such star whose mid-infrared spectrum lacks
such features. Combined radial velocity monitoring and adaptive optics imaging
find no evidence for stellar/sub-stellar companions within several hundred AU
of V488 Per.

Circumstellar disks (235) — Exoplanet systems (484) – Variable stars (1761)

††facilities: Subaru(COMICS), APF(Levy), Shane(Hamilton), Keck(NIRC2),
Herschel(PACS), TESS

##  1 Introduction

Many main sequence stars are now known to host substantial quantities of dusty
material in their inner planetary systems (e.g., Melis 2016; Moór et al. 2021;
Melis et al. 2021; Absil et al. 2021, and references therein). Where this
inner planetary system dust comes from, what drives its evolution, and what
its fate ultimately will be are matters of active research. Since this dust
could be indicative of processes that might influence the formation,
evolution, and habitability of Earth-like planets, it is essential to develop
a firm understanding of it. Systematic investigation of the dust in these
systems and their greater planetary system architecture will provide insight
into physical processes at play: e.g., how dust is generated and removed and
the impact of dynamical perturbers on the origin and evolution of the dust.

Exceptionally dusty main sequence stars are frequently found to reside in
multiple stellar systems raising the question of whether or not multiplicity
and chaotic dynamics contribute to generating these systems (e.g., Zuckerman
2015; Moór et al. 2021). For two older ($>$1 Gyr) systems with mid-infrared
excess emission indicative of inner planetary system dust, HD 69830 and BD+20∘
307, it does indeed seem like additional planetary or stellar companions play
a role in either destabilizing or corralling dust-producing rocky bodies
leading to the unusually high levels of dust (e.g., Zuckerman et al. 2008;
Payne et al. 2009).

Nesvold et al. (2016) study how a stellar-mass perturber orbiting exterior to
and inclined to a planetesimal disk can excite the disk by the Kozai-Lidov
mechanism resulting in dust-producing collisions. Moór et al. (2021) discuss
how a widely orbiting stellar companion may act via Kozai-Lidov mechanisms to
send outer planetary system bodies on inner planetary system-crossing orbits
where they may release dust via disruption through sublimation or collisions.

Planetary mass companions orbiting a star may undergo a dynamical instability
similar to the Late Heavy Bombardment hypothesized to have occurred in the
solar system (e.g., Gomes et al. 2005). Bonsor et al. (2013, 2014) investigate
such scenarios numerically, finding that sustained dust production in the
inner planetary system can only be achieved when a planet migrates into a
planetesimal belt. Fujiwara et al. (2009, 2012b) explore such a possibility in
two stars with dusty inner planetary systems, rejecting it as an explanation
for HD 15407A due to the lack of a detectable outer planetary system disk.
However, Lisse et al. (2012) strongly suggest such a model is appropriate for
$\eta$ Crv based on a detected outer planetary system disk and possible water-
ice features seen in its disk mid-infrared spectrum. Melis et al. (2021)
invoke instabilities in tightly-packed inner planetary systems of small
planets (Earth to Neptune-sizes) to produce the observed very dusty main
sequence stars.

Collisionally-produced dust particles with sizes $\lesssim$1 $\mu$m typically
generate solid-state emission features around 10 $\mu$m. Characterization of
solid-state emission features can provide insight into where the dust in
extreme debris disks originates from and what physical processes play a role
in its evolution (e.g., Honda et al. 2004; Song et al. 2005; Chen et al. 2006;
Lisse et al. 2007, 2008, 2009, 2012, 2017, 2020; Rhee et al. 2007, 2008;
Currie et al. 2011; Johnson et al. 2012; Olofsson et al. 2012; Melis et al.
2013; Ballering et al. 2014; Morlok et al. 2014; Mittal et al. 2015). For
example, highly energetic impacts that result in the production of crystalline
silicates or silica (e.g., Lisse et al. 2009) would be suggestive of
dynamically hot populations of mature rocky bodies pointing to the need for
eccentricity excitation mechanisms.

Thus far, most extreme debris disk stars have mid-infrared spectral features
from silicate species indicating large quantities of highly processed material
(crystalline and silica grains) in addition to less processed amorphous
species (e.g., Figure 1). However, it is worth noting three unusual systems
that might be host to rather different dusty material (Melis et al. 2013;
Lisse et al. 2017). Systems with atypical dust properties could provide
information about unique dust stoichiometry and hence composition or dust
production processes. Thorough characterization and modeling of mid-infrared
spectral features for the dustiest main sequence stars is essential in
developing a complete picture for the types of compositions at a mineralogical
level these systems host and the dynamical conditions that lead to their
production through collisional processes (e.g., Lisse et al. 2009, 2012; Meng
et al. 2014; Su et al. 2020).

We seek to obtain constraints on the overall planetary system architecture for
the dustiest main sequence stars known. V488 Per, a Solar-type star belonging
to the $\approx$80 Myr old $\alpha$ Persei cluster (Soderblom et al., 2014),
is of particular interest as it hosts what is currently the largest known
fractional infrared luminosity for a main sequence star of
$\tau$($=$LIR/Lbol)$\approx$16$\%$ (Zuckerman et al., 2012). Basic stellar
properties for V488 Per are given in Table 1. Zuckerman et al. (2012) suggest
that V488 Per is possibly host to two separate belts of dust with temperatures
of $\approx$820 K and 120 K, but the basis for the cooler dust component was a
single WISE excess measurement at 22 $\mu$m.

In this paper, we present new data that further characterize the dust and
planetary system architecture of V488 Per. Section 2 details the new data
sets, Section 3 discusses analyses based on measurements from the data, and
Section 4 discusses possible interpretations and future observations to
investigate them.

##  2 Observations

###  2.1 Optical Spectroscopy

Optical echelle spectra for V488 Per and a radial velocity standard of similar
temperature class, HR 124 (Nidever et al., 2002), were collected at Lick
Observatory with the Automated Planet Finder (APF) telescope and Levy
spectrograph (Vogt et al., 2014), as well as the Shane telescope and Hamilton
spectrograph (Vogt 1987; Pakhomov & Zhao 2013). We obtained these data to
search for doppler signatures from any companions to V488 Per at separations
that could not be probed with imaging techniques; the iodine cell was not used
due to the optical faintness of V488 Per for these telescopes (Table 1). Two-
dimensional raw frames are bias subtracted, flat-fielded, extracted into one-
dimensional spectra through straight summing of flux inside an aperture
defined for each individual echelle order, and finally wavelength calibrated
with comparison lamp spectra. Radial velocities are measured through cross-
correlation with HR 124 and corrected to the heliocentric reference frame.
Table 2 reports observation epochs, data quality, and measured velocities
while Figure 2 presents measurements and associated analysis.

###  2.2 TESS

To help assess the stellar inclination angle and hence better interpret
doppler mass limits by removing the sin$i$ ambiguity, we downloaded and
analyzed TESS lightcurve data for V488 Per and other $\alpha$ Per stars with
similar spectral type and spectroscopic projected rotational velocities
(Stauffer et al., 1985). All lightcurves are generated from Full Frame Imaging
data acquired in Sector 18 from 2019 Nov 03 to 2019 Dec 16. Data products are
MIT Quick Look Pipeline (QLP; Huang et al. 2020a, b) lightcurves as downloaded
from MAST. Some times with bad data are removed by hand before analysis.
Figure 3 shows TESS lightcurves used for this work.

###  2.3 Adaptive Optics

Diffraction-limited thermal-infrared adaptive optics (AO) imaging of V488 Per
was obtained at Keck Observatory with the facility AO system (Wizinowich et
al., 2000; Wizinowich, 2013) and NIRC2 camera on UT 2013 Feb 03. We conducted
observations using the L′ filter (central wavelength and bandpass of
3.776$\pm$0.700 $\mu$m) and the narrow camera setting, resulting in a plate
scale of 0.009952 ′′ pixel-1 (Yelda et al., 2010). We obtained these data to
further rule out possible contamination of V488 Per by a background object and
then to perform a shallow search for possible companion stars. The excess
emission level for V488 Per at a wavelength of $\approx$3.8 $\mu$m is just
over twice the stellar photosphere emission level (Figure 4), thus any
contaminating object that would be responsible for the apparent infrared
excess emission toward V488 Per would need to be of comparable brightness as
V488 Per (the star) alone.

Seven dithered exposures of 3.0 seconds each (0.3 seconds integration $\times$
10 coadds) were obtained resulting in a total integration time of 21 seconds.
Background-subtracted and flat-fielded individual images are registered, then
combined to produce a final image for analysis which is displayed in Figure 2.

###  2.4 COMICS Spectra

We collected COMICS (Kataza et al. 2000; Okamoto et al. 2003) R$\approx$170
mid-infrared spectroscopic data at the Subaru 8.2 m Telescope on UT 06 Nov
2014. The night featured seeing in the mid-infrared of $\approx$0.4′′ and a
precipitable water vapor value of $\sim$4 mm. Observations and data reduction
follow Su et al. (2020), important aspects of observations for V488 Per are
discussed here.

The observing strategy for V488 Per started with observations of the
calibrator star HD 21552 (K3 III), a sequence on V488 Per lasting $\approx$90
minutes of wall-clock time, a second visit to HD 21552 followed by another
$\approx$90 minute sequence of observations on V488 Per, then a final visit to
HD 21552. All exposures for both stars were chopped in an ABAB pattern keeping
all beams on slit. V488 Per is faint for ground-based mid-infrared astronomy
(Figure 4), so each saveset obtained for it had 222 total exposures or 111
chop pairs, the accumulation of which was sufficient to see the spectral
signal. By comparison, the bright calibrator only needed 24 total exposures or
12 chop pairs to obtain high ($\sim$50) signal-to-noise. During reduction,
each chop pair is differenced and examined for quality. Those chop pairs
showing strong residual structure across the detector (indicating rapidly
varying background conditions) are discarded. Chop pairs that are kept are
then rectified along the spatial axis such that night sky emission lines run
vertically only (initially they are slightly tilted horizontally). A second
pass of background removal is then done by subtracting at each dispersion
pixel location the median value of background-only pixels along the spatial
axis. At the end of assessment, roughly two-thirds of the first spectral
sequence on V488 Per was kept and roughly one-third of the second spectral
sequence.

Fully processed chop pairs in each sequence are median-combined and the
positive and negative spectral beams are extracted via a straight aperture
sum. Uncertainties on spectral samples are calculated by determining the rms
of background-only pixels and summing that in quadrature for the number of
pixels in the aperture and the Poisson noise on the total summed flux in the
aperture. This is done for each of the two target spectral sequences and for
spectra from each visit to the calibration star.

For each visit to the calibrator we scaled and combined positive and negative
beam spectra with a weighted mean, then used one spectrum from the set of
three visits to correct the telluric absorption in each positive and negative
spectrum for V488 Per. Telluric correction is done by dividing a science
spectrum by one of the calibrator visit spectra shifted in wavelength to
provide the best cancellation of the strong $\sim$9.5 $\mu$m ozone featue. For
each science spectrum, we selected the calibrator spectrum that resulted in
the lowest rms residuals from telluric features; in practice this was the
calibrator spectrum obtained in the middle of the two sequences on V488 Per.

Corrected science target positive and negative beam spectra in each sequence
were then scaled and combined via weighted mean. A wavelength scale is
determined based on a low-order polynomial fit to the position of known bright
sky emission lines. A Rayleigh-Jeans slope is used as an approximation for the
calibrator spectral shape and applied to the corrected science target spectra
to arrive at relative flux-calibrated spectra (i.e., the spectral shape is
robust, but the absolute flux level is not).

To obtain an accurate absolute flux scaling for the spectra for each sequence
we measured photometry from images of the target and calibrator. A short
imaging acquisition sequence with the N12.4 filter (bandpass of 12.4$\pm$1.2
$\mu$m) was obtained for V488 Per and the calibrator star HD 21552 in
chopping-only mode. Chop pairs were differenced to remove background emission
and aperture photometry performed on the science target and calibrator star.
We note that only a single point source was seen in the V488 Per field. With
an adopted flux density for HD 21552 of 9.11 Jy (extrapolated from the WISE-
measured 11.56 $\mu$m flux density of 10.48$\pm$0.11 Jy), we measure a 12.4
$\mu$m flux density for V488 Per of 60$\pm$16 mJy. Since this is consistent
within the errors with the WISE 11.56 $\mu$m flux measurement for V488 Per of
41.0$\pm$0.6 mJy, we adopt the WISE value to set the spectrum absolute flux
level. These absolute flux scales are accurate at the $\approx$10% level (this
error source is not included in the spectral uncertainties).

A final N-band spectrum for V488 Per is then obtained by averaging the two
separate sequences together. The final spectrum signal-to-noise level is
$\approx$6 near 11.5 $\mu$m.

###  2.5 Herschel Imaging

Simultaneous far-infrared observations at at 70 and 160 $\mu$m were obtained
with the ESA Herschel Space Observatory (Pilbratt et al., 2010) using the PACS
imaging photometer (Poglitsch et al., 2010). V488 Per was observed on UT 2012
Sep 11 (Proposal OT2_cmelis_3; OBSIDs 1342250847 and 1342250848) in Mini Scan
map mode. Scan legs of 3′ length were observed at medium speed with two
orientation angles of 70 and 110∘ in the array coordinates. High level image
products are obtained from the Herschel Science Center and aperture photometry
is performed on these images as described in Vican et al. (2016). We obtain a
70 $\mu$m flux density in agreement with the Herschel PACS Point Source
catalog (Herschel Point Source Catalogue Working Group et al., 2020) and adopt
their value; we report for V488 Per a 70 $\mu$m flux density of 66.5$\pm$6.5
mJy and 160 $\mu$m flux density of 22$\pm$10 mJy.

##  3 Analysis and Discussion

###  3.1 Search for Companions

Figure 2 reveals the overall stability of the radial velocity measurements
made at Lick Observatory. A periodogram search does not reveal any signals
hidden in the noise, and with no obvious signal present in the data we proceed
to estimate mass sensitivity limits. Radial velocity measurements given in
Table 2 are also consistent with values previously measured for V488 Per
(Stauffer et al., 1985; Mermilliod et al., 2008; Zuckerman et al., 2012). The
measurement from Mermilliod et al. (2008), taken on HJD 2449655.560 (UT 1994
Oct 30) and having a value of $-$0.31$\pm$0.15 km s-1, is incorporated into
our companion sensitivity analysis. The other two measurements are not
utilized as they either did not have information regarding time of observation
(Stauffer et al., 1985) or because of inaccuracies resulting from a much later
spectral type radial velocity standard used for cross correlation (Zuckerman
et al., 2012).

To estimate mass sensitivity limits as a function of orbital separation we
utilized The Joker (Price-Whelan et al., 2017) and adopted a stellar mass of
0.84 M⊙ (Table 1). The Joker is designed to characterize a two body system
with the requirement that the source behaves similar to a single-lined
spectroscopic binary and exhibits variability in radial velocity measurements.
It takes the input of multi-epoch radial velocity data with errors and by
Monte-Carlo analysis produces a suite of possible fits for the data. We
conducted a set of The Joker runs with a uniform prior distribution covering a
variety of orbital period ranges; in all runs, priors on the velocity semi-
amplitude and the systemic velocity were taken to be Gaussians with $\sigma$
of 3 km s-1 and 75 km s-1, respectively. Discrete orbital period range bins
are constructed out to a maximum period of 60,000 days ($\approx$30 AU for a
circular orbit). For each run, 103 priors are generated but only a subset
survive the rejection sampling step of The Joker. Figure 2 presents some
example orbital fits that are consistent with the available radial velocity
data. Five runs per period range bin are executed and a mass sensitivity for
that bin is calculated from the median of the maximum masses obtained from
each of the five runs. The resulting mass-period sensitivity curve from these
samplings is displayed in Figure 2; it is noted that mass sensitivities
obtained in this way are $m$sin$i$.

Radial velocity mass limits become increasingly insensitive with orbital
period, especially beyond the maximum time baseline covered as shown in Figure
2. To supplement mass sensitivities at larger orbital separations we make use
of the AO data. One other star was observed with the L′ filter on the same
night as V488 Per, but at a different time and significantly different airmass
and as such point-spread-function (PSF) subtraction was not fruitful. However,
in general the PSF of V488 Per matched that of the other observed star
indicating a single source. Figure 2 shows what we conclude is a single,
diffraction-limited point source in the L′ AO image.

There is no seeing-limited halo evident in the final image and thus the
sensitivity beyond the first airy ring is determined only by the total amount
of integration time covered at that position. The best sensitivity is obtained
in an annulus with inner radius just outside the first airy ring at 0.19′′ to
an outer radius of 4.72′′ from V488 Per. Adopting the Gaia EDR3 parallax
(Table 1) results in a distance to V488 Per of 173.5$\pm$0.4 pc and
separation-space probed by the L′ AO image of 33 to 820 AU.

Interpolating between the WISE W1 and W2 magnitudes, we obtain an L′ magnitude
for V488 Per of $\approx$9\. Using V488 Per as a comparison star, we estimate
that we are sensitive to point sources as faint as L′$\sim$13 magnitudes at
the 5$\sigma$ level in the best sensitivity areas of the AO image. Consulting
Leggett (1992) and Chabrier et al. (2000), and using the adopted age for the
$\alpha$ Per cluster of 80 Myr (see Section 1), we obtain mass limits of
$\sim$0.15$\,M_{\sun}$; we show this mass limit in relation to the doppler
monitoring mass sensitivities in Figure 2. This mass limit is more sensitive
than the $\sim$0.6 M☉ limit obtained from HST 1.3 $\mu$m imaging reported in
Patience et al. (2002).

We do not assess binarity of V488 Per beyond 800 AU; Moór et al. (2021)
discuss problems with the widely-separated candidate companions identified in
Zuckerman (2015).

####  3.1.1 Stellar Inclination

It is desirable to try and remove the sin$i$ ambiguity in the doppler mass
sensitivity limits. Without any detected companions (nor resolved imaging of
the circumstellar disks) it is not possible to assess the inclination angle of
the planetary system around V488 Per. As a proxy, we will use the inclination
angle of the stellar rotation axis, while noting that it need not be the case
that the two axes are aligned.

With a measurement of the rotation period of V488 Per from photometric
monitoring and the rotational broadening of the stellar photospheric
absorption lines from APF data we can obtain the stellar inclination angle.
From the UT 2017 Dec 08 APF spectrum we measure a $v$sin$i$ value of
4.5$\pm$1.5 km s-1 following the methodology of Strassmeier et al. (1990). To
obtain the stellar rotation period we analyzed TESS lightcurve data. However,
the TESS data reveal only mild variability for V488 Per, not the clear
periodic spot modulation that would be expected for a young, Sun-like star
(Figure 3).

The result from TESS stands in stark contrast to variability measurements for
V488 Per($=$AP 70) in the literature. Past ground-based observations covering
multiple rotational periods clearly detect rotational modulation of starspots
on the surface of V488 Per, finding rotational periods of 123.5 hours (or 5.15
days) in Stauffer et al. (1985), 6.4$\pm$0.1 days in Allain et al. (1996), and
5.98 days in Heinze et al. (2018) (ATO J052.0779+48.6632, ATLAS ObjID
166390520779906601).

To place V488 Per in context, we examined TESS lightcurve data for other
$\alpha$ Per stars with measured rotational periods in Stauffer et al. (1985)
and Allain et al. (1996) that had B$-$V colors and $v$sin$i$ comparable to
V488 Per (we considered stars with $v$sin$i$$\lesssim$10 km s-1 from Stauffer
et al. 1985). A comparison of these stars’ TESS lightcurves to that for V488
Per is shown in Figure 3. Each of the additional examined stars shows clear
periodic variability with periods on the order of 4-6 days.

Why does V488 Per not exhibit similar behavior as other $\alpha$ Per stars in
the TESS data? We believe the answer is starspot cycles, and the star AP 14
shown in Figure 3 demonstrates the evolution of spot modulation amplitude over
the TESS monitoring period ($\approx$27 days). In this interpretation, we just
happened to catch V488 Per with TESS at the minimum of its starspot cycle
which had a duration of at least $\sim$1 month. AP 14 was captured during its
decay phase which probably also took on the order of at least $\sim$1 month.
The other stars shown in Figure 3 were seen during times of near-maximum
activity.

Based on this interpretation, we adopt the average of literature-measured
rotational periods for V488 Per of 5.8 days. To calculate the stellar
inclination angle, we further adopt the stellar radius of 0.76 R⊙ from the
spectral energy distribution fit in Figure 4. With these values we calculate a
stellar inclination angle of 45$\pm$12∘. This suggests that the spectroscopic
mass sensitivity values shown in Figure 2 should be increased by a factor of
$\sim$1.4 if the stellar and planetary system inclination angles are the same.

###  3.2 Circumstellar Dust Properties

While the AO image is not capable of characterizing the dust around V488 Per,
it does effectively eliminate any possibility that the excess emission is due
to a background contaminating object. Any such object would be as bright as
V488 Per in the L′-band (Figure 4) and would be obvious even at separations
interior to the first airy ring. We conclude with high confidence that all
infrared excess emission seen toward V488 Per is associated with the star.

####  3.2.1 Dust Architecture

Figure 4 collects optical to far-infrared photometric measurements for V488
Per and fits them with a simplistic model consisting of a stellar photosphere
and two blackbody components. The Herschel data clearly confirms that there
exists a cooler dust population in the outer planetary system of V488 Per that
was hinted at previously from WISE 22.09 $\mu$m photometry (Zuckerman et al.,
2012). The blackbody temperature of this outer dust component is reasonably
constrained to $\approx$130 K while the inner dust population is fit
acceptably with a blackbody having temperature of $\approx$800 K.

We interpret the two dust populations as separate rings of material within the
V488 Per planetary system. This interpretation is motivated by past results
indicating separate inner and outer planetary system debris disks for other
multiple-blackbody component systems (e.g., Backman et al. 2009; Morales et
al. 2009; Su et al. 2009; Kennedy & Wyatt 2014; Vican et al. 2016). For the
luminosity of V488 Per (calculated with the spectral energy distribution-fit
temperature and radius in L$=$4$\pi$R2T4; see Table 1) this would place
blackbody-emitting grains having the above temperatures at orbital separations
of $\sim$0.07 AU and $\sim$2.7 AU. The widths of these two putative disks are
not well-constrained with the available data and can be better characterized
if the grain composition can be determined (see below) and/or if resolved
imaging can be obtained.

####  3.2.2 Grain Composition

Typically, strong solid-state emission features are seen in the mid-infrared
spectra of exceptionally dusty main sequence stars (e.g., Honda et al. 2004;
Chen et al. 2006; Lisse et al. 2008; Melis et al. 2010; Olofsson et al. 2012;
Lisse et al. 2017; Melis et al. 2021). V488 Per is thus unusual in not showing
any obvious features in its COMICS N-band spectrum (Figures 1 and 5). Figure 1
compares the COMICS mid-infrared spectrum of V488 Per with Spitzer IRS spectra
of exemplar extreme debris disk systems having well-detected solid-state
emission features. To date, stars having fractional infrared luminosities
$\gtrsim$1% all have strong solid-state emission features present in their
mid-infrared spectra that can be ruled out in the spectrum of V488 Per
(several new extremely dusty main sequence stars have yet to be observed with
mid-infrared spectrometers; e.g., those stars from Gaidos et al. 2019, Tajiri
et al. 2020, and Moór et al. 2021). This stands for all types of dust grain
species currently known, whether they be silica-rich (e.g., Chen et al. 2006;
Rhee et al. 2008; Melis et al. 2010; Fujiwara et al. 2012a), dominated by
amorphous and/or crystalline silicates (e.g., Honda et al. 2004; Song et al.
2005; Chen et al. 2006; Rhee et al. 2007; Olofsson et al. 2012; Melis et al.
2021), or stars with yet-to-be-identified emission features (e.g., Melis et
al. 2013; Lisse et al. 2017). When compared against less dusty stars
(fractional infrared luminosities $\sim$0.01%), it is possible that the noise
in the COMICS spectrum of V488 Per could hide weaker solid-state emission
features. This is evidenced by comparison to HD 69830 (e.g., Beichman et al.
2005) in Figure 1, and is similarly true for other such systems (e.g., Chen et
al. 2006; Moór et al. 2009; Mittal et al. 2015). While it is possible that
some weaker solid state emission features could be hidden in the COMICS
spectrum presented for V488 Per, it remains the case that it is unusual when
compared against other stars having huge amounts of inner planetary system
dust ($\tau$$\gtrsim$1%).

We discuss some possible explanations for why no strong solid state emission
feature is seen in the COMICS spectrum of V488 Per and elaborate on them
further here and in Section 4. To generate the observed featureless mid-
infrared spectrum, either 1) only a weak solid-state emission feature is
present as discussed above; 2) dust grains are larger than $\sim$10 $\mu$m in
size, at which point solid-state emission features near a wavelength of 10
$\mu$m become severely muted (e.g., Figure 5; see also Min et al. 2007 and
Section 4.3 of Olofsson et al. 2012); and/or 3) the dust composition could be
dominated by species which do not produce obvious mid-infrared solid-state
emission features, like amorphous carbon or metallic iron. Possibility 2)
seems highly unlikely given that the observed dust is almost certainly
produced via collisions of rocky bodies which should eventually be ground down
to radiation blow-out sizes ($\lesssim$0.1 $\mu$m for V488 Per; e.g.,
Artymowicz 1988) before being removed from the planetary system.

To explore possibility 1) in more detail, we constructed solid-state emission
feature models for V488 Per that would be consistent with the observed COMICS
spectrum. In these models, we utilize a blackbody component for the disk
continuum emission and an emission feature component computed from the linear
combination of dust absorption coefficients multiplied by a blackbody; the
solid angle of each component is adjusted to find agreement with the COMICS
spectrum. We use dust absorption coefficients from Min et al. (2007) for
silicates and silica grain species. Since emission from amorphous carbon or
metallic iron is indistinguishable from continuum emission in the 8-13 $\mu$m
wavelength region (e.g., Figure 2 of Chen et al. 2006; Figure 3 of Lisse et
al. 2008), we do not explore those grain species and cannot constrain their
presence. Figure 5 shows example dust emission models that are consistent with
the COMICS data. There are a wide range of models that could be consistent
with the data, but all such models have a total solid-state emitting dust
grain mass $\lesssim$1020 g (a couple times the mass of the asteroid Ida). In
comparison, other extremely dusty main sequence stars ($\tau$$\gtrsim$1%) have
been found to have masses for their solid-state emitting grains in the range
of 1021-1023 g (e.g., Chen et al. 2006; Lisse et al. 2008; Lisse et al. 2009).
Systems like HD 69830 ($\tau$$\sim$0.01% and whose mid-infrared solid-state
emission features could hide in the V488 Per COMICS spectrum) have solid-state
emitting grain dust masses that encompass the limits obtained for the models
shown in Figure 5, 1019-1021 g (e.g., Beichman et al. 2005; Chen et al. 2006;
Lisse et al. 2007, 2012).

If V488 Per is like other extremely dusty main sequence stars and hosts a
small grain population with mass of $\gtrsim$1021 g, then these small grains
must not produce strong solid-state emission features in the 8-13 $\mu$m
range. This limits the contribution from typically-seen silicates and silica
to $<$1020 g and requires an order of magnitude more mass of dust from species
like amorphous carbon and metallic iron. Given that inner planetary systems
(including our own) are found to be deficient in carbon (e.g., Jura 2006 and
references therein), we give preference to a scenario where these dust grains
are predominantly metallic iron.

##  4 Conclusions

We present an investigation into the overall architecture of the V488 Per
planetary system; Figure 2 shows companion sensitivity limits, while Figures 4
and 5 summarize dust properties.

Blackbody fits to infrared excess measurements indicate two dust populations
having temperatures of 800 and 130 K, or orbital separations of roughly 0.07
and 2.7 AU respectively. To constrain the true extent of these dust
populations within the V488 Per planetary system will require additional data.
Longer wavelength observations (e.g., with ALMA) can determine the outermost
extent of the cool dust population while optical/infrared interferometric
observations might be capable of assessing the presence of very close-in dust.

Assuming a separated two-dust-belt interpretation is appropriate, we limit any
companions within the gap between the belts to have mass $\lesssim$10 MJup.
This limit could be improved with precision radial velocity monitoring over a
few year duration. Stellar-mass companions are effectively ruled out for most
separations $<$800 AU from V488 Per. Deeper high spatial resolution imaging
(PSF full-width at half-max $\lesssim$50 mas) or sparse-aperture masking
interferometry could push to sub-stellar mass limits in this region.

The lack of any obvious solid-state emission features in the COMICS mid-
infrared spectrum could be suggestive of dust particles composed primarily of
metallic iron. If true, this might indicate the erosive bombardment and
subsequent grinding down of ejecta from something resembling the deep interior
of a differentiated planet. Clement et al. (2021) discuss how such a scenario
may have played out during the formation of Mercury in the solar system. It is
more likely to obtain such a scenario in exoplanetary systems with inner
planetary systems tightly packed with Super-Earth/Earth-type planets (e.g., Pu
& Wu 2015; Izidoro et al. 2019; Clement et al. 2021, and references therein),
and as such it is not unreasonable to suggest (based on the available data)
that V488 Per could potentially be in the process of forming a Mercury-like
planet. Planned GTO JWST/MIRI observations of V488 Per will be valuable in
assessing the presence and variability of any solid-state emission features
and thus further exploring this hypothesis.

C.M. acknowledges support from NASA ADAP grant 18-ADAP18-0233. B.K.
acknowledges support from the APS M. Hildred Blewett Fellowship. AWH
acknowledges NSF grant AST-1517655. We thank the anonymous referee for useful
comments that helped improve this paper. The authors wish to thank Bradford
Holden for assistance in scheduling the APF observations presented in this
paper. We thank George Rieke for useful discussion. Research at Lick
Observatory is partially supported by a generous gift from Google. Some of the
data presented herein were obtained at the W.M. Keck Observatory, which is
operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W.M. Keck Foundation. The authors wish to recognize
and acknowledge the very significant cultural role and reverence that the
summit of Maunakea has always had within the indigenous Hawaiian community. We
are most fortunate to have the opportunity to conduct observations from this
mountain. Similarly, we acknowledge that Lick Observatory resides on land
traditionally inhabited by the Muwekma Ohlone Tribe of Native Americans. This
paper includes data collected by the TESS mission, which are publicly
available from the Mikulski Archive for Space Telescopes (MAST). Funding for
the TESS mission is provided by NASA’s Science Mission directorate. This work
is based in part on observations made with Herschel, an ESA space observatory
with science instruments provided by European-led Principal Investigator
consortia and with important participation from NASA. Support for Herschel
work was provided by NASA through an award issued by JPL/Caltech. This
research has made use of NASA’s Astrophysics Data System, the SIMBAD database,
and the VizieR service.

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Figure 1:  Mid-infrared spectra of V488 Per and exemplar main sequence stars
with warm inner planetary system dust. Each spectrum is normalized by a
blackbody model fit to regions of the spectrum that do not have solid-state
emission features, typically near 7.8 $\mu$m and 13.5 $\mu$m. The signal-to-
noise ratio per pixel for the exemplar spectra range between roughly 100-200
and as such the error bars for those spectra are comparable to the thickness
of the lines used to plot them. The fractional infrared luminosity
($\tau$$=$LIR/Lbol) of each source shown is given in the legend. HD 113766
(e.g., Chen et al. 2006; Lisse et al. 2008) is an exemplar for amorphous and
crystalline solid-state emission features seen in stars with fractional
infrared luminosities $\gtrsim$1%; such sources present a range of solid-state
emission feature peak strengths of 2-5$\times$ the continuum level. HD 15407A
(e.g., Melis et al. 2010; Fujiwara et al. 2012a) is an exemplar for silica-
dominated disks around main sequence stars; such sources present solid-state
emission feature peak strengths $\approx$1.5$\times$ the continuum level. HD
69830 (e.g., Beichman et al. 2005; Lisse et al. 2007) is an exemplar for the
types of emission features seen around main sequence stars with lower
fractional infrared luminosities ($\sim$0.01%). Section 3.2.2 provides more
discussion.

Figure 2:  Radial velocity monitoring and adaptive optics imaging results and
mass sensitivity limits derived from them. Top Left Panel: Time series of
radial velocity measurements for V488 Per (black dots with error bars) with a
selection of representative The Joker allowable solutions overplotted (blue
dotted curves). t0 is HJD 2457448.75332 (UT 2016 Mar 01), the date of the
first observation of V488 Per at Lick Observatory. The median radial velocity
for all measurements shown is $-$0.423 km s-1. Top Right Panel: Keck AO L′
image of V488 Per; the first airy ring is clearly visible around the star.
North is up and East is left, no other sources are detected in the imaged
field of view. Bottom Panel: Mass sensitivity limits obtained as described in
Section 3; masses are $m$sin$i$ only for the spectroscopic curve covering
semi-major axes $<$30 AU, there is no inclination angle ambiguity for the AO
imaging mass sensitivity line shown for semi-major axes $>$30 AU. Dots are
from allowable solutions obtained with The Joker, the hatched region shows
companions that would not have been detected with AO imaging; above the solid
curve we expect we would have detected any such companions if they were
present.

Figure 3:  TESS light curves of V488 Per and other $\alpha$ Per stars having
similar B$-$V colors and $v$sin$i$ as V488 Per. The right column shows
periodograms calculated from these lightcurves; a marginal period around 5
days is possible for V488 Per while clear periods of 4-6 days are found for
the other stars. Of note is AP 14, which shows a reduction in the amplitude of
its periodic signal over the TESS monitoring period and ends with variability
levels comparable to V488 Per.

Figure 4:  Spectral energy distribution including the COMICS mid-infrared
spectrum of V488 Per. The thin black curve connecting optical and near-
infrared data points is a synthetic stellar atmospheric spectrum with
temperature and radius as indicated on the figure panel. The dotted curve is a
sum of two blackbodies having temperatures of 800 and 130 K. The thick black
curve is the sum of all the individual model components. Data points used to
inform the stellar photospheric model are AAVSO (Henden et al., 2015), Gaia
photometry (Gaia Collaboration et al., 2020) and 2MASS magnitudes (Skrutskie
et al., 2006). Excess measurements are from WISE (Cutri & et al., 2012), IRAC
from Zuckerman et al. (2012), and Herschel (in green, see text). The blue
curve between the IRAC 7.8 $\mu$m and WISE 11.56 $\mu$m measurements is the
COMICS N-band spectrum. Vertical lines in data points indicate the measurement
uncertainty. Some measurement uncertainties are smaller than the point sizes
on the plot.

Figure 5:  Example solid-state emission models that are consistent with the
V488 Per COMICS spectrum. In each panel, the V488 Per spectrum is shown in
grey (spectral samples and associated uncertainties), orange dotted curves are
800 and 130 K blackbodies, purple and blue dashed curves are amorphous and
crystalline silicates respectively, and the red dot-dashed curve is the sum of
all components shown. We do not include amorphous carbon or metallic iron in
these models as their emission is indistinguishable from that of the continuum
components; it is not possible to constrain amorphous carbon or metallic iron
with the available data (see Section 3.2.2). Top Panel: Small dust grain
emission model where the amorphous and crystalline silicates have a size of
0.1 $\mu$m. This model has a total mass of solid-state emitting grains of
$\approx$4$\times$1019 g, or the mass of the asteroid Ida. While this model is
a reasonable match to the data, we caution that the various wiggles seen in
the spectrum could instead be related to its low signal-to-noise ratio; e.g.,
such wiggles could be from low-level, low-frequency noise in the detector
electronics. Bottom Panel: Large dust grain emission model where the amorphous
and crystalline silicates have a size of 4.0 $\mu$m. The muting effect of the
larger grain sizes on the solid-state emission features is evident. For grain
sizes $>$5 $\mu$m no clear peaks from the solid-state transitions are present
and it is difficult to distinguish them from the continuum (see discussion in
Section 3.2.2). This model has a total mass of solid-state emitting grains
that is a few times the mass of Ida.

Table 1: V488 Per Stellar Parameters  Parameter | Value | Reference
---|---|---
Vmag |  12.89$\pm$0.05 | Henden et al. 2015
$\pi$ [mas] |  5.763$\pm$0.012 | Gaia Collaboration et al. 2020
Teff [K] | 4700 |  This paper, Figure 4
$R_{\star}$ [$R_{\sun}$] | 0.76 |  This paper, Figure 4
$L_{\star}$ [$L_{\sun}$] | 0.25 | This paper
$M_{\star}$ [$M_{\sun}$] | 0.84 |  Wright et al. 2011; Stassun et al. 2019
$i$ [∘] |  45$\pm$12 |  This paper, Section 3.1.1
Table 2: Lick Observatory Radial Velocity Measurements

UT Date | HJD | Instrument |  SNRa |  Vr
---|---|---|---|---
|  |  |  |  km s-1
2016 Mar 01 | 2457448.75332 | APF | 10.14 |  –0.383 $\pm$ 0.244
2016 Nov 01 | 2457716.90075 | APF | 18.03 |  –0.433 $\pm$ 0.160
2017 Aug 21 | 2457987.00913 | APF | 19.81 |  –0.578 $\pm$ 0.155
2017 Sep 18 | 2458014.75714 | APF | 21.09 |  –0.420 $\pm$ 0.158
2017 Oct 04 | 2458030.81506 | APF | 19.66 |  –0.530 $\pm$ 0.164
2017 Oct 30 | 2458056.78022 | APF | 21.20 |  –0.423 $\pm$ 0.158
2017 Dec 08 | 2458095.88214 | APF | 20.91 |  –0.490 $\pm$ 0.142
2018 Feb 08 | 2458157.68050 | APF | 19.31 |  –0.375 $\pm$ 0.142
2018 Oct 17 | 2458408.78883 | APF | 22.19 |  –0.436 $\pm$ 0.113
2018 Dec 13 | 2458465.72084 | APF | 11.68 |  –0.410 $\pm$ 0.257
2020 Aug 07 | 2459068.99811 | Hamilton | 18.60 |  –0.409 $\pm$ 0.244

aThe signal-to-noise ratio per pixel, measured near 6475 Å.


# Loan portfolio management and Liquidity Risk: The impact of limited
liability and haircut

Deb Narayan Barik  Department of Mathematics, Indian Institute of Technology
Guwahati, Guwahati-781039, India, e-mail<EMAIL_ADDRESS>Siddhartha P.
Chakrabarty  Department of Mathematics, Indian Institute of Technology
Guwahati, Guwahati-781039, India, e-mail<EMAIL_ADDRESS>Phone:
+91-361-2582606

###### Abstract

In this article, we consider the problem of a bank’s loan portfolio in the
context of liquidity risk, while allowing for the limited liability protection
enjoyed by the bank. Accordingly, we construct a novel loan portfolio model
with limited liability, while maintaining a threshold level of haircut in the
portfolio. For the constructed three-time step loan portfolio, at the initial
time, the bank raises capital via debt and equity, investing the same in
several classes of loans, while at the final time, the bank either meets its
liabilities or becomes insolvent. At the intermediate time step, a fraction of
the deposits are withdrawn, resulting in liquidation of some of the bank’s
assets. The liquidated portfolio is designed with the goal of minimizing the
liquidation cost. Our theoretical results show that model with the haircut
constraint leads to lesser liquidity risk, as compared to the scenario of no
haircut constraint being imposed. Finally, we present numerical results to
illustrate the theoretical results which were obtained.

Keywords: Optimization; Liquidity Risk; Limited Liability; Haircut

##  1 Introduction

The Basel regulations, as a part of its capital framework for the
international banking system, includes the Leverage Ratio as a key component
of its capital requirement structure [1]. The genesis of the work carried out
in [1] was to effectively analyze, the apparent contradiction between Leverage
Ratio based capital requirement and the view of the banks that Leverage Ratio
is incompatible with the modern risk management practices. The work concludes
that the Leverage Ratio restrictions enable supervisors elicit honest
reporting by the banks, especially when they (supervisors) have limited
punitive mechanism (for errant banks) at their disposal. Hulster [2] presents
an analysis of the benefits of adopting Leverage Ratio as a part of regulatory
framework, while identifying some of its inherent limitations. A formulation
to compute Leverage Ratio was given by $\text{Leverage Ratio}=\frac{\text{Tier
1 Capital}}{\text{Adjusted Assets}}$. The benefits of using Leverage Ratio
includes counter-cyclical measures, lesser regulatory arbitrage and simplicity
in terms of deployment and monitoring, while undesirable incentives and being
limited to balance sheet are a couple of shortcomings which were observed. An
interesting observation made in [3] was that, while Leverage Ratio can act as
an inducement for banks to take risk, this can be offset by the safety net of
higher capital. In particular, for highly leveraged banks, the inclusion of
the Leverage Ratio requirement leads to significant reduction in the levels of
distress probability. A contextual description of the financial fragility
resulting from being highly leveraged and the consequent inclusion and
benefits of Leverage ratio is provided for in [4]. In [5], the question of
impact of interest rates on banks’ leveraging tendencies is discussed. It is
shown that the extent of leverage, resulting from interest rate changes is
contingent on whether the capital structure is adjustable or fixed. The impact
of the non-risk based Leverage capital requirement on credit practices of
banks and their consequent stability is studied in [6]. The key finding
reported in this work is that Leverage Ratio can potentially act as an
incentive for banks to acquire highly risky credit portfolios and the authors
recommend the enhancement of the current Leverage Ratio requirements.

One of the key characteristics of credit assets held by a bank, namely, it
being considered as illiquid, in spite of existence of credit derivatives,
which enables transfers as well as sale of loan portfolios is examined in [7].
While this practice has been welcomed by the regulatory authorities, it does
not account for the possibility of greater liquidity acting as an inducement
for banks to take on newer risks, and paradoxically this increases the
instability and the likelihood of bank failures [7]. Another paradoxical
observation made in [8] finds evidence of higher risk-taking tendencies of
banks having lower funding liquidity (resulting from higher deposit rates),
especially during stressed market conditions. The authors of [9] carry out an
evaluation of how access to abundant levels of liquidity can lead to the
creation of asset bubbles, as was the case during the 2008 crisis. This
scenario of banks holding on to high levels of liquidity heightens
macroeconomic risk and acts as the genesis for crisis in the financial sector.
Ghenimi et al. [10] examine the relationship between credit risk and liquidity
risk and observe that (contrary to what one would expect) both these factors
do not exhibit reciprocal behaviour. Further, there is no time-delayed
association, even though they both contribute to the fragility of banks and
their consequent instability. The global financial crisis and the consequent
response of imposition of liquidity adequacy, as a part of the risk management
practices was intended to ensure the stability of the banking sector [11].

The safety net for banks, by way of limited liability has the potential for
the banks engaging in acquiring more risky portfolios, along with concurrent
undercapitalization [12]. This leads to the imposition of solvency constraints
in the banking system, which, in the context of globalization of the banking
system leads to a bulk of the cost of banking being borne by the domestic
customer base of the banks. Limited liability, from a historical narrative,
along with identification of its advantages and disadvantages are presented in
[13]. Limited liability offers the advantage of promoting and encouraging
investments by passive investors, while extending immunity to the managers of
such investments. In contrast, the concept of limited liability may encourage
indulgence in risky activities, stemming from poor monitoring (by
shareholders) of the risky activities being engaged in, by the firm. The
concept of limited liability, particularly its impact on the banking sector,
in terms of bailouts, is discussed in [14]. The paper highlights on the
dilemma of the role of the central bank as the lender of last resort (LOLR),
when balancing between being strongly punitive on the errant banks versus the
risk of systemic impact, resulting from the failure of the banks (who enjoy
immunity, by virtue of their limited liability status). The authors argue that
the central bank should make funds available in situations of stressed
macroeconomic conditions, as opposed to the adverse situations, resulting from
imprudent portfolio decisions by the banks, and that interventions by the
central banks should be contingent on these conditions.

From a mathematical perspective, optimization techniques play a vital role in
the process of determination of economic factors in various paradigms. In case
of Leverage Ratio [3], the authors consider a constrained decision problem,
which maximizes the expected profits (adjusted for payouts, survival and
investment costs) and subject to risk-adjusted capital, as well as a Leverage
Ratio constraint. In [9], the authors consider the optimization setup for a
bank owner’s problem of maximizing the expected profits, excluding any penalty
resulting from liquidity crunch, subject to constraints resulting from the
action of depositors. From the perspective of risk-return trade-off in case of
a portfolio of loans, the author in [15] considers a risk minimization
problem, akin to the classical Markowitz framework, which not only includes
the expected return and sum of weights constraint, but also a probabilistic
constraint, which ensures that banks achieve a return level above a threshold,
chosen in a manner so as to ensure that the minimum capital requirement is met
by the bank.

##  2 Motivation

In our earlier work [16], we had discussed at length, the aspects of
management of a loan portfolio, in the context of incorporating limited
liability. It is but natural, now to bring forth the consideration of
liquidity, the relevance and importance of which can be gauged from the recent
events, such as the collapse of the Silicon Valley Bank. To encapsulate the
dynamics of liquidity consideration, in the existent framework, we begin with
the enumeration of the necessary (and important) components towards achieving
this end, that is, incorporating liquidity in the model.

The incorporation of liquidity entails the construction of “at least” a three
(time) setup model ($t=0,1,2$), since the consideration of two time steps
would not create a situation of liquidity crunch, due to the absence of the
likelihood of deposit withdrawal at any intermediate time point of the loan
duration (of two time points). Having considered the (aforesaid) simplest
model setup, it is assumed that any liquidity requirement emerges as the
result of a fraction of deposits being withdrawn at the intermediate time
point $t=1$ (which for now is assumed to be known, but may be generalized to
being generated through a random process, such as the exponential
distribution). As a result of withdrawal demand of a fraction of the deposits,
the bank is likely to face the necessity of liquidating a commensurate
fraction of the loan assets (or the loan portfolio). Accordingly, in this
study, the goal is to develop (and of course, analyze) a two-step approach to
optimize the expected return on the loan portfolio, followed by an optimal
liquidation strategy. The timeline of the events on a time scale of $t=0,1,2$
is illustrated below in Table 1.

$t=0$ | $t=1$ | $t=2$
---|---|---
Bank collects money | A fraction of depositors | All the risky investments
through debt ($d$) and | withdraw their deposits | have matured. Hence the
equity ($e$) and then invests in | and consequently the bank has | bank either pays its liabilities
safe assets and risky assets. | to liquidate some assets. | or faces insolvency/bankruptcy.
Table 1: Timeline of events for the bank’s portfolio

##  3 Model Description

The deposit structure of the bank is assumed to follow the classical firm
value assumption (due to Merton [17]), where the banks’ assets are considered
equal to the sum of the money raised through equity, $e$ (from shareholders)
and the money raised through debt, $d$ (from depositors). Now, this amount is
assumed to be invested in three types of loans, namely, a safe loan $L_{0}$
and two risky loans $L_{1}$ and $L_{2}$ (where $L_{1}$ is less risky than
$L_{2}$). Accordingly, we have the relation:

$e+d=\sum\limits_{i=0}^{2}L_{i}.$

Now, coming to the quality of liquidity, one encounters the notion of high and
low liquidity, corresponding to easy and hard to sell assets, respectively. It
is obvious that more liquid assets have less haircut (a concept indicating the
extent of loss or price impact, as a direct result of the unwinding the assets
at a certain pace) and vice-versa (less liquid assets have to take more
haircut).

Accordingly, we let the constant $\gamma_{i}$ denote the haircut for loan
$L_{i}$ ($i=0,1,2$). It is obvious that the more risky the asset is, the more
haircut it will experience during the liquidation. Therefore,

$\gamma_{0}<\gamma_{1}<\gamma_{2}.$

Further, at the time $t=1$, when the liquidity requirement emerges, we assume
that a certain factor $\alpha_{w}$ (of $d$) is withdrawn and another fraction
$\alpha_{d}$ (of $d$) is deposited. This leads to the following observations:

  1. (1)

If $\alpha_{d}\geq\alpha_{w}$, then the new deposits are enough to meet the
withdrawal demand of the depositors and therefore there is no need to
liquidate any part of the loan portfolio.

  2. (2)

If $\alpha_{d}<\alpha_{w}$ then the new deposits prove to be insufficient to
meet the obligations to the depositors demanding withdrawal, which will
trigger the process of liquidation from the loan portfolio. Now, the dilemma
faced by the bank is as follows (summarized in Table 2):

    1. (A)

If the safe asset ($L_{0}$) is liquidated, then the likelihood of default risk
at $t=2$ increases, even though this exercise means less haircut.

    2. (B)

In contrast, if the riskier loans ($L_{1}$ and $L_{2}$) are liquidated then
the likelihood of default risk at $t=2$ decreases, but at the cost of greater
haircut.

Loans | Pros | Cons
---|---|---
$L_{0}$ | Lesser haircut | More default risk
$L_{1}$ and/or $L_{2}$ | Less default risk | Greater haircut
Table 2: Pros and cons for the safe asset and the risky assets

  3. (3)

The model setup assumes that the equity values remain unchanged at time $t=1$.

Suppose that we liquidate a fraction $\beta_{i}$ for loan $L_{i}$ for
$0\leq\beta_{i}\leq 1$ ($i=0,1,2$). Then we get,

$\sum\limits_{i=0}^{2}\beta_{i}(1-\gamma_{i})L_{i}=\left(\alpha_{w}-\alpha_{d}\right)d.$

Before we present the description of the models which are being proposed in
this work, we enumerate the various variables for the same in Table 3. Note
that Expected Loss=Probability of Default $\times$ Loss Given Default.

Variable | Description
---|---
$X$ | Realization of the loan portfolio
$\delta$ | Cost of equity
$\rho(x)$ | Risk of the loan portfolio $x$
$L$ | The total amount of loss from liquidating the whole position
$\theta_{1}$ | Upper bound on risk ($>0$)
$\theta_{2}$ | Lower bound on risk ($>0$)
$\beta_{i}$ | Liquidation strategy for the $i$-th asset at time $t=1$
$\gamma$ | Amount of loss due to liquidation
$k_{lev}$ | Leverage Ratio.
$K(x)$ | Internal Ratings Based (IRB) capital requirement for portfolio $x$
$e$ | Equity component of the bank’s portfolio.
$\eta_{i}$ | Loss Given Default (LGD) for the $i$-th loan
$EL_{i}$ | Expected Loss (EL) for the $i$-th loan
Table 3: Description of the model variables

######  Model 1.

The goal is to solve the maximization problem:

$\max_{x,e}\left[\mathbb{E}\left[\max\left(X-(1-e),0\right)\right]-\delta
e\right],$

subject to the constraints of:

  1. (A)

$0\leq x_{i}\leq 1\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i=0,1,2$
(short selling is not permissible).

  2. (B)

$\displaystyle{\sum\limits_{i=0}^{2}x_{i}=1}$.

  3. (C)

$e\geq\max\left(k_{lev},K(x)\right)$.

  4. (D)

$\rho(x)\leq\theta_{1}$ (upper bound on risk).

  5. (E)

$\displaystyle{\sum\limits_{i=0}^{2}x_{i}\gamma_{i}\leq L}$ (upper bound on
the total amount of haircut if the entire portfolio is liquidated).

######  Model 2.

Now we construct another model like the preceding one by removing the last
constraint, namely, the upper bound on to the total haircut amount, in the
event of liquidation of the entire portfolio. Accordingly, the goal is to
solve the maximization problem:

$\max_{x,e}\left[\mathbb{E}\left[\max\left(X-(1-e),0\right)\right]-\delta
e\right],$

subject to the constraints of:

  1. (A)

$0\leq x_{i}\leq 1\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i=0,1,2$
(short selling is not permissible).

  2. (B)

$\displaystyle{\sum\limits_{i=0}^{2}x_{i}=1}$.

  3. (C)

$e\geq\max\left(k_{lev},K(x)\right)$.

  4. (D)

$\rho(x)\leq\theta_{1}$ (upper bound on risk).

######  Model 3.

The formulation of this model deals with designing the strategy of liquidating
the problem of the loans portfolio. Let us denote the present value of the
$i$-th loan at time $1$ by $X_{i}^{(1)},\leavevmode\nobreak\ i=0,1,2$. Then
the goal is to solve the minimization problem:

$\min_{\beta}\left[\sum\limits_{i=0}^{2}\beta_{i}\gamma_{i}\right],$

subject to the constraints of:

  1. (A)

$0\leq\beta_{i}\leq x_{i},\leavevmode\nobreak\ i=0,1,2$.

  2. (B)

$\sum\limits_{i=0}^{2}\beta_{i}(1-\gamma_{i})X_{i}^{(1)}=\left(\alpha_{w}-\alpha_{d}\right)d$
(payment for the withdrawals by depositors).

  3. (C)

$\displaystyle{\sum_{i=0}^{2}\beta_{i}\text{EL}_{i}\geq\theta_{2}}$ (lower
bound on risk of liquidated portfolio, $\theta_{2}$).

######  Model 4.

Next we construct the model without the risk-lower bound. Accordingly, we get
the following the minimization problem:

$\min_{\beta}\left[\sum\limits_{i=0}^{2}\beta_{i}\gamma_{i}\right],$

subject to the constraints of:

  1. (A)

$0\leq\beta_{i}\leq x_{i},\leavevmode\nobreak\ i=0,1,2$.

  2. (B)

$\sum\limits_{i=0}^{2}\beta_{i}(1-\gamma_{i})X_{i}^{(1)}=\left(\alpha_{w}-\alpha_{d}\right)d$
(payment for the withdrawals by depositors).

The motivation of adding the last constraint in Model 3 is given as Remark 1
as follows.

######  Remark 1.

Banks will (naturally) try to minimize the loss from liquidating the loan
portfolio. Since the safe asset is the most liquid in our loan portfolio set-
up, hence liquidating this asset leaves the riskier assets in the portfolio
for the next step ($t=2$). Consequently, in case of any unfavorable
conditions, the bank has to face a significant loss, if the riskier loan are
not repaid in a timely manner. Keeping this in mind, we include the constraint
(C) in Model 3 to have a cap on liquidating the safe asset.

######  Remark 2.

The present value of the $i$-th loan is given by:

$X_{i}^{(1)}=\left(1+\frac{r_{i}}{2}\right)X_{i}^{(0)},$

and the final value of the loan is given by:

$X_{i}^{(2)}=\left(1+r_{i}\right)X_{i}^{(0)},$

where $r_{i}$ is the interest rate in the $i$-th loan, for $i=0,1,2$.
Obviously, $r_{1}<r_{2}<r_{3}$, for $i=0,1,2$.

##  4 Model Analysis

######  Theorem 1.

The solution of Model 1 exists.

###### Proof.

Let $S$ be the region defined by its constraints in the model 1. Since the
portfolio $x=(1,0,0)$, with equity being $100\%$ is feasible, therefore
$S\neq\phi$ (because the haircut for the safe loan is $\gamma_{0}=0$, the
expected loss is zero, and $100\%$ equity satisfies all the capital
requirement conditions). Further, since $S$ is a closed and bounded set in
$\mathbb{R}^{3}$, therefore it is a compact set. The objective function lies
in ${C}(\mathbb{R}^{3})$. So Weierstrass Theorem 111If $f(x)$ is continuous on
a nonempty feasible set $S$, which is closed and bounded, then $f(x)$ has a
global minimum in $S$. assures that the solution to the problem exists, which
we denoted by $\left(x^{*},e_{0}^{*}\right)$, where
$x^{*}=\left(x_{0}^{*},x_{1}^{*},x_{2}^{*}\right)$. ∎

######  Theorem 2.

The solution of Model 2 exists.

###### Proof.

The proof for the existence for the solution of Model 2 follows on the lines
of Theorem 1. ∎

######  Theorem 3.

The total amount of loss is more sensitive in case of illiquid asset.

###### Proof.

Total amount of loss due to liquidation is given by,

$L=\beta\gamma^{\top}=\sum\limits_{i=0}^{2}\beta_{i}\gamma_{i},$

where $\beta=\left(\beta_{0},\beta_{1},\beta_{2}\right)$ and
$\gamma=\left(\gamma_{0},\gamma_{1},\gamma_{2}\right))$. Now we see that
$\displaystyle{\frac{\partial L}{\partial\beta_{2}}\geq\frac{\partial
L}{\partial\beta_{1}}\geq\frac{\partial L}{\partial\beta_{0}}}$. Therefore
increasing the weights of the risky loans will increase the losses at higher
rate, as compared to the increase of losses resulting from increasing the
weight of the safe asset. ∎

######  Theorem 4.

Sensitivity of the solution of Model 1 with $L$ (upper bound on the haircut).

###### Proof.

Recall that, the constraint (E) of Model 1 is given by,

$\sum\limits_{i=0}^{2}x_{i}\gamma_{i}\leq L,$

where $L$ is the maximum limit on haircut. Accordingly, we consider the
equation:

$\sum\limits_{i=0}^{2}\frac{x_{i}\gamma_{i}}{L}=1\Rightarrow\sum\limits_{i=0}^{2}\frac{x_{i}}{L/\gamma_{i}}=1.$

It can be observed that, as $L$ decreases,
$\displaystyle{\frac{L}{\gamma_{i}}}\leavevmode\nobreak\ (\gamma_{i}>0)$
decreases. So the upper-bound of incorporating liquid investments reduces with
the decrement of $L$. If the optimal portfolio for the investment is
$\textbf{x}^{*}=({x}_{0}^{*},{x}_{1}^{*},{x}_{2}^{*})$, then its corresponding
haircut is $L^{*}$. Consequently, a decrease in the upper bound of the
constraint by an amount greater than or equal to $(L-L^{*})$ affects the
solution, and the new solution of Model 1 excludes illiquid assets. ∎

######  Theorem 5.

Using Limited Liability in the model reduces the incorporation of risky loans
in the portfolio.

###### Proof.

The proof has been presented in [16]. ∎

######  Theorem 6.

Importance of constraint $(C)$ in Model 3.

###### Proof.

The constraint is given by,

$\sum_{i=0}^{2}\beta_{i}\text{EL}_{i}\geq\theta_{2}.$

As we have already mentioned (in the proof of Theorem 3), we have
$\displaystyle{\frac{\partial L}{\partial\beta_{2}}\geq\frac{\partial
L}{\partial\beta_{1}}\geq\frac{\partial L}{\partial\beta_{0}}}$. Therefore
liquidating the most liquid asset causes the safe (risk-free) asset less
haircut. Nevertheless, in the event of bankruptcy, the bank has to face an
adverse scenario due to liquidation safe assets. Therefore, if we add this
constraint in the model, with the risk bound $\theta_{2}$ (which is less than
$\theta_{1}$, because the portfolio has risk measure less than $\theta_{1}$ by
Model 1), then a suitable threshold of this $\theta_{2}$ will restrict the
liquidation of safe assets and make it incumbent on the bank to liquidate the
illiquid assets (risky loans). The mathematical justification for the
preceding result is as follows. We first consider the plane given by the
following equation:

$\sum\limits_{i=0}^{2}\beta_{i}EL_{i}=\theta_{2}\Rightarrow\sum\limits_{i=0}^{2}\frac{\beta_{i}EL_{i}}{\theta_{2}}=1\Rightarrow\sum\limits_{i=0}^{2}\frac{\beta_{i}}{\left(\frac{\theta_{2}}{EL_{i}}\right)}=1.$

We observe that the upper half of this plane contains the feasible region for
Model 3. As the $\theta_{2}$ increases,
$\displaystyle{\frac{\theta_{2}}{EL_{i}}}$ also increases, provided
$EL_{i}>0$. Therefore the lower bound for liquidating the risky assets
increases, and hence the bank has to liquidate the more risky asset for higher
value of $\theta_{2}$.

From the constraint $(B)$ in Model 3, it is obvious that decreasing
$\beta_{0}$ results in increase of $\beta_{1}$ and $\beta_{2}$.
Differentiating the constraint $(B)$ with respect to $\beta_{0}$ we get,

$\left(1-\gamma_{0}\right)X_{0}^{(1)}=-\frac{\partial}{\partial\beta_{0}}\left(\beta_{1}(1-\gamma_{1})X_{1}^{(1)}+\beta_{2}(1-\gamma_{2})X_{2}^{(1)}\right).$

Let
$b=\min\left(\left(1-\gamma_{1}\right)X_{1}^{(1)},\left(1-\gamma_{1}\right)X_{1}^{(1)}\right)$.
Since the realizations are all positive, hence $b>0$. Therefore, we have,

$\frac{\partial}{\partial\beta_{0}}\left(\beta_{1}b+\beta_{2}b\right)<0\Rightarrow\frac{\partial}{\partial\beta_{0}}\left(\beta_{1}+\beta_{2}\right)<0.$

Now, we consider the worst case, in which both the risk loans have been
defaulted. Let
$\beta^{(3)}=\left(\beta_{0}^{(3)},\beta_{1}^{(3)},\beta_{2}^{(3)}\right)$ be
the solution of Model 3 and
$\beta^{(4)}=\left(\beta_{0}^{(4)},\beta_{1}^{(4)},\beta_{2}^{(4)}\right)$ be
the solution of Model 4 (without this risk constraint). Therefore
$\displaystyle{\beta_{0}^{(4)}\geq\beta_{0}^{(3)}}$. Now, in the worst case
scenario, the realization for the solution of Model 3 is better than the
realization for the solution of Model 4, provided:

$\left(\beta_{0}^{(4)}-\beta_{0}^{(3)}\right)X_{0}^{2}\geq\left(\beta_{1}^{(3)}-\beta_{1}^{(4)}\right)(1-\eta_{1})X_{1}^{0}+\left(\beta_{2}^{(3)}-\beta_{2}^{(4)}\right)(1-\eta_{2})X_{2}^{0},$

the LGD for the $i$-th loan is $\eta_{i},\leavevmode\nobreak\ i=0,1,2$.
Finally, we can reduce $\theta_{2}$ so that the above inequality will be
satisfied. ∎

##  5 An Example

In this Section, we construct an example in order to illustrate the
theoretical results presented in the preceding Section. For this purpose, we
consider three loans, comprising of one safe investment and two risky
investments, with one of them being riskier than the other. In this context,
the term riskiness refers to the credit worthiness of the debtor, which is
contingent on parametric values such as PD and LGD, some of which (the
parameter values) are available in [6, 18]. For the purpose of our
illustrative example, we have taken the values for return, PD and LGD to be
the same as in [16]. Further, the values of the haircuts of different types of
loans are listed in the last column of the Table 4. It may be noted that the
authors in [11] have discussed in detail about the haircuts applicable for
various class of assets. Finally we take $\delta=1.04$, which was motivated
from [6]. In summary, Table 4 enumerates all the details of these parameter
values pertaining to the loans.

We have taken a three-loan portfolio to illustrate the theoretical results
obtained. One of these three loans is safe, and the others are risky. All the
necessary parameter values are declared in Table 4 with leverage ratio,
$k_{lev}=4\%$.

Loan Type | Return | PD | LGD | Haircut
---|---|---|---|---
Safe Loan | $r_{rf}=3\%$ | $0$ | $0$ | $0\%$
Less Risky Loan | $r_{s}=9\%$ | $p_{s}=6.1\%$ | $lgd_{s}=10\%$ | $10\%$
More Risky Loan | $r_{r}=13.2\%$ | $p_{r}=12.2\%$ | $lgd_{r}=9\%$ | $20\%$
Table 4: Risk parameters for the three loans

We begin with the solution for Model 1, which with the inclusion of the
parameter values in Table 4, reduces to the model:

$\max_{x,e}\left[\mathbb{E}\left[\max\left(X-(1-e),0\right)\right]-1.04e\right],$

subject to the constraints of:

  1. (A)

$0\leq x_{i}\leq 1\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i=0,1,2$.

  2. (B)

$\displaystyle{\sum\limits_{i=0}^{2}x_{i}=1}$.

  3. (C)

$e\geq\max\left(0.04,K(x)\right)$.

  4. (D)

$x_{0}\times 0+x_{1}\times 0.0061+x_{2}\times 0.01098\left(\text{Expected
Loss}\right)\leq 0.012\left(=\theta_{1}\right)$

  5. (E)

$\displaystyle{x_{0}\times 0+x_{1}\times 0.1+x_{2}\times 0.2\leq
0.15\left(=L\right)}$

Further, in an analogous manner, Model 2, which does not include a cap on the
haircut becomes:

$\max_{x,e}\left[\mathbb{E}\left[\max\left(X-(1-e),0\right)\right]-1.04e\right],$

subject to the constraints of:

  1. (A)

$0\leq x_{i}\leq 1\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i=0,1,2$.

  2. (B)

$\displaystyle{\sum\limits_{i=0}^{2}x_{i}=1}$.

  3. (C)

$e\geq\max\left(0.04,K(x)\right)$.

  4. (D)

$x_{0}\times 0+x_{1}\times 0.0061+x_{2}\times 0.01098\left(\text{Expected
Loss}\right)\leq 0.012\left(=\theta_{1}\right)$.

Model 1 and Model 2 are continuous, non-differentiable problems (as the
objective function is not differentiable). However, it can be transformed into
a differentiable optimization problem, which has been elaborately discussed in
[16]. After solving the problem, we get the portfolio
$\left(2.91\%,44.18\%,52.91\%\right)$ (that is, investments of $2.91\%$ in
$L_{0}$, $44.18\%$ in $L_{1}$ and $52.91\%$ in $L_{2}$) from Model 1 and for
Model 2 we get $\left(0\%,27.78\%,72.22\%\right)$, with leverage ratio of
$4\%$, for both the portfolios. The results show that the inclusion of the cap
on the haircut reduces the incorporation of illiquid assets in its portfolio.
Hence Model 1 outperforms 2 from the perspective of liquidity risk management
222We have used the “scipy” package of Python to solve these models.

Next, we obtain the solution to the problem of liquidating assets, at time
$t=1$, to meet the claims of the depositors, both with and without a lower
bound on the risk. For this purpose, we need the solution of investment
decision at $t=0$, which in turn plays the role of the first constraint. We
first proceed with the solution of Model 1, that is,
$\left(2.91\%,44.18\%,52.91\%\right)$. Here, in order to solve for the model,
we have taken $\alpha_{w}=10\%$ and $\alpha_{d}=0$. In other words, $10\%$ of
the deposits are withdrawn and there is no further investment. Hence, Model 3
becomes:

$\min_{\beta}\left[\beta_{0}\times 0+\beta_{1}\times 0.1+\beta_{2}\times
0.2\right],$

subject to the constraints:

  1. (A)

$0\leq\beta_{i}\leq x_{i},\leavevmode\nobreak\ i=0,1,2,\leavevmode\nobreak\
\left(x_{0}=0.0291,x_{1}=0.4418,x_{2}=0.5291\right)$.

  2. (B)

$\beta_{0}(1-0)X_{0}^{(1)}+\beta_{1}(1-0.1)X_{1}^{(1)}+\beta_{2}(1-0.2)X_{2}^{(1)}=0.1\times
0.96$.

  3. (C)

$\displaystyle{\beta_{0}\times 0+\beta_{1}\times 0.0061+\beta_{2}\times
0.01098\geq\theta_{2}}$

We solve this problem with two different values of $\theta_{2}$, first for
$\theta_{2}=0.05\%$ and then for $\theta_{2}=0.1\%$. Before going to the
solution of Model 3, we solve Model 4, which gives the liquidation strategy
without lower bound on risk. With all these parameter values, the problem
becomes:

$\min_{\beta}\left[\beta_{0}\times 0+\beta_{1}\times 0.1+\beta_{2}\times
0.2\right],$

subject to the constraints:

  1. (A)

$0\leq\beta_{i}\leq x_{i},\leavevmode\nobreak\ i=0,1,2,\leavevmode\nobreak\
\left(x_{0}=0.0291,x_{1}=0.4418,x_{2}=0.5291\right)$.

  2. (B)

$\beta_{0}(1-0)X_{0}^{(1)}+\beta_{1}(1-0.1)X_{1}^{(1)}+\beta_{2}(1-0.2)X_{2}^{(1)}=\left(0.1\right)\times
0.96$.

Solving Model 4 we get $\left(2.91\%,7.07\%,0\%\right)$, that is, liquidation
happens for $L_{0}$ and $L_{1}$. Model 3, with $\theta_{2}=0.05\%$ gives the
liquidation portfolio of $\left(1.86\%,8.20\%,0\%\right)$ and for
$\theta_{2}=0.1\%$ we get $\left(0\%,3.93\%,6.93\%\right)$. This supports the
result established in Theorem 6, i.e., a lower bound on risk for the
liquidating portfolio reduces the risk for the remaining assets, and as a
result of which the chance of default is also reduced. Therefore the solution
from Model 3 performs better than Model 4, in the worst case scenario.

Next, we take the solution of Model 2 for solving the liquidation portfolio.
The formulation of the liquidation models is the same as discussed above,
except for the values of $(x_{i}:i=0,1,2)$. Solving Model 4 (without lower
bound on the risk) gives the portfolio of $\left(0\%,10.21\%,0\%\right)$, that
is, the liquidation of $10.21\%$ of less risky asset $(L_{1})$. We solve Model
3, first with a risk-lower bound of $r=0.05\%$ and then for $r=0.1\%$,
resulting in the liquidation strategies of $\left(0\%,10.21\%,0\%\right)$ and
$\left(0\%,3.93\%,6.93\%\right)$, respectively. Hence the model with a higher
risk-lower bound reduces the chance of default at the final time, as was
stated in Theorem 6. Now with both the inputs, it is clear that Model 3
outperforms Model 4 from a risk management perspective. We have taken two
inputs to solve these liquidation strategies, namely, the solution of Model 1
and the solution of Model 2. From the numerical results, it is clear that, as
an input, the solution of Model 1 plays a better role, since it produces less
probability of default at the final time, after liquidation in the
intermediate time. Therefore, in summary, solving the investment decision with
Model 1 outperforms Model 2 and in case of liquidation, Model 3 is more
advantageous than Model 4, which is evident from Tables 5 and 6

Model | Model used as input | Portfolio with $\theta_{2}=0.05\%$ | Portfolio with $\theta_{2}=0.1\%$
---|---|---|---
Model 3 | Model 1 | $(1.86\%,8.20\%,0\%)$ | $(0\%,3.93\%,6.93\%)$
Model 3 | Model 2 | $(0\%,10.21\%,0\%)$ | $(0\%,3.93\%,6.93\%)$
Table 5: Results for Model 3 Model | Model which is used as input | Portfolio
---|---|---
Model 4 | Model 1 | $(2.91\%,7.07\%,0\%)$
Model 4 | Model 2 | $(0\%,10.21\%,0\%)$
Table 6: Results for Model 4

Next, taking the solution of Model 1 as the first constraint of Models 3 and
4, we proceed to show the feasible region for the Problems 3 and 4,
respectively. We plotted three constraints of the Model 3 in three different
figures. The first constraint is presented in Subfigure 1(a). The interior of
the cube represents the set of admissible solutions. Subfigure 1(b) gives the
second constraint. The points on the surfaces are the feasible points,
implying that the bank will liquidate its assets equivalent to the amount to
be withdrawn by the depositors. Finally, Subfigure 1(c) shows the third
constraint. The surface presents the lower bound on risk. This model takes the
upper half of this surface for an admissible liquidating strategy. Thus the
intersection of these three is the feasible region for Model 3. Then we
present the feasible area of Model 4 in Figure 2. It excludes the lower bound
on risk. All the other descriptions are the same as the previous scenario. The
intersection of these regions is the feasible region for Model 4. The
constraint (E) in Model 1 is presented in Figure 3. Here, $A_{1}$ represents
an upper bound of $10\%$ in haircut, that is, the lower half of this surface
is feasible whereas $A_{2}$ represents the $15\%$ upper bound of the haircut.
Finally, Figures the $x$, $y$ and $z$-axes represent $L_{0}$, $L_{1}$ and
$L_{2}$, respectively.

((a)) First Constraint

((b)) Second Constraint

((c)) Third Constraint

Figure 1: The feasible region for Model 3

((a)) First Constraint

((b)) Second Constraint

Figure 2: The feasible region for Model 4 Figure 3: Upper Bound on Haircut

##  6 Conclusions

Liquidity is one of the major concerns in today’s bank’s portfolio decision,
especially in the context of maintaining solvency and avoiding a bank-run. The
inclusion of illiquid assets in the portfolio of bank, increases the
likelihood of it being unable to meet its obligations to the depositors. On
the other banks, by virtue of their very structure and design enjoy
protections of limited liability. Accordingly, this paper presents a novel
approach of incorporating the including liquidity constraint, while enjoying
limited liability protection, in case of the loan portfolio held by banks. We
also examine the benefits of applying a constraint on the haircut.

In this work, we have shown a three-time step model that optimizes profit with
limited liability at the initial time point of $t=0$. Here, we have
established the comparison between the two models, namely, Model 1) (with a
cap on the haircut) and Model 2 (without a cap on the haircut), with the other
constraints remaining identical. At $t=1$, a fraction of depositors claim
their money back. Therefore the model solves this problem of liquidating the
fraction of assets. Here we compare Model 3(which includes a risk-lower bound)
and Model 4 (which excludes a risk-lower bound), with all other constraints
being identical. Finally, at $t=2$, the bank either faces bankruptcy or makes
a profit after paying all the liabilities.

Moreover, our analysis shows that in the first comparison, the model with a
cap on the total haircut includes less illiquid assets in the portfolio, as
compared to the model without a cap on the total haircut. On the other hand,
the inclusion of limited liability produces a less risky portfolio, which fits
better into the actual scenario. In the case of the second comparison, a lower
bound on the risk resists the liquidation of safe loans, as compared to the
the model without a lower bound on the risk.

Therefore, from a practitioner’s point of view, the model with the cap on the
haircut and limited liability can help to make the right investment decision
because it reflects a more accurate scenario and produces better results.
Incorporation of limited liability reduces risk and upper bound on haircut
helps to survive in the stress scenario. In the intermediate time step model,
the liquidation strategy minimizes the haircut. It resists the liquidation of
safe assets, which increases the chance of default. Since it produces better
results by satisfying all the investment criteria, therefore it is undoubtedly
a more valuable model for decision-making.

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