In this post I’m going to simulate the previous result using R. I provided the R code to generate the results but I’m a pretty bad programmer so I’d encourage you to look elsewhere for programming methodology etc. but nonetheless my code does produce the intended result. Before starting I loaded the MASS library because it’s a quick way to make plots look less shitty if you’re not an R pro.
First, I simulated 1000 random draws from an exponential distribution with parameter 1/15 and assign the result to a variable called sample:
n <- 1000
sample <- rexp(n, rate=1/15)
Then I made a histogram of ‘sample’ using the truehist command (which you’ll need the MASS library to use:
Now I want to somehow count how many of the random waiting times are greater than the average, 15. The graph gives us an idea – it doesn’t look more than a third – but it’s not precise. To know for sure I create a list of only the values that exceeded 15 in the sample called upper_sample with the help of a logical vector that returns true if the corresponding value in ‘sample’ is greater than 15 and false otherwise:
above_avg <- sample > 15
upper_sample <- sample[above_avg]
truehist(upper_sample, main=”Sample Waiting Times > 15″)
The number of elements in the upper_sample vector divided by the number of trials gives me the probability that a component lasts more than 15 hours, which I named psuccess:
psuccess <- length(upper_sample)/n
psuccess
There were 377 observations in my sample, so the simulated probability that a component lasts more than 15 hours is 37.7%, about 1% off from the true value.
Rates on the U.S. 10 year treasury went down in 2014, from about 3% to 2.17%, despite widespread predictions that monetary policy would inevitably cause rates to increase. If pundits everywhere predicted this, why did the opposite materialize?
Global growth slowed down in 2014, dragging inflation down with it. Greece had another…episode, which reintroduced the possibility of a ‘Grexit’, pushing rates down further. So, by 2014’s end, the 10 year treasury rate had decreased by about 75 basis points, moving in the opposite direction from rates on high-yield debt. In other words, credit spreads widened in 2014, meaning investors demanded more compensation in order to take on credit risk (i.e. lend to corporations via bonds).
Intuitively, widening credit spreads are the result of investors demanding more compensation in exchange for lending to corporations and other risky entities, because they perceive them to be increasingly susceptible to risk in comparison to the U.S. government. Why would this be the case? Usually because the economic outlook isn’t so good – investors fear corporations are less credit-worthy due to their prospects and demand to be compensated accordingly. Furthermore, investors tend to seek the safest securities during times like these – i.e. ‘flight to quality’ occurs – namely U.S. government securities, which pushes down rates. This increases the credit spread, pushing risk free rates further away from rates on other debt.
The scenario described above is why in 2014 the average return on the 10 yr. treasury was about 10% despite the consensus forecast that they would perform poorly. Remember, the price of a bond is inversely related to the rate of interest, which is why the widening credit spread and corresponding decrease in rates pushed up the value of 10 year treasuries held by investors. To see why, think about the discount factor associated with a 10 year treasury at 3.02% versus 2.17%. In the first case (i.e. at the beginning of 2014), a $1000 par value bond is worth $741. When the applicable rate of interest falls to 2.17%, however, the corresponding price is $805.87. Clearly, the value of the 10 year US treasuries was impacted favorably by the decrease in rates, benefitting the investors who held them throughout the year. Moral of the story: interest rates are fucked up and hard to predict, and Squawk Box is almost never right.
Over the past four years, the S&P500 had both a significantly higher maximum and lower minimum return compared to the PowerShares Low Volatility Index. The S&P experienced many more extreme returns (+/- 1% daily return), suggesting that returns on SPLV fluctuate less than the market. The S&P also earned a lower average return with higher variance than SPLV.
Period 5/6/11 to 1/6/15 |
S&P 500 | SPLV |
Max Daily Return | 4.63% | 3.75% |
Min Daily Return | -6.90% | -5.18% |
Returns less than -1% | 98 | 62 |
Returns greater than 1% | 110 | 71 |
Average Daily Return | 0.04% | 0.06% |
Average Annual Return | 0.99% | 0.75% |
Standard Deviation of Daily Return | 10.98% | 14.03% |
Standard Deviation of Annual Return | 15.71% | 11.85% |
The table below is the same analysis for only the year 2014, during which the US equity market posted more gains.
Year 2014 |
S&P 500 | SPLV |
Max Daily Return | 2.37% | 2.00% |
Min Daily Return | -2.31% | -1.99% |
Returns less than -1% | 19 | 14 |
Returns greater than 1% | 19 | 13 |
Average Daily Return | 0.04% | 0.06% |
Average Annual Return | 0.72% | 0.60% |
Standard Deviation of Daily Return | 10.70% | 15.80% |
Standard Deviation of Annual Return | 11.34% | 9.55% |
The claim that the PowerShares Low Volatility ETF (SPLV) tracks the S&P with less variability in returns is corroborated by this simple analysis. The graph of daily close prices and trading volume below also seems to corroborate this – the S&P500 Index (Yellow) fluctuates around the steady-ish path followed by SPLV (Blue). The ETF misses out on some gains during the summer months, but outperforms later in the year.
Interestingly, the fund achieves its low volatility by being overweight in Healthcare and Financials, not the quintessentially low-risk sectors like Telecom or Utilities.
The Maximum Likelihood Estimate is one estimate of a parameter’s value. It basically answers the question: “Given my data, what’s the most likely value of an unknown parameter”. To use it, we need to compute the likelihood function by taking a sample of n observations from the same distribution, evaluating each of the n observations, (x_{1}, …, x_{n}), at the pdf p_{X}(x), and taking the product of the result for all n.
Note: The expression p_{X}(x; θ) is just the distribution of the population from which we’re sampling – usually the parameter is added with a semicolon to emphasize the fact that the distribution is characterized by the unknown parameter
The likelihood function is just the joint density of the random sample. Since samples are independent and identically distributed (iid), the joint pdf of all n sample observations is the product of the individual densities. This is the same principle that lets us multiply the P(A), P(B), and P(C) together to find P(A intersect B intersect C) when events A, B, and C are independent. Suppose we take a sample of size n = 3 from the distribution, and the resulting values are x_{1}, x_{2}, and x_{3}. What’s the probability associated with the three sample values? That is, what’s the joint density of the three sample values, p_{X}(x_{1}, x_{2}, x_{3})?
Generalizing this case for all n clearly gives us the result that the joint density of n randomly drawn sample values is the product of individual densities and the likelihood function is nothing more than the joint pdf of the sample – a multivariate probability density function of the values taken on my the n random variables in the sample.
The likelihood function is a function of the unknown parameter. The Maximum Likelihood Estimate for the unknown parameter is the parameter value that maximizes the likelihood function:
We use calculus to find this value, by first taking the derivative of the likelihood function with respect to the unknown parameter, then setting it equal to 0 and solving for the parameter. Don’t forget to verify conditions so as to make sure you are indeed finding a maximum.
This will usually involve complicated, messy math. To mitigate this, we sometimes work with the logarithm of the likelihood function and use properties of logs to simplify computations. This won’t chance our answer – taking the logarithm of some function won’t change the point at which the maximum value is achieved.
The value of the parameter that you end up with maximizes the probability of your sample values x_{1}, …,x_{n}. You could say it’s the value “most consistent” with the observed sample – the Maximum Likelihood Estimate.
It is reasonable to claim that when a soccer game has penalty kicks, these are one of the most important moments of the game and in that moment everyone tries to guess which side will the striker kick. Indeed penalty kicks is a type of game that game theory can solve.
To solve this game, we first need to think about the type of game, namely if it is a simultaneous game or a sequential game meaning that it is one in which the players effectively make their decisions at the same time or one in which the players take alternate turns to make their choices, respectively. In penalty kicks the players move simultaneously since the goalie cannot wait until the ball comes off the foot of the kicker to decide what to do. So, both players have to choose a side to play before the “game” starts. We will…
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Major news to help developing countries with the capital needed to go directly to wind and solar. By the way, the United States signed a treaty to do this back in the early 1990s, and never contributed a penny up to this point. Treaties have the force of federal law under the Constitution, don’t ya know, unless following it is something that’s optional ….
I was curious about the housing market after meeting a seemingly overzealous realtor on the train, and here’s what I found via calculated risk.
At the end of Q3 2014, the delinquency rate on 1 to 4 unit residential properties was 5.85% of all loans outstanding, down for the 6th consecutive quarter and the lowest rate since the end of 2007. The delinquency rate does not include loans in foreclosure, though they as well are at their lowest rate since the 4th quarter of ’07 at just under 2.5%. Though foreclosures have come down from the stratospheric levels reached during their peak in 2010, they’re still more common than they were before the crisis. Mortgages that are 30 and 60 days past due, on the other hand, have returned to approximately pre-crisis levels.
30-year fixed rate mortgage (FRM) rates are down 1 basis point (.01) from last week at 4.01%, roughly the same level as 2011 but lower than last year’s 4.46%. Obviously there isn’t “one” mortgage rate – the rate we’re talking about here is the one that applies to the most creditworthy borrowers in the best scenario possibly to receive a loan from the bank. Though all other mortgages are based on this rate, it’s not exactly a rate one should expect to be offered by a bank.
The relatively small difference between a mortgage quoted at 4.01% and 4.45% has a surprisingly large financial impact on the 30 year FRM. A $250,000, 30-yr. FRM at a 4.01% nominal annual rate compounded monthly (as is typically the case) necessitates a monthly payment of $1,194.98, whereas the same mortgage at 4.45% would require a monthly payment of $1,259.30. With the higher payment, the borrower pays an additional $23,155 in interest over the term of the mortgage.
Another post talks about subdued refinancing activity, which I’d guess is the result of relatively static mortgage rates as it’s generally only financially viable to refinance when rates have changed significantly. Banks could also be offering fewer refinancing options after the crisis, a reasonable assumption given their cautious resumption of lending post-crisis and the role that refinancing options played in exacerbating the housing bubble. I’m purely speculating, though, and I’ll look into this more later.
A widespread slowdown in the rate of housing price increases has been steadily taking hold since February of this year. Residential prices aren’t decreasing, they’re just rising at a slower and slower rate each month, and now sit 20% below their 2006 peak. This is not to say we should expect or even wish that housing prices should resume at 2006 levels, as such was clearly unsustainable – furthermore, though slow relative to preceding months, the (annualized) 6%+ experienced last month is still pretty strong and obviously outpaces inflation.