## Monday, March 5, 2018

### Perspectives

The most popular question I've been getting from readers is where I think the economy is headed and where I'm currently allocating my investable capital.

As always, I do not claim to know for certain what's going to happen, and even less certain when they will occur. Nevertheless, I'd like to share my perspectives and thought processes in hopes that it will be informative, and with a little bit of luck, profitable.

To start with, I want to clarify what I mean by "the economy", which consists of various interconnected markets. Roughly speaking, there is a market for any good or service, which includes tangible commodities like rice and gold, as well as intangible financial assets like stocks and bonds.

The price in a given market is determined by the laws of supply and demand. In the long-run (>5 years), price movements are caused by changes to both supply, such as manufacturing improvements, and demand, such as an increased need for computers. These changes are slowly driven by human productivity and innovation.

However, in the short-run (0-5 years), productivity and innovation is roughly constant. What matters most in the short-run is the impact of credit (i.e. borrowing) on demand (i.e. spending). Credit is the reason why the economy moves in cycles. Ray Dalio summarizes this concept in his 30-minute video "How the Economic Machine Works".

Borrowing allows people to spend more than they earn, at the cost of spending less than what they earn at some point in the future. It's good when it helps increase your future productivity, such as getting a car loan so you can go to work. But it's bad when it finances unproductive spending and potentially disastrous when it finances a bubble (e.g. sub-prime mortgage crisis).

The cost of borrowing is called the interest rate, and it varies greatly based on the borrower and the purpose of the loan. However, all interest rates (for US dollar denominated debt) are tightly coupled to the federal funds rate set by the Fed; colloquially called "the" interest rate. The federal funds rate is adjusted eight times a year, with the expressed goal of stimulating the economy with low interest rates during recessions and cooling the economy with higher interest rates to combat inflation.

Ever since the Great Recession, we've been in an era of unprecedentedly low interest rates and quantitative easing (QE), which is the injection of money into the economy through the purchase of bonds by the Fed.

These two factors have several effects on investments

1. a decrease in fixed-income yields (savings account interests, bond yields, etc)
2. low mortgage rates lead to increased housing affordability and house prices
3. an increase in stock market investments as a result of (1)
4. an increase in riskier, "alternative" investments as a result of (1) and (3)

In other words, it doesn't pay to save and it pays to borrow. And since your spending is someone else's income, they end up with greater spending power, and the cycle becomes self-reinforcing. The demand curve is fueled by this low interest environment, prices soar, and we have inflation.

In any inflationary environment, you want to avoid holding cash. In today's environment, bond yields are pitifully low, due to being artificially suppressed by low interest rates and quantitative easing. While interest rates are finally rising again due to improved economic signals (CPI, unemployment, etc), they still have another year or so before returning to any semblance of historic normalcy. I expect stocks to continue to rise during this period and perhaps even a while longer, as people continue to pour money into them.

Now that I've laid out the overall US economic landscape, I think it's time for some words of caution related to point (4) above. Over the past few years, there's been a dramatic increase in riskier investments, which I believe has contributed to inflated asset prices. I see this in a few different areas

• Private equity
• Venture capital
• Cryptocurrency / ICOs
• "High yield" bonds
• Student debt

I am not calling the end to these "bubbles", since as is often the case, momentum can keep the party going long after the music stops. At the same time, I'm not sticking around for when the lights come on to be caught with my pants down. These risk factors, combined with historically high asset prices, lead me to believe that the easy money has been made and that uncertainty going forward is high. Avoid investing in times and areas when it doesn't pay to take on the risk.

In summary, the investment allocation I'm using right now is the following

• Leaving existing stock investments (~60% of investable capital, mostly in retirement accounts)
• Putting cash and future income into 1 month US treasury bills (~25% and growing)
• Betting on market melt-up while hedging against a melt-down using ETF call options (~10%)
• Betting against certain highly overvalued tech stocks using put options (~5%)

## Sunday, January 21, 2018

### Best everyday credit cards for 2018

It's almost 2018. If you're not getting at least 2% cash back on every credit card purchase, you need to catch up with the times. This post will go over a few of the top "everyday usage" cards out there today. This means
• no annual fee
• no rotating categories
• high rewards in major spend categories
• cash back redemption
In particular, we will not take into account
• churning signing bonuses
• niche card benefits
• point:mile transfer value
While these rewards can be lucrative, taking full advantage of them is relatively time-consuming and the dollar value is very person-dependent.

We'll take a look at the following most popular cards that fit the above criteria and how they perform over the following major spend categories
• All purchases (not covered by the below categories)
• Travel (i.e. flights, hotel)
• Dining
• Gas
• Groceries
We've summarized the cash back rewards in the following table. Cards with special considerations are starred and will be discussed further below.
Card Name All Travel Dining Gas Groceries
Citi Double Cash 2% 2% 2% 2% 2%
Chase Freedom Unlimited* 1.5 - 2.25% 1.5 - 2.25% 1.5 - 2.25% 1.5 - 2.25% 1.5 - 2.25%
BoA Cash Rewards* 1 - 1.75% 1 - 1.75% 1 - 1.75% 3 - 5.25% 2 - 3.5%
BoA Travel Rewards* 1.5 - 2.63% 1.5 - 2.63% 1.5 - 2.63% 1.5 - 2.63% 1.5 - 2.63%
Uber Visa 1% 3% 4% 1% 1%
PNC Cash Rewards 1% 1% 3% 4% 2%
Costco Anywhere* 1% 3% 3% 4% 1%
Sam's Club Mastercard* 1% 3% 3% 5% 1%

The Chase Freedom Unlimited gives you a base 1.5% in Ultimate Rewards points. These points can be normally redeemed at 1 cent per point, but if you get the Chase Sapphire Preferred or Reserve cards, you can get a 25% or 50% redemption bonus respectively when booking travel. Many people plan on getting the Reserve card at some point, so spending on the Freedom Unlimited card can be worth up to 2.25% back.

The Bank of America suite of cards have a redemption bonus depending on your Preferred Rewards status. Preferred Rewards status can be achieved by maintaining a total cash plus investment balance with Bank of America and Merill Edge. Simply having a Bank of America checking or savings account gives you a flat 10% bonus. A $20,000 balance gives you Gold status which has a 25% bonus,$50,000 gives you Platinum status which has a 50% bonus, and $100,000 gives you Platinum Honors status which has a whopping 75% bonus. Note for the Cash Rewards card, the bonus is applied at time of redemption as opposed to purchase, so if you plan on achieving one of these tiers in the future, the bonus is worth considering now when choosing that card. The Costco and Sam's Club cards are great, but require a membership with the respective warehouse store. If you already shop at these locations, it's definitely worth taking a look. Otherwise, it's probably not worth the membership fee just to get these cards. We've written a simple calculator for you to determine which combination of cards is optimal for your spending needs. How much do you spend (e.g. yearly) on your credit cards in the following categories? What version of the Chase Sapphire card do you plan on holding? Which tier of the Bank of America Preferred Rewards do you plan to have? ## Tuesday, December 19, 2017 ### Backdoor and mega-backdoor Roth IRA tldr: If you hold investments in a taxable account, but haven't maxed out your 401k, HSA, backdoor Roth, and mega-backdoor Roth, you are losing a lot of money to the tax man! In "Taking advantage of tax advantaged accounts" part 1 and part 2, we saw that retirement accounts make a huge difference in growing your investments due to their tax advantages. However, many of these accounts such as the traditional IRA and Roth IRA contain several restrictions as far as who can partake in these advantages. This post goes into how to get around these restrictions to maximize your retirement contributions (100% legal). First, a little background: for a traditional IRA, a single filer can only take a full deduction if you're modified adjusted gross income is$63,000 or less. Between $63,000 and$73,000, you can only take a partial deduction, and after $73,000 you can't take any deduction. A non-deductible traditional IRA is actually at a tax disadvantage compared to a regular taxable account because it can no longer claim the lower capital gains tax. You can contribute to the Roth IRA up to$120,000 AGI. Between $120,000 and$135,000, your contribution limit starts phasing out. After $135,000, you can no longer contribute to a Roth IRA. In any given year, the total contributions to all of your traditional and Roth IRA accounts cannot exceed$5,500. This puts yet another limit on how much you can take advantage of these accounts even if you meet the income thresholds.

The first technique is fairly well-known and is called the "Backdoor Roth IRA" or more officially as a "Roth IRA conversion". This uses a loophole where any traditional IRA can be converted to a Roth IRA, as long as you pay the requisite taxes. Remember, a traditional IRA may be deductible (i.e. tax deferred) whereas a Roth IRA is taxed immediately, thus when you do the conversion you must make sure you pay the taxes that a Roth IRA requires. However, you should only be using this technique if you're income is above $120,000 anyways (otherwise you should just contribute to a Roth IRA directly). This means your traditional IRA contribution was non-deductible and you already paid taxes on that amount, which means that you don't need to pay any additional taxes on the conversion. The conversion process is relatively simple with most brokerages (see your brokerage documentation for details). Two things to keep in mind about the process • If you have taxable balances in any of your IRA accounts, you are subject to the associated pro-rata rules and must pay taxes on the conversion, even if you are converting an after-tax contribution. • You should do the conversion immediately after your traditional IRA contribution, otherwise you will have to pay taxes on the capital gains. • You are still subject to the$5,500 IRA contribution limit.
The second technique is called the "Mega-backdoor Roth IRA", and is much less well-known, but can increase your tax-advantaged contributions by tens of thousands of dollars. This technique requires your employer to offer after-tax 401k contributions. The background is the following: the IRS limits your traditional and Roth 401k contributions to $18,000 (+employer match) per year. Some employers will additionally offer an "after-tax" 401k plan that is normally not very useful since it doesn't have any tax benefits (the tax treatment is comparable to a non-deductible traditional IRA). However depending on your 401k plan, you can convert those after-tax 401k contributions to a Roth 401k or roll it over to a Roth IRA. The process is more involved since you should do the conversion after every after-tax 401k contribution, which typically occurs every pay check. However, a few minutes of work every paycheck is a small price to pay for the associated tax-advantage. The total contribution limit for your 401k accounts is a whopping$55,000, which means (assuming a 50% employer match) you can put an additional $55,000 -$18,500 - $9,250 =$27,250 into a tax sheltered account every year. If you forget and do the conversion late and the account grows in the meantime, you'll just have to pay a small tax on the growth.

The mega-backdoor withdrawal does not count towards your IRA contributions, so you can (and should) take advantage of both the backdoor and mega-backdoor techniques. The combination of the two brings your tax-advantaged limits from ~$20,000 to ~$60,000 per year.

## Sunday, December 10, 2017

In part 1, we derived the growth rate of assets in various types of accounts. In this second part, we'll use this information to come up with some rules of thumb as to how you can optimally allocate your assets.

tldr: max out your 401k and HSA with stocks, then your Roth/deductible IRAs with stocks, put the rest in taxable accounts. Note, this is contrary to what many popular sources claim: that you should put tax-inefficient assets like bonds into tax advantaged accounts.

Since we're interested in maximizing our tax advantage, we'll look at the tax advantage of each of these account types, for both stocks and bonds, in absolute dollar terms.

We'll use the same illustrative example presented in part 1
• $1,000 pre-tax contribution amount • 28% income tax bracket • 28% retirement income tax bracket • 15% capital gains tax rate • 8% stock growth • 3% fixed-income yield • 20 year investment horizon • 50% employer 401k match Under these assumptions, we get the following account balances after the 20 year investment horizon for stocks Account Type Withdrawal Amount Advantage Regular$2961
HSA $4661 +$1700
Roth and Traditional 401k $5034 +$2073
Roth and Traditional+Deductible IRA $3356 +$395
Traditional, non-deductible IRA $2618 -$343

And similarly for bonds
Regular $1104 HSA$1806 + $702 Roth and Traditional 401k$1951 + $847 Roth and Traditional+Deductible IRA$1300 + $196 Traditional, non-deductible IRA$1138 + $34 The above table gives you a sense of how far$1000 (pretax) of stocks will go versus bonds. Once you have decided on a stock/bond allocation, you should allocate those dollars starting from the most advantaged places and moving to the least advantaged.

For example, let's say we plan on putting (in terms of pre-tax dollars) $50,000 into stocks and$50,000 into bonds. The above table would suggest maxing out our 401k first with stocks, so we put in $18,500 there. Then we should max out our HSA with stocks, so we put$3,400 there. Then (assuming we're eligible) we should max our our Roth/deductible IRA with stocks, so that's another $5,500. If we were not eligible for the Roth/deductible IRA, then we should max out our non-deductible IRA with bonds. Finally, once we've exhausted our tax-advantaged accounts, everything else should go into our taxable account. If we do this, our$100,000 investment will grow to $249,553 over the course of 20 years. If instead we had put bonds into our tax-advantaged accounts, it would only have grown to$222,384. In this scenario, this roughly corresponds to a 12% difference by simply putting assets in the right account.

The results are pretty robust to the input parameters chosen. I've made this spreadsheet (make a copy to edit) to play around with the parameters yourself if you don't believe me.

## Sunday, October 1, 2017

Different investment accounts have various tax implications, which affect your returns in deceptively dramatic ways. We'll take a look at two types of assets:
• Growth - e.g. stocks
• Fixed-income - e.g. bonds and CDs
and how they perform in the following investment accounts
• "Regular" taxable accounts
• HSA
• Roth IRA
First, let's summarize the tax differences between the two types of assets and the different investment accounts (recommended read). Then using this information, we'll derive formulas for calculating the post-tax return for various combinations of assets and account types (if you want to check my work). Finally, we'll provide some concluding remarks on the results.

Growth assets, such as non-dividend paying stocks, derive their value primarily from growth in price. For example, Tesla stock grew from $200/share to$350/share in the past year, for a gain of $150/share. We call these capital gains, and they are taxed at a special rate called the capital gains tax rate, which is typically much lower than your income tax. Fixed-income assets, such as bonds, CDs, and dividend-paying stocks, derive their value primarily from regular payments. This means that when you buy and hold the asset, the issuing entity will pay you some amount of money at set points in time. This payment goes under various names, such as dividend, coupon, interest, etc. But it is normally counted as income and taxed at your current income tax rate. Now let's go into some of the tax implications of the (non-exhaustive) list of investment accounts that you can buy these assets in. What I'm going to call a "regular" taxable account is just a typical account that you would open with a brokerage and fund using after-tax money (e.g. from your bank account). Fixed-income is taxed the year it is received, and capital gains are taxed the year the asset is sold. A health savings account (HSA) is a tax-advantaged account. You contribute with pre-tax money, pay no taxes on growth, and no taxes upon withdrawal when used for qualifying medical expenses. A traditional 401k account is a retirement benefit optionally provided by your employer. It is funded by pre-tax money that is deducted from your paycheck. As part of the benefit, your employer may choose to additionally contribute to your 401k account, typically in the form of a percentage match (e.g. for every dollar you put into your 401k, your employer will put in 50 cents). Taxes are simple: you don't pay income tax on the money you put in and you pay income taxes when you withdraw. One non-obvious consideration is that your income tax rate in retirement will likely be different than your current income tax rate. A Roth 401k is nearly identical to the traditional 401k, except you fund it using after-tax money with no taxation at withdrawal. This means you pay taxes now at your current income tax rate instead of later at your retirement tax rate. There's an additional twist to the Roth 401k, and that is employer matches are pre-tax and placed into a traditional 401k. A Roth IRA is like a Roth 401k, except there is no employer match. You put in after-tax money and withdraw without taxes. A traditional, deductible IRA is like a traditional 401k, except there is no employer match. You put in pre-tax money and get taxed at withdrawal. You can also think of it as a Roth IRA but tax-deferred. A traditional, non-deductible IRA and a traditional, deductible IRA are really the same account (it's just called a traditional IRA). However, depending on your income level, your contributions may or may not be tax-deductible. This difference has big implications, so it's worth treating the two cases separately. So how does this differ from a Roth IRA, since both are funded using after-tax money? In a Roth IRA, you don't pay any taxes at withdrawal. However, in a traditional, non-deductible IRA, you pay income tax on the earnings. Okay! Now we're ready to figure out how these factors impact our returns. Let's start by defining a few variables • $$C$$ - pre-tax contribution amount • $$PTM$$ - present income-tax multiplier, equal to one minus your present tax rate • $$RTM$$ - retirement income-tax multiplier, equal to one minus your retirement tax rate • $$CGR$$ - capital gains tax rate • $$i$$ - annual percentage yield, corresponds to the growth rate or fixed-income yield • $$T$$ - number of years • $$EM$$ - employer 401k match percentage For a regular, taxable account, a pre-tax contribution of $$C$$ gets income taxed, meaning that we really only start with $$C \times PTM$$ dollars in our account. If we invest in growth stocks, due to compounding growth after $$T$$ years, we will have $$C \times PTM \times (1+i)^T$$ dollars worth of assets. However, when we sell the assets, the capital gains are taxed at $$CGR$$. So we lose $$C \times PTM \times [(1+i)^T - 1] \times CGR$$ to taxes. Thus we are left with $\boxed{C \times PTM \times [(1-CGT) \times (1+i)^T + CGT]}$ Now let's consider a fixed-income asset in our taxable account. In this situation, we still start with $$C \times PTM$$ dollars, but our yield also goes down to $$i \times PTM$$. This is becomes the yield is also taxed by income tax, which lowers the compounding rate. This means after $$T$$ years, we end up with $\boxed{C \times PTM \times (1+ i \times PTM)^T}$ In a tax-advantaged account, like a 401k, no taxes are paid on neither growth nor yield, so the calculations will be identical for growth and fixed-income assets. In an HSA, things are quite simple since there are no taxes. Your pre-tax contribution $$C$$ gets the full benefit of compound growth $$(1+i)^T$$, and you get to spend the full resultant amount tax-free on health expenses $\boxed{C \times (1+i)^T}$ In a traditional 401k, we get our full pre-tax contribution of $$C$$ plus the employer match, so we start off with $$C \times (1+EM)$$. We then have compound growth of $$(1+i)^T$$ and then at withdrawal we pay an income tax of $$RTM$$. This leaves us with $\boxed{C \times (1+EM) \times RTM \times (1+i)^T}$ A Roth 401k behaves similarly, except we pay taxes up front on our contributions. So from our contributions, we have $$C \times PTM \times (1+i)^T$$. Now for our employer contributions which is treated as a traditional 401k, we have $$C \times EM \times RTM \times (1+i)^T$$. This gives us $\boxed{C \times (PTM + EM \times RTM) \times (1+i)^T}$ A Roth IRA behaves like a Roth 401k without the employer match portion, so we have $\boxed{C \times PTM \times (1+i)^T}$ A traditional, deductible IRA is like the Roth IRA but taxed-deferred. $\boxed{C \times RTM \times (1+i)^T}$ A traditional, non-deductible IRA behaves like a regular taxable account, but with a regular income tax instead of a capital gains tax. The account starts off with a taxed-contribution of $$C \times PTM$$ and grows tax-free so we get a factor of $$(1+i)^T$$. However the earnings are taxed at like income tax at withdrawal time, so we lose $$C \times PTM \times [(1+i)^T - 1] \times RTM$$ to taxes. This leaves us with $\boxed{C \times PTM \times [RTM \times (1+i)^T + (1-RTM)]}$ I've summarized the results in the following table Account/Asset Type Withdrawal Amount ($$\times C$$) Regular + Growth $$PTM \times [(1-CGT) \times (1+i)^T + CGT]$$ Regular + Fixed-Income $$PTM \times (1+ i \times PTM)^T$$ HSA $$(1+i)^T$$ Traditional 401k $$(1+EM) \times RTM \times (1+i)^T$$ Roth 401k $$(PTM + EM \times RTM) \times (1+i)^T$$ Roth IRA $$PTM \times (1+i)^T$$ Traditional, deductible IRA $$RTM \times (1+i)^T$$ Traditional, non-deductible IRA $$PTM \times [RTM \times (1+i)^T + (1-RTM)]$$ Just taking a glance at the formulas above, we can draw some obvious conclusions about the tax-effectiveness of various accounts. But to make it even more obvious, let's work out a numerical example based on some reasonable assumptions •$1,000 pre-tax contribution amount
• 28% income tax bracket
• 28% retirement income tax bracket
• 15% capital gains tax rate
• 8% stock growth
• 3% fixed-income yield
• 20 year investment horizon
• 50% employer 401k match
For stocks, we'll see the following result
Regular $2961 HSA$4661 +57%
Roth and Traditional 401k $5034 +70% Roth and Traditional+Deductible IRA$3356 +13%
Traditional, non-deductible IRA $2618 -11% And for bonds Account Type Withdrawal Amount Advantage Regular$1104
HSA $1806 +64% Roth and Traditional 401k$1951 +77%
Roth and Traditional+Deductible IRA $1300 +18% Traditional, non-deductible IRA$1138 +3%

With the formulas and numerical results in mind, we can draw the following conclusions
• there's a huge advantage in investing in your HSA and 401k accounts.
• the HSA has the biggest tax-advantage, but the employer match is usually enough to compensate or overtake the HSA.
• there is a significant tax advantage to be had in your deductible IRA accounts, since you're not paying any tax on the growth or yield.
• There is no difference between a traditional deductible vs Roth 401k/IRA if your income tax rate stay constant. However a lower retirement tax rate will favor traditional whereas a higher retirement tax rate will favor Roth.
• A non-deductible traditional IRA performs worse than a regular, taxable account for stocks since capital gains are taxed as ordinary income.
• The tax advantage is greater for fixed-income assets than for growth assets, because a tax-advantaged account improves the compounding ability of fixed-income assets
Of course, the real-world picture isn't quite so simple. There are restrictions, loopholes, and other considerations when using these tools. This post should convince you of the benefits of thinking about tax advantaged accounts and hopefully provide a basic framework for doing so. A deeper treatise on this will be the subject of a future post.

See part 2 to see how to make use of this information (with a tldr).

## Monday, February 10, 2014

The most common way of contributing towards your 401k is by setting aside a percentage of each paycheck. With a bi-weekly paycheck (once every two weeks), to max out the annual contribution limit of $17,500 (as of 2014), you would put in$673.08 per paycheck. While this strategy has many benefits in its simplicity and amortization, it is not the most optimal in terms of maximizing the long-term value of your retirement account.

Time is your most valuable asset in both saving and investing. If you are certain about how much you will contribute this year, then it is better to make that contribution as early on in the year as possible. This will give you a little extra time to let that money grow.

How much growth? Let's compare the two extreme examples: loading your 401k at the beginning of year versus loading it all at the end of the year. The difference between the two is a whole year of compounding. At a 10% growth rate, a front-loading a $10,000 contribution would net you an extra$1,000 by the end of the year. Assuming a consistent growth rate, that extra $1,000 will become over$2,593 in 10 years and over \$17,000 in 30 years. And not only that, but you'll be able to reap the same rewards each year.

 Example graph of net 401k value using each of the three contribution strategies assuming the same total yearly contributions.

If you compare front-loading to an amortized contribution over the course of a year, the benefit is approximately half of the above - still a very significant amount.

However, there are a few drawbacks that come with this more aggressive strategy:
1. You must know how much you will contribute ahead of time.
2. You must have an adequate amount of money saved up at the beginning of the year since your paycheck will be significantly diminished.
3. Negative economic growth will also be amplified.

## Tuesday, January 28, 2014

### Optimizing the asset allocation of your portfolio (part 1)

Suppose you have $$n$$ investment opportunities, each with its own rate of return distribution. How should you allocate your resources so that you maximize your long-term return?

At first glance, it seems optimal to put everything into the investment with the highest average ROI. It is the best performer after all and so we'd expect it to do just well in the future. The issue with this allocation strategy is that it is highly susceptible to gambler's ruin. That is to say, one bad day or year in that particular investment can completely wipe your whole portfolio out. It is this multiplicative nature of the rate of return that makes investing both a highly lucrative and a highly volatile business.

So what is the correct allocation strategy so that you minimize your risk and maximize your overall return? The answer is in the generalization of the Kelly criterion.

For this first part, let's restrict the problem to that of one investment opportunity. That is to say, you have the choice of what fraction $$f$$ of your portfolio to put into this one investment (keeping the rest in cash). It turns out that the optimal solution is of the form $f = \frac{\mu}{\sigma^2}$ where $$\mu$$ is the mean rate of return and $$\sigma^2$$ is the standard deviation.

Suppose we start out with $$V$$ dollars and this investment has a randomly distributed rate of return of $$R$$ over a given time period. We wish to find the allocation fraction $$f$$ that maximizes our expected long-run rate of return. Let $$r_1, r_2, \dots$$ denote the portfolio return for each time period. Then our asset value after $$t$$ periods is $V_t = V \times (1 + r_1) \times (1 + r_2) \times \dots \times (1 + r_n)$ As usual, multiplication is difficult, so let's maximize the expected log value $\log V_t = \log V + \sum_{i=1}^t \log(1 + r_i)$ Taking the expectation of this (letting $$X$$ be a random variable representing our portfolio return), we get \begin{align*} E[\log V_t] &= \log V + \sum_{i=1}^t E[\log(1+X)] \\ &= \log V + t \times E[\log(1+X)] \end{align*} Since $$\log V$$ and $$t$$ are constant, we simply need to maximize $$E[\log(1+X)]$$. Expressing $$X$$ in terms $$f$$ and $$R$$: \begin{align*} 1 + X &= (1-f) + (1 + R) \times f \\ &= 1 + fR \\ E[\log(1+X)] &= E[\log(1 + fR)] \end{align*} To simplify this further, we will use the second-order Taylor expansion of the logarithm $$\log(1+x) = x - \frac{1}{2} x^2 + O(x^3)$$. Thus we have that \begin{align*} E[\log(1 + fR)] &= E\left[ fR - \frac{1}{2} (fR)^2 + O((fR)^3) \right] \\ &= E[R] f - \frac{E[R^2]}{2} f^2 + O(f^3) \end{align*} To maximize this, we take the derivative with respect to $$f$$ and set it equal to 0 \begin{align*} 0 &= \frac{\partial}{\partial f} E[\log(1 + fR)] \\ &= E[R] - E[R^2] f + O(f^2) \end{align*} To a first-order approximation, we have that $\boxed{f \approx = \frac{E[R]}{E[R^2]}}$ i.e. you should allocate according to the ratio of the first and second raw moments of the distribution of returns. A quick sanity check verifies this approximation since a higher mean and lower variance leads to a higher allocation fraction.

If you have the third-moment, you can solve the quadratic to go up to a second-order approximation.

Also note that there are two other critical points for the boundaries: $$f=0$$ and $$f=1$$, which may be the correct solutions for some extreme distributions.

## Obligatory Disclaimer

The author is not a financial adviser, tax accountant, or lawyer and disclaims any and all liability for the contents of this blog. The information reflects the author's personal research and experience, which may contain errata.