Monday, February 10, 2014

Front-loading your 401k

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.

Monday, January 6, 2014

Dividend Discount Model

This is part of a series on valuation techniques.

The fundamental reason why stocks are a vehicle for investment is that they represent a fractional ownership of a company and thus allow you to partake in that fraction of the profits. These profits, called dividends, are typically distributed once per quarter (i.e. four times a year) and are directly proportional to the number of shares that you own. If we have perfect information of future dividends, then we can compute the present value of a share of the company via discounting.

Suppose I have a constant cost of capital (also called the discount rate) of \(r\), i.e. the opportunity cost of 1 dollar over one year is \(1+r\) dollars. And for simplicity, let's say dividends are distributed yearly, starting tomorrow, at \(D_0, D_1, D_2, \dots\) dollars per share. Then the value (to me) of a share is \[ V = D_0 + \frac{D_1}{1+r} + \frac{D_2}{(1+r)^2} + \dots \] If the dividends are constant at \(D\), then this simplifies to a simple geometric series \[ \begin{align*} V &= D \left(1 + \frac{1}{1+r} + \frac{1}{(1+r)^2} + \dots\right) \\ &= \left(\frac{1}{1 - \frac{1}{1+r}}\right) D \\ &= \boxed{\left(\frac{1+r}{r}\right) D} \end{align*} \] If instead the dividends grow linearly at a rate of \(m\), then we have that \[ V = D + \frac{D+m}{1+r} + \frac{D+2m}{(1+r)^2} + \dots \] Then we use the standard technique for simplifying such expressions \[ \begin{align*} \left(\frac{1}{1+r}\right) V &= \frac{D}{1+r} + \frac{D+m}{(1+r)^2} + \dots \\ \left(1 - \frac{1}{1+r}\right) V &= D + \frac{m}{1+r} + \frac{m}{(1+r)^2} + \dots \\ \left(\frac{r}{1+r}\right) V &= D + \frac{m}{r} \\ V &= \boxed{\left(\frac{1+r}{r}\right) \left(D + \frac{m}{r}\right)} \end{align*} \] Finally, let's consider the case where the dividends grow exponentially at a rate of \(g\) \[ \begin{align*} V &= D + \frac{(1+g) D}{1+r} + \frac{(1+g)^2 D}{(1+r)^2} + \dots \\ &= D \left(1 + \frac{1+g}{1+r} + \frac{(1+g)^2}{(1+r)^2} + \dots \right) \\ &= \boxed{\left(\frac{1+r}{r-g}\right) D} \end{align*} \] It is worth noting that these computations only reflect the value of a stock for a given person's or organization's discount rate. The actual price of a stock is a function of supply and demand, i.e. the distribution of values as computed by everyone in the market.

Furthermore, having perfect knowledge of future dividend distributions is, of course, impossible. However, it can be reasonably approximated for certain classes of stocks, such as blue chips. For example, energy companies like Pepco (POM) and PG&E (PCG) have had very consistent dividends over the course of their lifetimes and can be expected to continue such trends in the future.

Perhaps also of interest, we assumed that the first dividend would be distributed the very next day. This reflects the maximum value of the stock to me. The minimum value is achieved the day after a dividend distribution. And the difference between these two values is given by \(D_i\) (i.e. the value will fall by \(D_i\) after the dividend is distributed). This can give rise to some arbitrage opportunities if the market is inefficient at such pricing.

Obligatory Disclaimer

The author is not qualified to give financial, tax, or legal advice and disclaims any and all liability for this information.