Extra material: Optimal Growth Portfolios with Risk Aversion#

Among the reasons why Kelly was neglected by investors were high profile critiques by the most famous economist of the 20th Century, Paul Samuelson. Samuelson objected on several grounds, among them is a lack of risk aversion that results in large bets and risky short term behavior, and that Kelly’s result is applicable to only one of many utility functions that describe investor preferences. The controversy didn’t end there, however, as other academic economists, including Harry Markowitz, and practitioners found ways to adapt the Kelly criterion to investment funds.

This notebook presents solutions to Kelly’s problem for optimal growth portfolios using exponential cones. A significant feature of this notebook is the inclusion of a risk constraint recently proposed by Boyd and coworkers. These notes are based on recent papers such as Cajas (2021), Busseti, Ryu and Boyd (2016), Fu, Narasimhan, and Boyd (2017). Additional bibliographic notes are provided at the end of the notebook.

# install dependencies and select solver
%pip install -q amplpy numpy pandas

SOLVER_CONIC = "mosek"  # ipopt, mosek, knitro

from amplpy import AMPL, ampl_notebook

ampl = ampl_notebook(
    modules=["coin", "mosek"],  # modules to install
    license_uuid="default",  # license to use
)  # instantiate AMPL object and register notebook magic

Financial Data#

We begin by reading historical prices for a selected set of trading symbols using yfinance.

While it would be interesting to include an international selection of financial indices and assets, differences in trading and bank holidays would involve more elaborate coding. For that reason, the following cell has been restricted to indices and assets trading in U.S. markets.

# run this cell to install yfinance
%pip install yfinance --upgrade -q
import matplotlib.pyplot as plt
import numpy as np

import pandas as pd
import datetime
import yfinance as yf

# symbols as used by Yahoo Finance
symbols = {
    # selected indices
    "^GSPC": "S&P 500",
    "^IXIC": "Nasdaq",
    "^DJI": "Dow Jones Industrial",
    "^RUT": "Russell 2000",
    # selected stocks
    "AXP": "American Express",
    "AMGN": "Amgen",
    "AAPL": "Apple",
    "BA": "Boeing",
    "CAT": "Caterpillar",
    "CVX": "Chevron",
    "JPM": "JPMorgan Chase",
    "MCD": "McDonald's",
    "MMM": "3 M",
    "MSFT": "Microsoft",
    "PG": "Proctor & Gamble",
    "XOM": "ExxonMobil",
}

# years of testing and training data
n_test = 1
n_train = 2

# get today's date
today = datetime.datetime.today().date()

# training data dates
end = today - datetime.timedelta(int(n_test * 365))
start = end - datetime.timedelta(int((n_test + n_train) * 365))

# get training data
S = yf.download(list(symbols.keys()), start=start, end=end)["Adj Close"]

# compute gross returns
R = S / S.shift(1)
R.dropna(inplace=True)
[*********************100%***********************]  16 of 16 completed
# plot
fig, ax = plt.subplots(2, 1, figsize=(8, 6), sharex=True)

S.divide(S.iloc[0] / 100).plot(ax=ax[0], grid=True, title="Normalized Prices")
ax[0].legend(loc="center left", bbox_to_anchor=(1.0, 0.5), prop={"size": 8})

R.plot(ax=ax[1], grid=True, title="Gross Returns", alpha=0.5).legend([])

fig.tight_layout()
../../_images/140b9694831c8a2e3fa59bd16cdb645ad8f94e0cbc9fc3073cfdb5545cb95eb0.png

Portfolio Design for Optimal Growth#

Model#

Here we are examining a set \(N\) of financial assets trading in efficient markets. The historical record consists of a matrix \(R \in \mathbb{R}^{T\times N}\) of gross returns where \(T\) is the number of observations.

The weights \(w_n \geq 0\) for \(n\in N\) denote the fraction of the portfolio invested in asset \(n\). Any portion of the portfolio not invested in traded assets is assumed to have a gross risk-free return \(R_f = 1 + r_f\), where \(r_f\) is the return on a risk-free asset.

Assuming the gross returns are independent and identically distributed random variables, and the historical data set is representative of future returns, the investment model becomes

\[\begin{split} \begin{align} \max_{w_n \geq 0}\quad & \frac{1}{T} \sum_{t\in T} \log(R_t) \\ \text{s.t.}\quad \\ & R_t = R_f + \sum_{n\in N} w_n (R_{t, n} - R_f) & \forall t\in T\\ \end{align} \end{split}\]

Note this formulation allows the sum of weights \(\sum_{n\in N} w_n\) to be greater than one. In that case the investor would be investing more than the value of the portfolio in traded assets. In other words the investor would be creating a leveraged portfolio by borrowing money at a rate \(R_f\). To incorporate a constraint on the degree of leveraging, we introduce a constraint

\[\sum_{n\in N} w_n \leq E_M\]

where \(E_M\) is the “equity multiplier.” A value \(E_M \leq 1\) restricts the total investment to be less than or equal to the equity available to the investor. A value \(E_M > 1\) allows the investor to leverage the available equity by borrowing money at a gross rate \(R_f = 1 + r_f\).

Using techniques demonstrated in other examples, this model can be reformulated with exponential cones.

\[\begin{split} \begin{align} \max_{w_n}\quad & \frac{1}{T} \sum_{t\in T} q_t \\ \text{s.t.}\quad & (R_f + \sum_{n\in N}w_n (R_{t,n} - R_f), 1, q_t) \in K_{exp} & \forall t \in T \\ & \sum_{n\in N} w_n \leq E_M \\ & w_n \geq 0 & \forall n\in N \\ \end{align} \end{split}\]

For the risk constrained case, we consider a constraint

\[\mathbb{E}[R^{-\lambda}] \leq R_f^{-\lambda}\]

where \(\lambda\) is a risk aversion parameter. Assuming the historical returns are equiprobable

\[\frac{1}{T} \sum_{t\in T} R_t^{-\lambda} \leq R_f^{-\lambda}\]

The risk constraint is satisfied for any \(w_n\) if the risk aversion parameter \(\lambda=0\). For any value \(\lambda > 0\) the risk constraint has a feasible solution \(w_n=0\) for all \(n \in N\). Recasting as a sum of exponentials,

\[\frac{1}{T} \sum_{t\in T} e^{- \lambda\log(R_t)} \leq R_f^{-\lambda}\]

Using the \(q_t \leq \log(R_t)\) as used in the examples above, and \(u_t \geq e^{- \lambda q_t}\), we get the risk constrained model optimal log growth.

Given a risk-free rate of return \(R_f\), a maximum equity multiplier \(E_M\), and value \(\lambda \geq 0\) for the risk aversion, risk constrained Kelly portfolio is given the solution to

\[\begin{split} \begin{align} \max_{w_n, q_t, u_t}\quad & \frac{1}{T} \sum_{t\in T} q_t \\ \text{s.t.}\quad & \frac{1}{T} \sum_{t\in T} u_t \leq R_f^{-\lambda} \\ & (u_t, 1, \lambda q_t) \in K_{exp} & \forall t\in T \\ & (R_f + \sum_{n\in N}w_n (R_{t,n} - R_f), 1, q_t) \in K_{exp} & \forall t \in T \\ & \sum_{n\in N} w_n \leq E_M \\ & w_n \geq 0 & \forall n \in N \\ \end{align} \end{split}\]

The following cells demonstrate an AMPL implementation of the model with the Mosek solver.

AMPL Implementation#

The AMPL implementation for the risk-constrained Kelly portfolio accepts three parameters, the risk-free gross returns \(R_f\), the maximum equity multiplier, and the risk-aversion parameter.

%%writefile kelly_portfolio.mod

param Rf;
param EM;
param lambd;
    
# index lists
set T;
set N;
param Rloc{T, N};
    
# decision variables
var q{T};
var w{N} >= 0;

# objective
maximize ElogR:
    sum {t in T} q[t] / card(T);

# conic constraints on return
var R{t in T}
    = Rf + sum {n in N} w[n]*(Rloc[t, n] - Rf);
s.t. C{t in T}:
    R[t] >= exp(q[t]);

# risk constraints
var u{T};
s.t. USum:
    sum {t in T} u[t] / card(T) <= Rf**(-lambd);
s.t. RT{t in T}:
    u[t] >= exp( -lambd*q[t] );
    
# equity multiplier constraint
s.t. WSum:
    sum {n in N} w[n] <= EM;
Overwriting kelly_portfolio.mod
def kelly_portfolio(R, Rf=1, EM=1, lambd=0):
    ampl = AMPL()
    ampl.read("kelly_portfolio.mod")

    ampl.param["Rf"] = Rf
    ampl.param["EM"] = EM
    ampl.param["lambd"] = lambd

    # index lists
    ampl.set["T"] = [str(t) for t in R.index]
    ampl.set["N"] = [str(n) for n in R.columns]
    ampl.param["Rloc"] = {
        (str(t), str(n)): R.at[t, n]
        for i, t in enumerate(R.index)
        for j, n in enumerate(R.columns)
    }

    ampl.option["solver"] = SOLVER_CONIC
    ampl.solve()

    return ampl
def kelly_report(ampl):
    # print report
    s = f"""
Risk Free Return = {100*(np.exp(252*np.log(ampl.get_value('Rf'))) - 1):0.2f}
Equity Multiplier Limit = {ampl.get_value('EM'):0.5f}
Risk Aversion = {ampl.get_value('lambd'):0.5f}

Portfolio
"""
    w = ampl.get_variable("w").to_dict()
    Rvar = ampl.get_variable("R").to_dict()

    s += "\n".join([f"{n:8s} {symbols[n]:30s}  {100*w[n]:8.2f} %" for n in w.keys()])
    s += f"""
{'':8s} {'Risk Free':30s}  {100*(1 - sum(w[n] for n in w.keys())):8.2f} %

Annualized return = {100*(np.exp(252*ampl.get_value('ElogR')) - 1):0.2f} %
"""
    print(s)

    df = pd.DataFrame(
        pd.Series([Rvar[str(t)] for t in R.index]), columns=["Kelly Portfolio"]
    )
    df.index = R.index  ## original index are the DateTimeStamps

    fix, ax = plt.subplots(1, 1, figsize=(8, 8))
    S.divide(S.iloc[0] / 100).plot(
        ax=ax,
        logy=True,
        grid=True,
        title="Normalized Prices",
        alpha=0.6,
        lw=0.4,
        ls="--",
    )
    df.cumprod().multiply(100).plot(ax=ax, lw=3, grid=True)
    ax.legend(
        [symbols[n] for n in R.columns] + ["Kelly Portfolio"],
        bbox_to_anchor=(1.05, 1.05),
    )

    d = S.index[-1]
    print(d)

    for n in w.keys():
        y = 100 * S[n].iloc[-1] / S[n].iloc[0]
        print(n, 100 * S[n].iloc[-1] / S[n].iloc[0])
        ax.text(d, y, n)
# parameter values
Rf = np.exp(np.log(1.0) / 252)
EM = 1
lambd = 10

m = kelly_portfolio(R, Rf, EM, lambd)
kelly_report(m)
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115058867
0 simplex iterations
22 barrier iterations

Risk Free Return = 0.00
Equity Multiplier Limit = 1.00000
Risk Aversion = 10.00000

Portfolio
AAPL     Apple                              57.78 %
AMGN     Amgen                               0.00 %
AXP      American Express                    0.00 %
BA       Boeing                              0.00 %
CAT      Caterpillar                        18.97 %
CVX      Chevron                             0.00 %
JPM      JPMorgan Chase                      0.00 %
MCD      McDonald's                          0.00 %
MMM      3 M                                 0.00 %
MSFT     Microsoft                           0.00 %
PG       Proctor & Gamble                    0.00 %
XOM      ExxonMobil                          0.00 %
^DJI     Dow Jones Industrial                0.00 %
^GSPC    S&P 500                             0.00 %
^IXIC    Nasdaq                              0.00 %
^RUT     Russell 2000                        0.00 %
         Risk Free                          23.25 %

Annualized return = 32.44 %

2022-06-03 00:00:00
AAPL 313.3368230987512
AMGN 154.57373693975774
AXP 143.59720714356118
BA 40.08344979462141
CAT 193.1338141274621
CVX 168.98770245792917
JPM 130.31933675829177
MCD 129.6343379131071
MMM 97.33192901788738
MSFT 211.75261984245606
PG 144.29053167324705
XOM 159.39669011089973
^DJI 126.61551678143734
^GSPC 142.9882963168018
^IXIC 155.1611360900202
^RUT 124.34379721298355
../../_images/2256a012d5792eb578a41fff86940845aad776dd7d72d68fc6ba089dd47fd6e9.png
S.head()
AAPL AMGN AXP BA CAT CVX JPM MCD MMM MSFT PG XOM ^DJI ^GSPC ^IXIC ^RUT
Date
2019-06-07 46.121952 155.557022 114.483406 347.400238 112.699974 101.451523 96.646111 187.392517 142.644150 126.295914 98.509666 60.087990 25983.939453 2873.340088 7742.100098 1514.390015
2019-06-10 46.711361 155.619110 115.948608 347.498413 113.859032 102.127975 97.690834 183.580475 144.082504 127.449310 98.464378 60.353874 26062.679688 2886.729980 7823.169922 1523.560059
2019-06-11 47.252258 154.768219 116.487427 343.108063 115.253525 101.192619 97.991859 185.367935 144.613327 126.968742 99.062119 60.297466 26048.509766 2885.719971 7822.569824 1519.109985
2019-06-12 47.101887 155.636841 115.353081 340.849030 115.090538 100.374207 96.743507 186.890930 144.698929 126.382423 99.333824 59.644863 26004.830078 2879.840088 7792.720215 1519.790039
2019-06-13 47.092175 156.620621 115.192368 342.646423 115.153931 100.975510 96.982559 186.489700 144.561996 127.180168 100.447807 60.168556 26106.769531 2891.639893 7837.129883 1535.800049

Effects of the Risk-Aversion Parameter#

lambd = 10 ** np.linspace(0, 3)

results = [kelly_portfolio(R, Rf=1, EM=1, lambd=_) for _ in lambd]
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514729021
0 simplex iterations
12 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514730217
0 simplex iterations
11 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514731433
0 simplex iterations
12 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514732444
0 simplex iterations
12 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514732116
0 simplex iterations
13 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514731363
0 simplex iterations
13 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514730112
0 simplex iterations
13 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514731633
0 simplex iterations
13 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514732135
0 simplex iterations
14 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514730977
0 simplex iterations
14 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.00151473216
0 simplex iterations
19 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514731721
0 simplex iterations
18 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001514732466
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001510090893
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001437622267
0 simplex iterations
27 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001295714115
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001158395522
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001032061346
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.000916678938
0 simplex iterations
36 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0008119779914
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0007174855375
0 simplex iterations
29 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0006326317078
0 simplex iterations
57 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0005567612961
0 simplex iterations
68 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.000489173801
0 simplex iterations
46 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0004291643027
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0003760314216
0 simplex iterations
30 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0003291057879
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0002877495441
0 simplex iterations
49 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0002513785857
0 simplex iterations
46 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0002194350789
0 simplex iterations
43 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0001914243691
0 simplex iterations
39 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0001668951972
0 simplex iterations
104 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0001454355608
0 simplex iterations
34 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0001266777712
0 simplex iterations
61 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.0001102993894
0 simplex iterations
50 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 9.601835593e-05
0 simplex iterations
51 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 8.353869664e-05
0 simplex iterations
54 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 7.267431243e-05
0 simplex iterations
120 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 6.320673287e-05
0 simplex iterations
74 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 5.496267527e-05
0 simplex iterations
81 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 4.780118333e-05
0 simplex iterations
121 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 4.15554229e-05
0 simplex iterations
95 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 3.612044892e-05
0 simplex iterations
119 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 3.137887756e-05
0 simplex iterations
244 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 2.726071662e-05
0 simplex iterations
197 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 2.368800934e-05
0 simplex iterations
294 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 2.059797531e-05
0 simplex iterations
168 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 1.789750665e-05
0 simplex iterations
123 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 1.555107149e-05
0 simplex iterations
242 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 1.349274934e-05
0 simplex iterations
200 barrier iterations
import matplotlib.pyplot as plt
import numpy as np

fig, ax = plt.subplots(2, 1, figsize=(8, 4), sharex=True)

ax[0].semilogx(
    [m.get_value("lambd") for m in results],
    [100 * (np.exp(252 * m.get_value("ElogR")) - 1) for m in results],
)
ax[0].set_title("Portfolio Return vs Risk Aversion")
ax[0].set_ylabel("annual %")
ax[0].grid(True)

ax[1].semilogx(
    [m.get_value("lambd") for m in results],
    [
        [w[str(n)] for n in R.columns]
        for m in results
        for w in [m.get_variable("w").to_dict()]
    ],
)
ax[1].set_ylabel("weights")
ax[1].set_xlabel("risk aversion $\lambda$")
ax[1].legend([symbols[n] for n in R.columns], bbox_to_anchor=(1.05, 1.05))
ax[1].grid(True)
ax[1].set_ylim(0, EM)

fig.tight_layout()
../../_images/eb82fafbb9776023a4a60ad2195d9466cc6b84d745e384b7db7943e3f71c1e65.png

Effects of the Equity Multiplier Parameter#

EM = np.linspace(0.0, 2.0)

results = [kelly_portfolio(R, Rf=1, EM=_, lambd=10) for _ in EM]
MOSEK 10.0.43: MOSEK 10.0.43: unknown (0, problem status: 0), max number of iterations reached
0 simplex iterations
400 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 7.127764616e-05
0 simplex iterations
15 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0001417540358
0 simplex iterations
13 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0002114242562
0 simplex iterations
12 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0002802885279
0 simplex iterations
12 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0003483468736
0 simplex iterations
12 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0004156018916
0 simplex iterations
13 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0004820568272
0 simplex iterations
14 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0005477046862
0 simplex iterations
14 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0006125490061
0 simplex iterations
15 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0006765882375
0 simplex iterations
15 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.000739824828
0 simplex iterations
18 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0008022547016
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0008638806781
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0009247042986
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0009847215819
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001043934055
0 simplex iterations
25 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001078172194
0 simplex iterations
29 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001104816542
0 simplex iterations
27 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.00111506199
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115066018
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.00111506275
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115058768
0 simplex iterations
27 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115062686
0 simplex iterations
25 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115058794
0 simplex iterations
60 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115064775
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115059022
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115061998
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115064832
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115063757
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115062221
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115062519
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115063848
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115063227
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115061374
0 simplex iterations
30 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115059603
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115060952
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115058514
0 simplex iterations
39 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115060287
0 simplex iterations
18 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.0011150608
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115060207
0 simplex iterations
33 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115064884
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115064279
0 simplex iterations
31 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115061648
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115062476
0 simplex iterations
19 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115058494
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115061564
0 simplex iterations
19 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115063624
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115060886
0 simplex iterations
19 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115060828
0 simplex iterations
21 barrier iterations
import matplotlib.pyplot as plt
import numpy as np

fig, ax = plt.subplots(2, 1, figsize=(8, 4), sharex=True)

ax[0].plot(
    [m.get_value("EM") for m in results],
    [100 * (np.exp(252 * m.get_value("ElogR")) - 1) for m in results],
)
ax[0].set_title("Portfolio Return vs Equity Multiplier")
ax[0].set_ylabel("annual return %")
ax[0].grid(True)

ax[1].plot(
    [m.get_value("EM") for m in results],
    [
        [w[str(n)] for n in R.columns]
        for m in results
        for w in [m.get_variable("w").to_dict()]
    ],
)
ax[1].set_ylabel("weights")
ax[1].set_xlabel("Equity Multiplier")
ax[1].legend([symbols[n] for n in R.columns], bbox_to_anchor=(1.05, 1.05))
ax[1].grid(True)
ax[1].set_ylim(
    0,
)

fig.tight_layout()
../../_images/66bd7cdd6ae560c4088662e127c77556bf587f3637fc9ab2a38b4cf0e60e46e1.png

Effect of Risk-free Interest Rate#

Rf = np.exp(np.log(1 + np.linspace(0, 0.20)) / 252)

results = [kelly_portfolio(R, Rf=_, EM=1, lambd=10) for _ in Rf]
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001115058867
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001109229733
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001103644806
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001098315886
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001093231488
0 simplex iterations
25 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001088400657
0 simplex iterations
19 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001083803179
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001079457309
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001075345323
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001071468947
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001067825963
0 simplex iterations
19 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001064420867
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001061247108
0 simplex iterations
19 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001058296152
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001055577603
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001053077952
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001050797566
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001048747589
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001046908631
0 simplex iterations
20 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001045285188
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001043880982
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001042686638
0 simplex iterations
21 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001041703608
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001040928355
0 simplex iterations
45 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001040359883
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001040000051
0 simplex iterations
26 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.00103984141
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001039879451
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001040122591
0 simplex iterations
26 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001040556867
0 simplex iterations
45 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001041194597
0 simplex iterations
27 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001042019612
0 simplex iterations
29 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001043041453
0 simplex iterations
28 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001044258079
0 simplex iterations
33 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001045652626
0 simplex iterations
29 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001047237922
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001049012414
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001050967859
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001053110487
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001055426877
0 simplex iterations
24 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001057925246
0 simplex iterations
23 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal, stalling; objective 0.00106060001
0 simplex iterations
30 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001063452812
0 simplex iterations
39 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001066479571
0 simplex iterations
40 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001069678622
0 simplex iterations
32 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001073048927
0 simplex iterations
27 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001076589141
0 simplex iterations
26 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001080298143
0 simplex iterations
25 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001084143277
0 simplex iterations
22 barrier iterations
MOSEK 10.0.43: MOSEK 10.0.43: optimal; objective 0.001088103524
0 simplex iterations
39 barrier iterations
import matplotlib.pyplot as plt
import numpy as np

fig, ax = plt.subplots(2, 1, figsize=(8, 4), sharex=True)

Rf = np.exp(252 * np.log(np.array([_.get_value("Rf") for _ in results])))
ax[0].plot(Rf, [100 * (np.exp(252 * m.get_value("ElogR")) - 1) for m in results])
ax[0].set_title("Portfolio Return vs Risk-free Rate")
ax[0].set_ylabel("annual return %")
ax[0].grid(True)

ax[1].plot(
    Rf,
    [
        [w[str(n)] for n in R.columns]
        for m in results
        for w in [m.get_variable("w").to_dict()]
    ],
)
ax[1].set_ylabel("weights")
ax[1].set_xlabel("Risk-free Rate")
ax[1].legend([symbols[n] for n in R.columns], bbox_to_anchor=(1.05, 1.05))
ax[1].grid(True)
ax[1].set_ylim(
    0,
)

fig.tight_layout()
../../_images/fce26198524c90f614c6ac72fbcee29f90214cfd609f4fed39fd3434bc89bead.png

Extensions#

  1. The examples cited in this notebook assume knowledge of the probability mass distribution. Recent work by Sun and Boyd (2018) and Hsieh (2022) suggest models for finding investment strategies for cases where the distributions are not perfectly known. They call them “distributionally robust Kelly gambling.” A useful extension to this notebook would be to demonstrate a robust solution to one or more of the examples.

Bibliographic Notes#

Thorp, E. O. (2017). A man for all markets: From Las Vegas to wall street, how i beat the dealer and the market. Random House.

Thorp, E. O. (2008). The Kelly criterion in blackjack sports betting, and the stock market. In Handbook of asset and liability management (pp. 385-428). North-Holland. https://www.palmislandtraders.com/econ136/thorpe_kelly_crit.pdf

MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (2010). Good and bad properties of the Kelly criterion. Risk, 20(2), 1. https://www.stat.berkeley.edu/~aldous/157/Papers/Good_Bad_Kelly.pdf

MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (2011). The Kelly capital growth investment criterion: Theory and practice (Vol. 3). world scientific. https://www.worldscientific.com/worldscibooks/10.1142/7598#t=aboutBook

Carta, A., & Conversano, C. (2020). Practical Implementation of the Kelly Criterion: Optimal Growth Rate, Number of Trades, and Rebalancing Frequency for Equity Portfolios. Frontiers in Applied Mathematics and Statistics, 6, 577050. https://www.frontiersin.org/articles/10.3389/fams.2020.577050/full

The utility of conic optimization to solve problems involving log growth is more recent. Here are some representative papers.

Cajas, D. (2021). Kelly Portfolio Optimization: A Disciplined Convex Programming Framework. Available at SSRN 3833617. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3833617

Busseti, E., Ryu, E. K., & Boyd, S. (2016). Risk-constrained Kelly gambling. The Journal of Investing, 25(3), 118-134. https://arxiv.org/pdf/1603.06183.pdf

Fu, A., Narasimhan, B., & Boyd, S. (2017). CVXR: An R package for disciplined convex optimization. arXiv preprint arXiv:1711.07582. https://arxiv.org/abs/1711.07582

Sun, Q., & Boyd, S. (2018). Distributional robust Kelly gambling. arXiv preprint arXiv: 1812.10371. https://web.stanford.edu/~boyd/papers/pdf/robust_kelly.pdf

The recent work by CH Hsieh extends these concepts in important ways for real-world implementation.

Hsieh, C. H. (2022). On Solving Robust Log-Optimal Portfolio: A Supporting Hyperplane Approximation Approach. arXiv preprint arXiv:2202.03858. https://arxiv.org/pdf/2202.03858