Tag Archives: expected

Portfolio Allocations: Bet Sizing

The math that dictates optimal portfolio allocations is complicated and an overly simplistic approach introduces a lot of unnecessary risk. The math of “gambling” and the math of investing share a lot of similarities. I believe the math presented below is equally applicable to both worlds. While EV (Expected Value) is a critical concept, it is meaningless without the concept of EG (Expected Growth). Chip Kelly is not the guy the “Kelly Criterion” is named after, but his presence creates interesting football betting opportunities. As a guy who many would consider to be a “professional gambler,” the concept of bet sizing has been something that I have spent a lot of time thinking about. I firmly believe trading stocks and derivatives for great portfolio managers is not all that different from playing poker for elite poker players. Individual investments for a portfolio manager and individual bets of a poker player (or elite sports handicapper) may be extremely risky, but the entire set of investments that make up a portfolio or long series of bets over time by a “professional gambler” are likely to yield a high return with a relatively low risk over the long run. This article is in response to an Instablog written by one of the most interesting contributors on this site, Chris DeMuth . He gives a relatively simple methodology of how to allocate capital. He presents the basic premise that he will invest around 1.25% of his portfolio in an investment he likes and will increase his position in the stock if he continues to love it as the price declines. He will continue to add to this position until a maximum of 10% of his portfolio is allocated to the individual investment. Adding to an investment that is becoming more undervalued relative to its fair market value makes a lot of sense. However, the exact portfolio allocations he suggests seem to be quite arbitrarily chosen instead of meticulously calculated. Based on my user name, it is probably clear that I have spent a lot of time in my life thinking about the fancy math of endeavors most would consider to be reckless gambling. I would like to introduce the idea of the Kelly Criterion, the most fundamental formula for elite sports gamblers. You can read about it here . What this magic formula does is tell you how much of your portfolio (bankroll) you should invest (bet) on a particular investment in order to maximize the growth of your portfolio given your estimate of the probability of winning and the odds received on the wager. The formula is written below: Every investor (bettor) is familiar with the concept of EV (Expected Value). Everyone knows that positive EV bets are wonderful. However, there is a corresponding concept that is much less well understood. It is the idea of Expected Growth, and frankly, it is equally important to understand as Expected Value when thinking about portfolio allocations. Let’s consider an investment where you are allowed to bet on a game of flipping quarters. The odds of picking a winning bet are 50% when flipping a quarter one time. Let’s also assume that in this generous game that for each flip of the quarter, you are getting a +200 payout. For those of you not familiar with common sports gambling notation, this means that you are given 2-1 on your wager. If you wager $1 on this bet and lose, you will lose $1. However, if you win, you will receive back $3 ($1 for your initial investment and a $2 profit). Better yet, let us assume that there are no caps on how much we are allowed to bet. This is a wonderful game that I would love to play forever everyday if it were readily available. Let’s now further assume you have a bankroll of $1,000,000. You are allowed to play this game only two times. In this game, there are four distinct possible outcomes (the sample space) that each have the same probability of occurring. The 4 possible outcomes are as follows: Win both the first and second bets. Win the first bet, lose the second bet. Lose the first bet, win the second bet. Lose both the first and second bets. Let’s assume that you are conservative and wager 1% of your bankroll on each coin flip. These are the possible outcomes of the size of your bankroll after playing the game of 2 quarter flips. Bankroll = $1,000,000 x (1 + (2 * 0.01)) x (1 + (2 * 0.01)) = $1,040,400 Bankroll = $1,000,000 x (1 + (2 * 0.01)) x (1 – (1 * 0.01)) = $1,009,800 Bankroll = $1,000,000 x (1 – (1 * 0.01)) x (1 + (2 * 0.01)) = $1,009,800 Bankroll = $1,000,000 x (1 – (1 * 0.01)) x (1 – (1 * 0.01)) = $980,100 Since each of these results is equally likely, the Expected Value of the outcome of these sequential bets is a profit of $10,025 (or 1.0025%). Expected Growth (EG) is a little bit more tricky. In order to figure it out (without having the formula in front of you), you must see what the expected outcome is. Since the odds of the game are always 50/50 for each coin flip, the expected outcome is simply winning once for each time you lose. Since we are flipping the coin twice, the expected outcome is to win once and lose once. The order that you win or lose doesn’t matter as the bankroll ends up at the same number either way in this game. The bankroll, based on the calculations above, in the expected outcome is $1,009,800, which is a profit of $9,800 (or 0.98%). This 0.98% is the EG. For those that are interested in generalized equations, here they are: Bankroll after Expected Outcome = (Initial Bankroll) * (1 + (Decimal Odds – 1) * (Bet Size / Initial Bankroll))p * (1 – (Bet Size / Initial Bankroll))(1 – p) EG = (1 + (Decimal odds – 1) * (Bet Size / Initial Bankroll))p * (1 – (Bet Size / Initial Bankroll))(1-p) – 1 EV = ((p * Decimal Odds) – 1) * (Bet Size / Initial Bankroll) where p is the probability of winning Let’s look at a fun example of playing that original game but instead of betting 1% of your bankroll on each coin flip, you want to make a bet with a higher EV and bet 90% of your bankroll. (For those of you still reading at this point, that is far greater than what the Kelly criterion says you should bet.) The 4 possible outcomes are as follows again: Win both the first and second bets. Win the first bet, lose the second bet. Lose the first bet, win the second bet. Lose both the first and second bets. Bankroll = $1,000,000 x (1 + (2 * 0.90)) x (1 + (2 * 0.90)) = $7,840,000 Bankroll = $1,000,000 x (1 + (2 * 0.90)) x (1 – (1 * 0.90)) = $280,000 Bankroll = $1,000,000 x (1 – (1 * 0.90)) x (1 + (2 * 0.90)) = $280,000 Bankroll = $1,000,000 x (1 – (1 * 0.90)) x (1 – (1 * 0.90)) = $10,000 Since each of the 4 outcomes is equally likely, that yields an EV of $2,102,500 or a profit of $1,102,500 (or 110.25%). However, the EG here is a loss of 74%! That means that although you would be making bets with higher expected value, you would end up with (significantly) worse expected growth. In fact, you now expect a significant shrink in the size of your portfolio (or bankroll). Extending this logic out further, if you were to go “all in” on every single bet even if the coin were weighted in a manner such that you win 99.999999999% of the time, the EG = -100% if you are allowed to flip the coin infinite many times despite the fact that your EV would be exploding to infinity. The math of investing isn’t as simple as winning and losing as in the case, I present above. You get to input distributions of possible results with probability distributions of those results. In the end, you get to the indisputable truth that BET SIZING MATTERS (portfolio allocation sizing matters). The math gets way more complicated than this and frankly, I don’t truly understand it yet. The key takeaways from this fun math end up being quite intuitive: The better the investment (in terms of expected return and likelihood of success), the greater the percentage of your portfolio that you should allocate to this investment. If you overbet (oh what a horrible screen name to have for this discussion) or underbet, you will not maximize your expected growth. If you underbet in a +EV situation, you still will expect to grow your portfolio. If you overbet in a +EV situation, expected growth of your portfolio can be positive or negative. The penalty of underbetting (in general) is less severe than the penalty of overbetting. While I think the guidelines Chris DeMuth lays out are not necessarily all that bad in practice, I would caution taking an overly simplistic approach to a (very) complex problem.

The Natural Gas Market And UNG Heat Up Again

Summary The price of UNG rallied in recent weeks – it’s close to $15. The natural gas storage injection was 111 Bcf last week – below market expectations. Despite the recent recovery in natural gas market, the market still expects a high buildup in storage, which may indicate the price of UNG will remain low this year. The natural gas market continues to show signs of recovery, as the price of the Henry Hub increased by 9.6% since the beginning of the month. Moreover, the price of the United States Natural Gas ETF (NYSEARCA: UNG ) also rose by 10.5% during May. Despite the recovery in the natural gas market, it seems that the future progress of UNG isn’t too optimistic considering the latest Energy Information Administration report . The recent updated short-term outlook for the natural gas market still shows that the price of the Henry Hub is still expected to remain below $3 for the year. This is mostly related to the expected higher-than-normal storage buildup. According to the EIA, the current estimate is that the natural gas storage will rise to 3,890 Bcf by the end of the injection season in October, bringing the net injection to 2,420 Bcf, which will be the second highest injection season. Based on the current storage level, the injections to storage will need to be only, on average, 5% higher than the 5-year average for the storage to reach those levels by the end of October, as you can see in the chart below: Source of data taken from EIA The recent rally in the price of UNG and natural gas was partly driven by deviations between the storage buildup and the market expectations. The chart below shows the relationship between the percent change in the price of natural gas on Thursday – the day the EIA releases its weekly report – and the deviation in storage change from market estimates. (click to enlarge) Source of data: EIA The chart suggests a negative relation between the two data sets – the linear correlation is -0.59 – but since we have only a handful of data points for this injection season, this relation should be taken with a pinch of salt. In the past week, the demand for natural gas has picked up mostly due to higher consumption in the power sector. Considering the price of natural gas is low for the season, we could keep seeing higher demand for natural gas in the power sector. In the next couple of weeks, the weather is expected to be warmer than normal mainly in the East Coast. This could also drive the demand for natural gas in the power sector higher – after all, the cooling degree days are projected to be higher than normal this week. The supply didn’t move much, as the production remained nearly unchanged the past week. According to the latest update by Baker Hughes (NYSE: BHI ), the number of rigs rose again by 2 to 223 – which is still 32% below the level back in 2014. Last week’s injection to storage was 111 Bcf, which was 6 Bcf below market expectations. If the injections to storage were to remain lower than estimates, this could provide a short-term back-wind for UNG. But, over the coming months, the expected higher buildup could indicate the price of UNG isn’t going much higher than its current level. For more see: Natural Gas is Still Floating…Barely Disclosure: The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. (More…) The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.

Despite Slow First Quarter, Duke Energy Remains A Safe Dividend Play

Summary Duke Energy’s first quarter 2015 EPS of $1.24 beat estimates by $0.10, while Revenue of $6.06 billion missed expectations by $240 million. The company’s residential retail energy market declined as a result of more efficient energy practices and the company’s international business segment declined due to issues n Brazil. I believe Duke Energy stock presents a safe dividend play with opportunity for slow stock appreciation going forward. On May 1, 2015, Duke Energy Corporation (NYSE: DUK ) reported their first quarter of 2015 earnings results and provided an update on their four financial objectives for 2015 and beyond-(1) current year earnings guidance, (2) long-term earnings growth, (3) dividend growth, and (4) balance sheet strength. In this article, I will review the company’s four financial objectives and analyze their progress in obtaining them. Achieve 2015 earnings per share within guidance range of $4.55 and $4.75 Capital expenditures are expected to fall within the range of $7.4 and $7.8 billion for the year. In the first quarter of 2015, the company had $1.45 billion in capital expenditures putting the annualized projection to $5.8 billion. While the capital expenditures projection is lagging behind projections, management expects the economic development usage of the expenditures to result in almost 3,000 new jobs as the company makes commitments to pursue alternative energy generation sources. The company saw retail load growth of 0.5% to 1.0% for the year. The weather normalized retail growth rate decrease 0.2% year-over-year largely due to the 2014 polar vertex. The strong performance in the industrial market was offset by the disappointing residential market performance. The residential market experienced lower usage year-over-year due to changes in energy efficiency and conservation, polar vertex in 2014, and higher use of multi-family housing. There were 700M average shares outstanding at 12/31/2015. The company had 708M outstanding shares at 3/31/2015, up from 707M at 12/31/14. The company does not have any planned equity issuances through 2017. We saw $65 per barrel average Brent crude price for 2015. Oil price projections have remained consistent to projections as the expected Brent crude oil prices have increased from EIA’s February 2015 report of $57.56 to $61 in May 2015’s report . The joint venture, National Menthol Company (NMC), which runs through 2032, is 25% owned by Duke Energy. NMC’s earnings are positively correlated with crude oil prices and an approximate $10 per barrel change in the average annual price of Brent crude oil has roughly a $0.01 to $0.02 EPS impact annually. There was an exchange rate of approximately 2.85 BRL/US dollar. The exchange rate has increased above this expected rate to $3.01 on 5/13/2015 as the Brazilian economy struggles and the US economy rebounds. The continued drought conditions, struggling Brazilian economy, and weaker foreign currency exchange rates are the largest factors behind the $0.13 year-over-year quarterly earnings per share decline in the company’s international segment. The ongoing drought in the country has caused the company to dispatch higher cost thermal generation instead of the low cost hydro generation. Additionally, the struggling economy has caused the company to lower demand growth for 2015 between 0% and 2%, which is much lower than the greater than 3% seen over the past several years. Deliver earnings per share growth of 4% to 6% through 2017 There was retail load growth of 1% going forward. The company has been stagnant with a 0.6% retail load growth from 2012 and 2014. As seen by the decrease in the first quarter of 2015, I think it is going to be very difficult for the company to achieve a 1% growth going forward. I think it is going to be difficult to achieve because of the lower energy usages in homes. I don’t see this trend reversing and allowing this 1% growth rate to be achieved. The company expects total wholesale net margin to increase due to the new 20-year contract with NCEMC at Duke Energy Progress (began in 2013) and 18-year contract with Central EMC at Duke Energy Carolinas growing to a load of 900MW in 2019 from 115MW in 2013. FY2015’s total wholesale net margin is expected to be approximately $1.1 billion with an anticipated 5% compound annual growth rate. The regulated earnings base growth is expected to follow the $2 billion growth trend in 2015 that was seen in 2014. Continue growing the dividend within a 65% to 70% target payout ratio On May 7, 2015, Duke Energy declared a quarterly cash dividend of $0.795 per share, in line with previous quarterly dividends. Management expects the dividend to rise to $3.24 per share in 2015 (almost 2% increase year-over-year). With the Company achieving a payout ratio close to 70% and management’s commitment to paying out a quarterly dividend to investors, I do not see the company’s current 4% dividend yield to be at risk. Management has paid 89 consecutive years of dividends with increases coming the past 7 years. This is largely possible due to the Company’s strong balance sheet and no planned equity issuances through 2017. In addition, the company announced a strategically tax-efficient way to repatriate $2.7 billion back to the U.S. during the fourth quarter 2014 earnings call, which will help fuel the dividend increases going forward. Maintain strong, investment-grade credit ratings. While the company’s credit rating was recently upgraded by S&P, I believe there are three primary risks for the company going forward. The exposure to Brazil is a significant risk for the company’s future, which was seen in the 2014 financial results. In 2014, there was a decrease in sales volume as well as higher purchased power costs due to the interruptions in the hydrology production. Per the earning’s call, they are assuming normal hydrology despite the rainy season starting slowly. Brazil is a major story to follow for Duke Energy in 2015 and beyond as the Company is predicting EPS growth from this business segment despite recent downward trends in profits there as well as the Brazilian economy. I think the company will have difficulty increasing the retail load growth to 1% given the increased technologies and social initiatives to decrease electric use. Oil prices will continue to be a wild card going forward. Forecasting a price on such a volatile asset is a difficult task. If oil prices continue to fluctuate widely, it will significantly impact the company’s bottom line. Conclusion Duke Energy faces some difficult obstacles including a slowing Brazilian economy, lower residential energy usage, and volatile oil prices; however, I believe that the company gave conservative and very obtainable estimates in each of the key assumptions used to allow them to meet their financial objectives for FY 2015 and beyond. While I don’t see Duke Energy being a rapid growth story going forward which can be seen in the lagging capital expenditures, I do believe they have the ability to present slow stock appreciation with the safety of a consistent dividend. Disclosure: The author is long DUK. (More…) The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.