CFA Level II Foreign Currency Translation

As usual the accounting portion of the CFA is taking me the longest to learn. In fact it currently accounts for 38% of the hours I've studied so far and I'm still having trouble consistently scoring over 70% on Stalla's tests.

In an effort to hammer some of the concepts home I've decided to mind map some of the decision rules, so here is the Foreign Currency Translation section in 3 slides.

Choosing the right method

Differentiating between Current and Temporal

 How to balance (what to plug)

Guide to my shorthand.
BS = Balance Sheet
IS = Income Statement
DNE = Does Not Exist
NI = Net Income
REb = Retained Earnings Beginning
Div = Dividend
Dep = Depreciation
MV = Market Value
G/L = Gain or Loss
Inf = Inflation
CHG = Change in

Common Financial Modeling Mistakes

A working list of things that everyone assumes right, that is actually wrong.

Beta Lever / Unlever
Right out of the gate students are taught about the perils of leverage. M&M one - leverage irrelevance, M&M two - full leverage is optimal, M&M three -  optimal leverage rests on earnings volatility assumptions (ie bankruptcy risk). Let's focus on M&M three, as this is closest to reality.

Theory suggests the optimal capital structure should be found by taking the limit of WACC, where Ke changes with respect to D/A via beta, and that debt should get more expensive under higher leverage ratios. Bankrupcy costs are nonlinear, changing from 1% debt to 2% debt has no effect, going from 90% debt to 91% debt has material effect. The graph below outlines what theory might expect a typical optimal capital structure curve to look like.

Here my bankruptcy multiplier would be suitable for a pretty stable firm, perhaps a utility company. Riskier firms would have higher multiples, and the minimum would be found closer to the left axis.

Now here is the catch

This is what is approximated by the beta unlever, lever formula (that is based off of M&M two) where Beta_levered = Beta_Unlevered*(1+(D/E)(1-t)). Notice this is a linear relationship.

Now look at this.

This is what my finance professor suggests is found in practice (highly recommend taking Prof. Jarrell, his class is where the bulk of this post comes from). The main takeaway from above is that bankruptcy costs seem only to kick in at the end, and that firms actually have a wide region of leverage ratios in which to achieve their minimal WACC.

So what's the problem? Analysts are constantly using the lever unlever formula to adjust a firms beta and thus their WACC in comparable analysis. This crops up in all sorts of places, but nowhere is it worse than WACC estimation for private firms. Private firms looking to IPO will usually have very different capital structures than the firms in the industry they are about to enter. IB analysts don't have a cost of equity because the firm has no publicly traded stock, so the discount rate for the company comes nearly entirely from peers. Trouble is they adjust for the different leverage using the beta unlever, lever formula. Relatively debt light firms will be heavily penalized and relatively debt heavy firms will get a unwarranted premium. Analysts also do this in relative valuation when firms have different leverage ratios.

The simple rule should be, when it's close don't adjust, when it's not close, try not to use it. I am very interested to see if these horror stories pan out in reality when I enter the industry. My hope is that a PhD has taken the time to develop a more robust leverage adjustment ratio that accounts for bankruptcy costs, just like the CAPM now is commonly used with add-on's like size risk, liquidity risk, and country risk.


Terminal CapEx and Depreciation Assumptions
A firm must build faster than it falls apart right? It's funny how many DCF's assume this not to be the case with implied depreciation larger than CapEx in perpetuity.

This is what I take to be the best DCF framework, (again it is also what is taught by Jarrell). When you have granular information about specific revenues and expenses doing cash flows by division or product definitely has value, however the general spirit of the valuation should be as above.

I have also had some success with the CFO - Chng in Op Cash + CFI(usually negative) - Pref Divs + Pref issuance + Debt = Free Cash Flow to Equity. The major downside with this method is how much information is impounded into CFO and CFI, here the burden really lies on your economic intuition in your pro forma statements you are pulling from. The trouble with the terminal forecast is it often sticks out as unreasonable when put next to the year that proceeds it, because it needs to encompass all years after forecasting, not just the "steady state" year after the growth phase. The CFO model lends itself to be a function of projection, with a black box effect on many crucial assumptions. This is what the forecast period is for, the terminal period is altogether a different beast and needs to be delt with appropriately.

In the forecast period your margins will lie heavily on company guidance and the extension of historical trends. Your CapEx and Depreciation assumptions can be stripped out the footnotes and MD&A. Getting the forecast period right is not rocket science and most practitioners will end up with similar results.

The terminal period however sees some interesting solutions and some very wrong solutions.

Error 1
The real egregious error that can be made is to forecast NCF in the terminal period as simply the NCF of the last year in the forecast period times the perpetuity growth assumption. Here you imbed all of the forecast assumptions into the terminal period, including assuming all terminal years will have the exact CapEx/Depreciation relationship as the last year.

Error 2
Some will crawl up to NOPAT and multiply it by the perpetuity growth and then use the plowback method. The plowback method is derived from economics. A firm will need to spend it's steady state growth (real) divided by its return on investment (real also) on PPE, and the rest of its NOPAT income can be returned to shareholders. The exact derivation of the formula can be found on the internet, or in Jarrell's class notes. The reason why forecasting off of NOPAT is wrong is because you imbed the profitability and efficiency assumptions from the forecast period into the terminal period.

Solution 1
The above solution works by forecasting revenue to grow at perpetuity and then making purposeful assumptions all the way down to NCF. COGS and SG&A margin can be gleaned by looking at mature firms in the industry. Change in working capital is forecasted the same way it was in the forecast period; steady state ((A/R+Inv-A/P)/Rev) multiplied by the change in Revenue from year to year. Finally the plowback ratio is used to end at NCF and then you find perpetuity figure through the Gordon Growth Model using a mature discount assumption.

Solution 2
Not shown above, but often used is to simply forecast the terminal cash flow using multiples valuation. (NCF of the firm * AveNCF of industry/Average MVE = Terminal MVE). The trouble here is the quality of the peer multiple (quality = standard deviation within the peer group), and its propensity to change overtime (volatility or the standard deviation of the median over time). Of course as with most things in finance, it is usually a good idea to do them both, and do a sensitivity analysis on top of each method.

Trading Size

Check out this video, I was reminded of it while studying for my trading exam this Saturday. After taking Burnside's class I have a lot more appreciation for what this teddy bear does, and the hurdles he faces. 

Types of Traders

Traders come in three main types (so say the academics), Noise, Informed, or Liquidity. Informed traders trade because they have valuable information about a security that isn't impounded in the price yet, and they place the position accordingly. Noise traders are traders who think they are informed, but are mistaken and are actually taking a bet against the true price. Liquidity providers are traders who step in and buy and sell a security with the goal to only make a spread between their buys and sells, or they are traders who place limit orders to buy or sell at certain prices. The former liquidity trader is usually a market maker, and the latter can be anybody,. 

The size trading teddy say's he's a liquidity trader, but what he really means is that he is a informed trader. Chances are most of his orders are "market orders" and so he is actually liquidity taking, not liquidity providing. 

Motivations to Trade

There are five main reasons people trade in the market.

• Information: Investors may have "slow ideas" based on fundamental analysis that suggest the stock is mispriced, and they will buy or sell hoping it reverts to the true price over a period of time. Traders may have "fast ideas" based on a wide variety of methods that may suggest the stock is either temporarily mispriced, or is about to move in a certain direction, and they will buy or sell hoping to catch that move.
• Liquidity: Most liquidity trades are made for reasons that have nothing to do with future outlook of the stock. Index’s that track a benchmark need to trade to avoid tracking error, mutual funds need to rebalance holdings, hedge funds need to adjust their holdings to keep certain risks neutral. Any trade that is of this nature can be labeled a liquidity trade.
• Noise: As mentioned above the market is full of traders who think they know something but actually don't. Their volume helps informed traders move size, and so they are an essential part of the markets efforts to decrease transaction costs. 
• Tax: Many individuals and funds sell at strategic times to push capital gains taxes around or to avoid dividends.
• Agency Conflict: Mutual funds hate to show their holder's stocks that have had big negative returns, and so before the report goes out many funds sell their losers. This is called "window dressing"

Transaction Costs

I had always thought of transaction costs as that commission fee I pay every time I buy or sell a stock. Turns out that is part of it, but there are two other costs that can be much harder to see that also play a role. The total transaction cost or "implementation shortfall" is made up of commissions and fees, the execution cost, and the opportunity cost. Execution cost can be thought of the difference between the average price per share you receive and the midpoint between the bid and ask right before you traded, and opportunity cost can be thought of the difference between the midpoint when you decided you wanted to trade the stock, and the midpoint right before you started trading. For small traders these extra costs are small, but they can be enormous for large traders, and the trouble with trading is that if you do well you get bigger.

Think of it this way. Let's say you have decided to spend a lot of time researching oil junior stocks to look for mispricing’s, you figure (correctly) that this small cap sector of the market won't have as much coverage and the chance to find a mispriced stock will be easier. Let's say you find a stock that is priced at $1.00/$1.05 (bid/ask) and you think it should be priced at $1.25. You want to buy 10000 shares of it. On paper you expect to make (1.25-1.025)*10000=$2250, but how much might trading cost you? As an oil junior the stock is thinly traded, average volume of 10000 per day, so you know there is no way you can buy all 10,000 shares at once. Let's say you decide to break it into 4 blocks, and trade in 4 consecutive days. Possible trades could go as follows:

• Day 1 Stock at 1.00/1.05, Buy 2500 @ 1.075
• Day 2 Stock at 1.05/1.10, Buy 2500 @ 1.10
• Day 3 Stock at 1.12/1.17 Buy 2500 @ 1.20
• Day 4 Stock at 1.15/1.20 Buy 2500 @ 1.20

On day 2 and 4 I assume enough liquidity that you can fill at the ask, but on days 1 and 3 you need to eat into the limit order book to fill the entire order. The spread is assumed at 0.05 which is a bit wide, but definitely not unheard of for small thinly traded securities. The costs break down as follows, your average price is $1.14375 and so the total implementation shortfall (assuming no commission) is (1.14375-1.025)*10000 = $1187.5. This is more than half of your expected profit, and we have only gotten half way through the transaction, we still have to sell! The opportunity cost is captured by how the security moved up as you scaled in over 4 days (very typical with heavy buying), and the execution cost is captured by the difference between the execution price and the midpoint. 

Problems with Trading Size

• Latent Demand Problem - When you are trading a significant portion of a stocks average daily volume, it can be tough to find enough counterparties to fill your trade. In searching for these parties you are forced to show the market your intended order size, or experience 'order exposure'.
• Order Exposure - Leakage about your order causes the market to move away from you because traders expect your liquidity needs to creep through the limit order book and they want the best price, and traders may think you are informed (what do you know that makes you want to take such a large position). 
• Price Discrimination - Traders will assume that because you are trading a large block of stock, you will trade another block shortly afterwards. This is very common in sell orders, panicking investors (informed or noise) will try to hide their costs by breaking up huge sell orders into smaller blocks, so traders learn to treat a block of any size with suspicion.
• Asymetric Information - If you know you will suffer the costs described above and you are still willing to trade size, you must know something (or so the market assumes). Even before doing their own homework, the market will side with your bet and the price will move away from you, because traders will assumed you are informed. 

The Winners Curse and Where Trading Still Works

As you can see from above, if you are a successful trader transactions costs will quickly become the nemesis of your strategy. As you get larger you will have trouble with your block orders, and if you start to gain a reputation of being successful the market will move as soon as you place your first order. This winners curse has an interesting selection aspect to it, if you are too good you kill your strategy, if you are bad you go bankrupt, if you are just okay you might be able to float along undetected at a reasonable size and scratch a living. 

There are ways to trade size, and a huge mutual find called Dimensional Fund Advisers manages to do it in the toughest of markets. Talking about how they do it is a blog within itself, but the essence is simple. They invest entirely on a passive strategy that rests on the principles on the Fama-French 4 factor model; assets have risk that is characterized by beta, size, value, and momentum. Small companies are riskier than large companies, and value is riskier than growth. The genius of the strategy from a trading sense, is they are proudly uninformed. They don't care what they buy (to a certain extent) and so they happily take hard to trade small cap stocks off active investors hands for a small haircut, when otherwise the market would fleece both with transaction costs. DFA gets the stock with negative transaction costs (profit), and the active investor saves a bundle on shortfall. The real interesting part is that through the 90's DFA made more money on this trading strategy, than it make on its investment strategy - and it's a passive fund!

What schools have traction with i-banks?

I ran across this graph for recent analyst hiring by a major BB IBD for North America. It's interesting how much of a chance the 'little guys' actually have.

Connecting Valuation with Entrance Timing

My Investment Management and Trading Strategies class has moved into the trading strategies portion of the curriculum, and we have been talking a lot about the tradeoff between execution costs and opportunity costs. This discussion has helped flesh out some of the classic arbitrage examples behavioral investors like to point out, as we quantify the frictions in the efficient market system which can give rise to these mispricings. While the big mispricings are interesting to study, I think the magic lies in all of the small mispricings.

It struck me in class that the biggest weakness of any valuation activity is the time period of applicability. On one had you have the trade-off every science deals with. The more exact the measurement, the more time the measurement takes, and thus the staler it gets. But finance struggles with another application hurdle; even if you can solve for the 'true' price of an asset instantaneously, you have no idea when the market price will move to that value or if in fact it ever will. You are subject to two independent risks, the market can stay wrong or get 'wronger' (move against you), or your true price estimate becomes untrue as information enters the market about the asset and the valuation changes.

You might say, sure but we do know that the market moves to the true price in the long run. I'd agree, but that true price is always changing. If the market price of X is higher than you think it should be, you might short X. Let’s say X is trading at 15, and you think it's worth 12. Your expectation is that X will drop from 15 to 12. You might even hedge your short with a long on the S&P to protect from systematic changes in the stock’s value, and only short the 'firm specific' valuation of the stock. What is to say the following doesn't happen: positive idiosyncratic information enters the market, and the market and true price rise to 17. The S&P doesn't change (by much) because only the stock reacted to the firm specific information, your short is a loss of $2, and your hedge didn't protect you. Even with absolute knowledge about the true price, you can lose money.

I think this is why some very successful asset managers seem to know very little about valuation, or sometimes even seem to care about it. Successful investment seems to be much more about the interaction between the market price and the true price over time. Decent money managers can get by with only knowing how one of the sides works. Traders only focus how the market price changes over time, 'deep value' investors focus intently on the derivation of the true price and somewhat haphazardly jump in when they feel the spread between the market price and the true price is big enough to warrant the risk of jumping in. A true master of investments should know both.

I'm trying to develop a checklist for investments that will focus my efforts on balancing these two fields. It will need to balance both fundamental and behavioral considerations. As luck would have it some much brighter minds are working on accomplishing exactly that!

MSF Class Quality, Simon Graduate School of Business, University of Rochester

I'm going to detail these as it applied to my interests and background. I am a direct from undergrad applicant, where I did a Bachelors of Commerce, and I am heading into Investment Banking. Thus I am biased against foundation courses as they are repeats for me, and I selected more banking courses than modeling courses.

All ranks are out of 5.
Content - Is the content itself interesting and at a masters level of comprehension.
Pace - How fast do you move through the material.
Real World Applicability - How likely will this class prepare you for your interviews and job.
Workload - Is it a 7 day a week grind?
Difficulty - How much re-tracing do you do on homework. How often do you not 'get it'.
Exams - How hard are the exams.
Cohort - How good are the classmates you are competing on the curve with.
Engagement - How pumped was I to be in class.
Professor: Who taught me.

Summer: 5 weeks, August to September
Corporate Budgeting - Mandatory
Content 3
Pace 5
Real World Applicability 3
Workload 4
Difficulty 2
Exams 3
Cohort 4
Engagement 5
Professor: Irfan Safdar
Bottom line: A good warm up for a challenging year ahead, it takes some getting used to Sunday nights at Simon!

This course is a total review of 2nd and 3rd year undergrad finance taught at breakneck speed. Although nothing's new it is useful to have it all put together in a short period, and is an excellent refresher for anyone doing a the fall banking recruitment push. The exams were challenging primarily because the midterm was a take home and several students grouped up so the curve was extremely competitive.

Core Statistics - Mandatory
Content 2
Pace 2
Real World Applicability 1
Workload 1
Difficulty 1
Exams 2
Cohort 2
Engagement 2
Professor: PhD Student
Bottom line: A baseline 'to little to fast' stats course that is mind numbing for ex stats grads and too fast for first timer's.

All the basics, t tests, regressions, binomial distributions, bayes theorem. It was almost comical to be taught time series econometrics in 5 slides after taking two entire courses on it in undergrad. This course is the vegetables you have to eat before the steak is brought out.

Core Economics - Mandatory
Content 3
Pace 1
Real World Applicability 1
Workload 1
Difficulty 1
Exams 3
Cohort 2
Engagement 3
Professor: PhD Student
Bottom line: Micro, Macro in 5 weeks.

A five week course somehow still felt painfully slow. However economics is always interesting, and these exams were surprisingly challenging, they really spread the curve out.

Communications - Mandatory Continues Until Spring
Content 1
Pace 1
Real World Applicability 4
Workload 1
Difficulty 1
Exams Nil
Cohort 1
Engagement 3
Professor: Dan Struble, Bob M.
Bottom line: Public speaking, business writing, leadership.

It seems fruitless but being video taped and critiqued on your presenting skills is an activity that can only happen in school.

Fall, 10 weeks, September to December
Investments - Mandatory
Content 3
Pace 2
Real World Applicability 3
Workload 3
Difficulty 3
Exams 3
Cohort 4
Engagement 3
Professor: Anzhela Knyazeva
Bottom line: It's Rochester so Investments means EMH.

This investments class does not extend beyond undergrad investments, but has some useful case based homework. Beware the Knyazeva sisters can get pretty tricky on their exams.

Institutional Finance - Mandatory
Content 4
Pace 3
Real World Applicability 5
Workload 4
Difficulty 3
Exams 3
Cohort 4
Engagement 5
Professor: Diana Knyazeva
Bottom line: Banking

I really enjoyed this class because it was the first departure from curriculum I had already received in my undergrad, and it cowed very closely to real world applications.

Corporate Financial Accounting - Or Financial Accounting 1
Content 2
Pace 1
Real World Applicability 2
Workload 1
Difficulty 1
Exams 1
Cohort 2
Engagement 2
Professor: Heidi Tribunella
Bottom line: Intro accounting.

Taking this class was a mistake. If you have any undergraduate accounting you definitely need to switch into the upper accounting course. This course does intro accounting, intermediate 1 and intermediate 2 at a surface level.

Corporate Finance - Mandatory
Content 3
Pace 2
Real World Applicability 2
Workload 4
Difficulty 3
Exams 3
Cohort 4
Engagement 4
Professor: Wei Yang
Bottom line: Stakeholder Game theory

This is Corporate Finance as you probably haven't been taught it before. You discuss all the different stakeholders, things like bankruptcy costs and debt overhang, but you do it all in a game theory framework. Homework, exams, and lectures are all structured around multistep games, with payoffs and NPV's. This added complexity makes the tuggle war between debt holders, shareholders, and management more interesting as you can quantify their actions and then the firms value as a result.

Winter, 10 weeks, January to March
Economic Theory of Organizations - Mandatory
Content 3
Pace 3
Real World Applicability 3
Workload 2
Difficulty 2
Exams TBD
Cohort 3
Engagement 3
Professor: Michael Raith
Bottom line: MBA course in an MS degree

I'm not entirely sure why we are learning this material in an MSF degree, but I think it's because Simon has been a research powerhouse in this area. It's all about incentive plans, intrinsic motivators, and org structure, all through economic maximization's. 

Cases In Finance - Optional
Content 4
Pace 3
Real World Applicability 5
Workload 3
Difficulty 3
Exams 3
Cohort 3
Engagement 5
Professor: Gregg Jarrell
Bottom line: Story time with some investment banking and cut throat 'negotiations'

The prof for this class is a riot, lots of great finance stories about half of which have a useful takeaway message. So far I've learned: never mess with the government, the best clients are billionaire's facing jail time, always act innocent even if your not, and bankers aren't really financiers, they are salesmen. Aside for the war stories, I have picked up some great 'things to remember' nuggets on the major valuation models. Jarrell is full of pointers on what mistakes professionals always make in their modeling and how to avoid them. For instance the much loved, beta unlever-lever technique for comparable valuation is flawed at it's core because it ignores the bankruptcy effect on the cost of equity.  The negotiations are a great game theory experiment, grades from 60 to 100 need to go out, and it all depends on what 'deal' you get from your peers. Every winner is matched with a loser, great class politics!

The test's are not hard however any mistakes are costly. Jarrell drops many useful hints in class, and non-Finance MBA's are nice to have on the curve come test time, because negotiations can be bloody and indiscriminant in their grade split ups.

Investment Management & Trading Strategies - Optional
Content 4
Pace 4
Real World Applicability 5
Workload 4
Difficulty 3
Exams 2
Cohort 3
Engagement 5
Professor: Daniel Burnside
Bottom line: A current mutual fund chief economist shows where EMH bends and where it breaks.

The prof for this course is very good. Down to earth, but extremely bright, lots of class engagement. The course starts a bit slow as you rehash EMH, but soon you are talking about anomalies, alpha creation, and I think the last half of the course is market structure and trading strategies. You are worked, but its mostly qualitative assignments.

The midterm is very similar to the practice midterm.

Financial Statement Analysis - Optional
Content 5
Pace 5
Real World Applicability 5
Workload 5
Difficulty 5
Exams 5
Cohort 4
Engagement 5
Professor: Charles Wasley
Bottom line: The hardest and most worthwhile course I have ever taken in my university career.

Do yourself a favor and sign up in the evening class with MBA's to make the curve a little easier because this class will punish you. The prof is a slave driver, and extremely strict (extra case assignment for anyone whose cell phone rings). I write the exam tomorrow morning, and after studying all week for it I can easily say it will be very difficult. The takeaways are huge though, FSA is an absolutely crucial skill for anyone who wishes to be an analyst, and this guy is the best I've seen. If you come to Simon for one course, make this it.

Be very weary as to how Wasley answers his qualitative questions on practice solutions, if you are not giving his 'right answer' on the exam to qualitative questions credit is very hard to come by.

Spring, 10 weeks, March to June
Fixed Income - Optional
Content 4
Pace 3
Real World Applicability 3
Workload 3
Difficulty 3
Exams 4
Cohort 3
Engagement 4
Professor: Wei Yang
Bottom line: A Fixed Income class that quickly moves beyond bonds and into structured products. Yang's exams always extend beyond the material and can have some very challenging questions.

Accounting For Management and Control - Required
Content 3
Pace 2
Real World Applicability 3
Workload 3
Difficulty 3
Exams 3
Cohort 2
Engagement 2
Professor: Jerold Zimmerman
Bottom line: A required managerial accounting class that can easily be switched out for another elective if the student wishes. The prof is good, but the class is almost entirely geared towards corporate management problems.

Entrepreneurial Finance - Optional
Content 5
Pace 4
Real World Applicability 4
Workload 3
Difficulty 3
Exams 3
Cohort 4
Engagement 5
Professor: Boris Nikolov
Bottom line: A very good follow up to Cases in Finance. The prof is young and the content covered in class is refreshingly technical. The class picks up after the business plan has been completed and goes all the way to IPO, focusing on forecasting and financing stages.

Thinking about Market Timing Strategies

In my 4th year undergrad I tried to create a crash predicting model based off of asset correlations. The idea was that as markets crash, unrelated securities correlate. The model failed for two reasons, by the time correlations really got underway the market had lost a lot, and markets correlate on the way up too.

But now that it's mid term week, I have a lot of "free time" on my hands and the bug bit again. This time I was going to make a much simpler model, based off of only variance. If I wasn't in the market I would be investing at the risk free rate, but as soon as I hit my buy signal I would plow everything I had in marketable securities into the market and keep it there. When my buy signal was tripped off I wouldn't sell any securities, but any new cash would be invested at the risk free rate until I hit a buy signal again.

Thesis
I predict that I can accurately enter market bottoms, by buying only on occasions where volatility is at its highest. I predict that these timed buys will yield profits that outweigh the opportunity cost of not investing in the market at all times.

Methodology
I have two time periods, from 1955 to present, and from 2000 to present. I have monthly S&P 500 and risk free rate data. I assume no transaction costs (or that I have a lot of capital). For presentation purposes I assume I have $1 of capital to invest each month, it can either be invested at the risk free rate, or buy a $1 stake of the S&P 500.

Variance has been calculated at the 6 month average Z score (average S&P value over last 6 months / standard deviation of S&P over last 6 months). The Z score is needed to standardize the non-randomly dispersed sample data (the index increases over the time period).

The buy signal is the manipulated variable, and is compared against the ratio (current 6 month Z score / average 6 month Z score for entire period). For instance if [buy signal] < Zcurrent/Z_ave then buy the market. Note that when markets are volatile the ratio is small, and when calm the ratio is large, bounded at 0. Also note that because we use the average 6 month Z from the whole period this is an ex post study, although it wouldn't be hard to estimate ex ante.

As stated above, if we don't have a buy signal we put $1 in a risk free investment. This $1 will be compounded monthly and receive additional $1 investments each month until a buy signal is reached. When a buy signal is reached the entire money market balance is invested in the S&P never to be sold. If the following month is also a buy signal, $1 is invested in the index at that months value, never to be sold. The first period that there is no buy signal, we invest $1 into a now empty money market account and the process starts again.

Results

1955 to Present
Before we get to the model results it is useful to look at the comparative returns between the S&P and the risk free rate.

This graph shows what dollar return you would get if you invested in either the S&P or the risk free rate some time in history. For instance in 1965 you would actually make more money on a dollar invested at the risk free, than you would in the S&P if you held them both to today. Thus although it is clear equity returns trump risk free return, there are exceptions, and abstaining from the market may provide good returns if timed correctly.

So let's put the strategy to work.

It sucked. The red line shows the number of trades, the blue line shows the return minus what you would make if you bought and held from 1955 ($6645 when investing a dollar a month). This is what a lot of the literature says, you can't beat the market, you need to be in it all the time. The horizontal access shows the buy signal value; because we only buy when the market is risky, the far left returns are when we hardly buy at all, and the far right returns are when we are almost always in the market (buy signal almost always on).

Just for fun I flipped the buy signal methodology. Now I want to buy when the market is at it's stablest, and invest in the risk free rate when it's at it's riskiest. Theoretically this doesn't make sense, risk should bring reward; but sometimes investments is about being contrarian so let's see what happens.

Would you look at that. It works! This graph works backwards to the one above, results on the left are when we are in the market the most, and the results on the right are when we are in the market the least. It looks like somewhere on the right hand side we hold off till an optimal moment, and then invest making an extra $3351 over the market return of $6645. I dug into it and below we see where that transaction actually takes place.

It just so happened that the last 6 months of 1983 were incredibly stable, Z/Z_ave peaked at 131, and we hit a buy signal in January of 1984 after investing only in risk free securities for 29 years. We dumped $1208 of compounded assets into the S&P, invested a further $1 in the S&P the next month, and then stuck to the money markets for the next 27 years to get a total return of $9995. This looks great but it is a data mining result in my opinion. If we drop our buy signal for 3.6 to 3.0 we make a $10 loss as shown below. It's to jumpy to be predictable.

We see here that buying earlier, in 1974, and later in 1993, although visually benign, totally wreaks the return profile with a slightly negative abnormal return.

2000 to 2011
So if it doesn't work in the last 55 years, how about the last 11?

Expected results this time, the low volatility strategy that worked above never significantly outperforms and the high volatility strategy that didn't work above, does create major positive returns with buy signal thresholds between 0.1 and 0.5. On the graph this looks like a very narrow window of opportunity, but we must remind ourselves that ratios compress outcomes between 1 and 0 and so there is a fair bit of room to work with.(As the numerator shrinks the marginal change in the ratio is less per unit change in the numerator.)

So what does the buy timing look like.

Graph error here and below. It should be "less than" and it's not 3.5% is 3.3%

We see in this strategy we abstained from the market up until volatility reached it's peak in the crash of 2009. The 6 month window worked in our favor here, if we were using a shorter window we would have likely bought in higher. This resulted in a $53 dollar over performance against the buy and hold which yielded $146 after 11 years of $1 monthly investments. This works out to a cumulative annual return of ~3% over the benchmark, not bad at all. However again this is a pretty optimal data set. It is surprising how much return we have to give up in order to grab 2002 as well.

We lose more than 1% in annual return, by grabbing both dips (which is what we want in an ex ante mindset). This loss is of course the direct result of the 2009 crash, even the best market timer was behind in 09; risk free would have been better.

Takeaways
It's not time to set up a hedge fund, the results are mined and ex post. It is however an interesting result that we can think about in the future. When the 6 month Z score is 3 times smaller than the historical Z score (30.74) we have a historical suggestion that we have reached a bottom.

Timing strategies - when they work - are chunky, you fall from a winning position to a losing position with small deviations of your buy signal. Considering this buy signal will need to be derived from ex ante data, the margin for estimation error is very slim. 

Contrarian strategies may work. Surprisingly a 1955 investor who waited for extremely stable markets to invest, and avoided rocky markets, could have done extremely well if their buy signals were just right.