text: It is a date after the security is traded to the buyer that is after the issue date. Another word for yield. A high yield may have resulted from a falling market value of the security, which decreases the denominator value used in the formula and increases the calculated yield value even when the security's valuations are on a decline. The end result of the many differences in yield is … This is an approximation of the elastic limit of the steel. to supply or produce something positive such as a profit, an amount of food or information: an attempt to yield increased profits The investigation yielded some unexpected results. Yield is not, however, total return. Find more ways to say yield, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. Here are our top picks. © 2021 TheStreet, Inc. All rights reserved. A yield to maturity of a bond is the internal rate of return on a bond's cash flow, including the cost of the bonds, period payments from the bonds, if any, and the return of the principal at redemption. Yield describes an investment owner’s entire return or a part of the income. Maturity (required argument) – This is the maturity date of the security. The yield on cost calculation would look like this: If the investment made $10 during the year, and its current yield were calculated, it would look like this: As you can see, if a stock price increases and the dividend remains the same, the current yield will be lower than when the stock was originally purchased. It varies only slightly in quality...Read More ›, As an investor who loves to buy, hold and collect returns from my investments, I usually hate the idea of having to sell one of the stocks...Read More ›, Bill Gates and Warren Buffett are constantly make headlines, but their individual worth pale in comparison to the...Read More ›, Ronald Reagan, the 40th President of the United States (1981-1989), was a busy man. This publication is intended to provide definitions and explanations of these important terms for cattle producers and entrepreneurs that are interested in Yield refers to the earnings generated and realized on an investment over a particular period of time. Gains on stock prices, accruing profits to the business, also accrue profits to shareholders. Join 1,000+ other subscribers. A project’s location determines the amount of sunlight, or irradiance, … What Is Buffett's "Big Four" Sleep-At-Night Strategy? For example, a yield on bonds, such as the coupon yield is the annual interest paid on the principal amount of the bond. While individual high-yield bonds can be purchased, high-yield bond mutual funds offer more investment diversity. Here’s everything you need to know to get started today. Like yield, as it is a ratio, return is usually quoted as a percentage. It’s calculated by dividing the annual dividend by the current stock price. The percentage is based on the amount invested, the current market value, or the face value of the investment security. In other words, it is a measure of the cash flow an investor is getting on the money invested. Amortization Schedule Calculator: Find My Mortgage Repayment Schedule. The current yield would be ($50)/($900), or 5.56%. This is because yields react inversely to stock prices. Yield can also refer to the amount of usable product after it has been processed (peeled, cooked, butchered, etc.) ... and would mean ruling out about 1000 split votes on the amendments. 3. Understanding of the First Pass Yield. The appreciation of the share price plus any dividends paid in a year, divided by the original price of the stock, is the total return. The precise yield and floor space requirements of the Follow The Sun? Prior to starting InvestingAnswers, Paul founded and managed one of the most influential investment research firms in America, with more than 2 million monthly readers. The percentage is based on the amount invested, the … Learn how to create tax-efficient income, avoid mistakes, reduce risk and more. Yield is the term for earnings generated and realized on an investment over a specific period of time, expressed in a percentage. However, a high yield in either stocks or bonds can be the result of a falling market value of the security, decreasing the denominator value, even when the security's valuations are declining. 9 the profit or return, as from an investment or tax. Loan Interest Calculator: How Much Interest Will I Pay My Lender? Prior to starting InvestingAnswers, Paul founded and managed one of the most influential investment research firms in America, with more than 2 million monthly readers. Read More ›, By submitting this form you agree with our Privacy Policy, Please tell us what you liked about our content and how it helped you. . The percent yield is the ratio of the actual yield to the theoretical yield, expressed as a percentage. While yields of various investments do not explain the reasons for the gains and losses, they may mask declines in the underlying value of the assets or the effects of inflation. It is a ratio that defines how much a company pays in dividends or interest to investors each year, relative to the purchase price of the security. Yield is deﬂned as the ratio of the number of products that can be sold to the number of products that can be manufactured. Crop yield is the measurement most often used for … Is This The Ultimate Value Investing Model? To calculate Total Return, the purchase price is subtracted from the current price and added to the dividend, then is divided by the original purchase price: So, at is most basic, if an investor purchases a stock for$100, which pays an annual dividend of $3, and sells it for$153, the capital gain plus dividend of the stock would be $56. ], [Use our Yield to Maturity (YTM) Calculator to measure your annual return if you plan to hold a particular bond until maturity.]. Yield is the term for earnings generated and realized on an investment over a specific period of time, expressed in a percentage. The current yield refers to the annual payments divided by the current market price. There are various types of yield, and the method of calculation depends on the particular type of yield and the … Yield can mean different things in varying situations. Yield on Cost = Div/Purchase Price or Current Yield = Div/Current Price. It is usually computed on an annualized basis, though quarterly and monthly yields can be reported as well. Definition and meaning Yield is the the amount in cash (in percentage terms) that is generated by an investment. Yield on cost can be calculated by dividing the annual dividend paid and dividing it by the purchase price. = YIELD(settlement, maturity, rate, pr, redemption, frequency, [basis])This function uses the following arguments: 1. Among the potential higher-yielding investments are high yield bonds, Canadian Income Trusts, Master Limited Partnerships, Dividend Paying Stocks, Preferred Stocks, Real Estate Investment Trusts, and High Yield Bonds. Yield is the term for earnings generated and realized on an investment over a specific period, expressed in a percentage. Total return is a more comprehensive measure of return on investment, which factors in interest, dividends and capital gains. Definition: In financial terms, yield is used to describe a certain amount earned on a security, over a particular period of time. If, however, you buy the same$1,000 bond at a premium of $1,100, the current yield will be ($50)/($1,100), or 4.54%. solarintegrated.net. Because you paid a premium for a bond with the same fixed dollar amount of interest, the current yield is lower. As interest rates rise, investors may find themselves reconsidering their stock-to-bond allocations. How to Calculate Yield In general, yield is calculated as follows: Periodic Cash Distributions / Total Cost of Investment = … AN Irish trawler was barred from fishing in UK waters by a patrol boat in the first post-Brexit fishing clash. A former actor and governor of California, Reagan negotiated a treaty with the Soviet Union to eliminate...Read More ›, Mortgage rates are at historic lows. It is typically expressed on an annual basis as a percentage of the investment’s cost or current market value. Therefore, the total return would be: [($153-$100)+3]/$100 = 0.56 = 56% on the initial investment. tracking system was proven by a yield report of a renown, independent institute. Rate (required argument) – The annual coupon rate. Professional recipes should always state a yield; for example, a tomato soup recipe may yield 15 L, and a muffin recipe may yield 24 muffins. While there, Paul authored and edited thousands of financial research briefs, was published on Nasdaq. In the case of bonds, the yield on your investment and the interest rate your investment pays are sometimes, but by no means always, the same. However, in the stock tables, yield (“Yld”) is a reference to what percentage that particular dividend is of the stock price. Take advantage now by finding the most attractive rates in your local area. Action Alerts PLUS is a registered trademark of TheStreet, Inc. Cancel anytime. solarintegrated.net. Meanwhile, high-yield bonds tend to be offered by companies that present more risk than others. Dividend yield is a stock's annual dividend payments to shareholders expressed as a percentage of the stock's current price. This number (typically a fraction) is multiplied by 100 to give a percent. While typical values can range from 1,000 kWh/kWp to over 2,000 kWh/kWp, the actual value is driven by many factors, including: Location. Settlement (required argument) – This is the settlement date of the security. The stock may also pay a dividend of $2 per share, during the year. For stock investments, two kinds of yields are generally watched - yield on cost, and current yield. Paul has been a respected figure in the financial markets for more than two decades. If the price you pay for a bond is higher or lower than par, the yield will be different from the interest rate. The 8 Best Reasons to Invest in a Roth IRA, 5 Tips to Know Before Using an Online Brokerage, How to Become Financially Independent in 5 Years or Less, Target Date Funds: The Hassle-Free Way to Invest in Your 401k or IRA, 10 Outrageously Weird Craigslist Jobs You Won't Believe Are Real, Everything Beginners Need to Know About Commodity Investing, Wondering When to Sell a Stock? The difference between yield on cost and current yield is that, rather than dividing the dividend by the purchase price, the dividend is divided by the stock's current price. fourteen States for the Acreage report, released June 28, 2019, were re-contacted in July. Read More ›, Whether you want a$750 sign-up bonus, 6% cash back, or a 0% introductory APR, we’ll help you find the perfect credit card for your needs. Paul has been a respected figure in the financial markets for more than two decades. Read More ›, Looking to secure a low-interest loan with minimal fees? Favorable weather yielded … That's where yield differs from total return. [Use our Yield to Call (YTC) Calculator to measure your annual return if you hold a particular bond until its first call date. First pass yield (FPY), also known as throughput yield (TPY), is defined as the number of units coming out of a process divided by the number of units going into that process over a specified period of time. If an investor buys a stock at $100 per share, and the stock price rises to$120 in a year, the stock investment gain is $20. Percent yield is the percent ratio of actual yield to the theoretical yield. Yield in culinary terms refers to how much you will have of a finished or processed product. 4. Yield refers to the cash return to the owner of a security or investment. We'll never sell or share your email address. Learn more about TheStreet Courses on investing and personal finance here. Read More ›, Looking for a dependable bank that charges low fees and pays up to 10X higher interest than the national average? $\text{Percent Yield} = \frac{\text{Actual Yield}}{\text{Theoretical Yield}} \times 100\%$ Percent yield is very important in the manufacture of products. How Many Years Will It Take to Save a Million Dollars? production yield definition in English dictionary, production yield meaning, synonyms, see also 'production line',production platform',pair production',mass production'. Yield strength is the maximum stress that can be applied before it begins to change shape permanently. It is sometimes figured up as an IRR, or Internal Rate of Return, or alternatively as a ratio. A significant rise in yield without a higher stock price could indicate a company is paying a dividend without a commensurate rise in earnings, which could also suggest problems in the near future for the business. Return is the gain or loss an investment makes over a certain period of time. Yields vary with different types of investments in securities, the duration of the investment, and the return on it. Yield is only a part of total return. The term yield may refer to slightly different aspects of a return for variable types of investments. 10 the annual income provided by an investment, usually expressed as a percentage of its cost or of its current value. When you're investing, you'll want to know what sort of money you can expect and will earn on your assets, over a specified period of time. The Northern Celt's skipper was told … It is calculated to be the experimental yield divided by theoretical yield multiplied by 100%. Investment managers find this ratio useful to define the optimal asset allocation in the portfolio of the client. It is the date when the security expires. In this chapter, we are going to discuss yield loss mechanisms, yield analysis and common physical design methods to improve yield. Specific yield (or simply “yield”) refers to how much energy (kWh) is produced for every kWp of module capacity over the course of a typical or actual year. The percent yield formula is a way of calculating the annual income-only return on an investmentReturn on Investment (ROI)Return on Investment (ROI) is a performance measure used to evaluate the returns of an investment or to compare efficiency between different investments. If the actual and theoretical yield are the same, the percent yield is 100%. Sign up for our weekly newsletter and get our most popular content delivered straight to your inbox. Current yield, in contrast, is the interest or dividends divided by the current market price. com, Yahoo Finance, and dozens of other prominent media outlets, and appeared as a guest expert at prominent radio shows and i... Yield refers to the cash return to the owner of a security or investment. europarl.europa.eu. A 10-year bond yield combines the bond’s interest income and its capital gain or loss to calculate an average annual rate of return. Introducing TheStreet Courses:Financial titans Jim Cramer and Robert Powell are bringing their market savvy and investing strategies to you. Compound Savings Calculator: How Much Should I Save Each Year? Earnings yield is a measure of a company’s earnings relative to its market cap.Owners of a stock can consider a company’s earnings yield as a measure of total return on their investment into the company. With our courses, you will have the tools and knowledge needed to achieve your financial goals. Yield is most important to income investors. Generally, yield is calculated by dividing the dividends or interest received on a set period of time by either the amount originally invested or by its current price: For a bond investor, the calculation is similar. Receive full access to our market insights, commentary, newsletters, breaking news alerts, and more. Money Market vs Savings: Which Account is Best for You? In general, yield is a return on the money you invest. Use This 4-Minute Checklist, 10 Things You Didn't Know About Carlos Slim. Yield measures the cash flow an investor receives on the amount invested. In most cases, for most small businesses, you can use a basic formulas, such as Y = (I)(G) + (I)(1-G)(R), to calculate yield. In general, yield is calculated as follows: Periodic Cash Distributions / Total Cost of Investment = Yield. Enrich your vocabulary with the English Definition dictionary Estimated typical cost of modern 300mm or 12inch wafer 0.13 „m process fabrication plant is$2-4 As another example, stock investment gains can come in two forms: in terms of price rise, and in terms of a dividend. Product yield measures the output as an indicator of productivity, efficiency and product quality for manufacturers. A rising stock price and rising dividend should result in a consistent or marginal rise in yield. The outlook is...Read More ›, By definition, a commodity is a product that tends to come from the earth or is grown. Current yield is the coupon yield on a bond at a specific point in the time before the bond maturity. If stress is added to the metal but does not reach the yield point, it will return to its original shape after the stress is removed. High-yield bonds have been referred to as "junk" because the companies have to pay a higher yield to attract investors. Yield is a part of the total return generated from holding an invested in financial security over a year. Yield improvement is the most critical goal of all semiconductor operations as it reflects the amount of product that can be sold rela-tive to the amount that is started. Much time and money is spent improving the percent yield for chemical production. Factors that impact specific yield. The approach seems complicated only because it requires an understanding of the system by which bonds earn money. We will answer the question 'What is yield management?’ YM is a strategy based on selling to the right customer, at the right time, for the right price. Important to the understanding and application of yield and quality grades is the definition of beef quality, cutability and undesirable carcass traits. If you have a question about Yield, then please ask Paul. Also known as the yield to maturity, it assumes the 10-year bond’s interest payments to be reinvested at the same rate. In equities, yields on preferred shares are similar to bond yields. Yield includes interest earned, or dividends received from holding a particular security over the specific period. Using the yield is a convenient way of comparing the returns on various financial investments. In finance, the yield on a security is a measure of the ex-ante return to a holder of the security. Only good units with no rework or repairs are counted as coming out of an individual process. That is, incremental increases in yield (1 or 2 percent) signifi- Traditionally, yield is the proportion of correct items (conforming to specifications) you get out of a process compared to the number of raw items you put into it. This is also why stocks with less growth potential often offer higher dividend yields to investors than stocks with high growth potential. The nature and valuation - whether the valuation is fixed or fluctuates - results in yields being classified as known, or anticipated. Homeowners Insurance: Protect Your Investment, Travel Insurance: Protection from Your Worst Trip Nightmares, How to Pick the Best Life Insurance Policy. the yield on these shares is 15 per cent at today's market value. It refers to the interest or dividend earned on debt or equity, respectively, and is conventionally expressed annually as a percentage based on the current market value or … But it ignores capital gains. As an example, say an investor has put $100 into a stock that paid$1 as an annual dividend. Mortgage Calculator: What Will My Monthly Principal & Interest Payment Be? If a yield appears extremely high, it could be an indication that either the stock price is going down or the company is paying a high dividend, or both. The Best Stock To Profit From America's 'New Competitive Advantage', Simple Savings Calculator: See How You Can Grow Your Money, Calculate Cost of Monthly Used Car Loan Payments, Your Guide to Mortgages and Finding the Best Rates, Your Guide to Home Equity Loans and HELOC, 90,000 Reasons Why You've Got The Wrong Mortgage, Your FICO Score: 5 Things You Didn't Know Could Hurt It, 5 Secrets to Surviving the Mortgage Process, 5 Owner Financing Options for Home Buyers, 5 Devastating Mistakes That Turn 0% Credit Cards into Nightmares, Lower Your Credit Card Interest Rate with These Magic Words, How to Find a Personal Loan with the Best Rate, Using Leverage and Debt to Juice Your Investment Strategy, Good Debt: The 5 Best Reasons to Borrow Money, How to Create Your Own Loan Amortization Calculator, A Full Breakdown of Savings Accounts to Help You Find the Best Rates, Money Market Accounts & How to Find the Best Rates, Your Guide to CD Accounts and Finding the Best Rates, How to Find the Right Checking Account for You, How to Avoid Paying Bank Fees Once and For All, 7 Ways to Hack Proof Your Online Bank Account, How To Protect Your Assets When Your Bank Fails. The U.S. 10-year Treasury yield hit its highest level … For example, the dividend yield is the total payments in a year from the preferred shares divided by the principal value of the preferred shares. The yield won't include variations observed in the price of the security, like changes in the share price on its way to $120. Excessive rainfall had led to planting delays and challenges at the time of the survey, leaving a portion of acres still to be planted ... Yield, and Production – States and United States: 2018 and Forecasted . Yield is also the single most important factor in overall wafer processing costs. ROI measures the return of an investment relative to the cost of the investment. Read More ›, If you want to lower your insurance bills, then these reliable providers offer some of the best rates on home, auto, and life insurance. Crop Yield: A crop yield is a measurement of the amount of agricultural production harvested per unit of land area. 2. Thus, yield is a major decision-making tool used both by companies and investors. It is measure applied to common, preferred stock, convertible stocks and bonds, fixed income instruments, including bonds, including government bonds and corporate bonds, notes and annuities. The dividends on preferred stocks are paid at a fixed rate, and companies are required to pay preferred stockholders dividends before common stockholders, which can make them attractive for high-yield investors. It's expressed as a percentage based on the … Even people with jobs are depressed about the nation's unemployment statistics. Because higher yields are often an indicator of higher risk, a number of high yield investments attract those with more risk appetite than aversion. | Meaning, pronunciation, translations and examples Yield definition: If you yield to someone or something, you stop resisting them. As an example, if you invest$900 in a $1,000 bond that pays a 5% coupon rate, your interest income would be ($1,000 x 5%), or \$50. The traditional calculation of yield is often employed on the final inspection step of a process to measure the effectiveness of the overall process. The first pass yield at an operation is calculated as quantity completed on the first pass through the operation divided by quantity started at the operation on the first pass within the specified timeframe. At InvestingAnswers, all of our content is verified for accuracy by.


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# Using recursion to find the trace of gamma matrices

All you need to know about the gamma matrices for this problem is that there are four of them, $\gamma_{\mu}$ with $\mu = 0, 1, 2, 3$, and that the trace of the product of n gamma matrices (n even) obeys the following recursive relation:

$Tr(\gamma_{\nu_1}\gamma_{\nu_2}...\gamma_{\nu_n}&space;)&space;=&space;\begin{cases}&space;4g_{\nu_1,\nu_2}&space;&&space;\text{if}\&space;n&space;=2&space;\\&space;\sum_{k=2}^{n}&space;(-1)^{k}g_{\nu_1,\nu_k}&space;Tr(\gamma_{\nu_2}...\widehat{\gamma}_{\nu_k}...\gamma_{\nu_n})&space;&&space;\text{if}\&space;n&space;>&space;2&space;\end{cases}$

Here, $\widehat{\gamma}_{\nu_k}$ signifies absence of $\gamma_{\nu_k}$ and $g_{\nu_1,\nu_k}$ is a tensor with two indices.(BTW, when n is odd the result is zero). Here's my attempt at the code:

TrGamma[z__] :=
If[Length[z] == 2,
4*Subscript[g, z[[1]], z[[2]]] ,

(*Else*)
Module[ {x = 0, n = Length[z]},
For[i = 2, i <= n, i++,
Module[{y = Delete[z, {{1}, {i}}]},
x = x + (-1)^i*Subscript[g, z[[1]], z[[i]]]*TrGamma[y]]
];
x
]
]

This gives the correct answer for n=2 and n=4:

In[4]:= TrGamma[{Subscript[\[Nu], 1],Subscript[\[Nu], 2]}]
TrGamma[{Subscript[\[Nu], 1],Subscript[\[Nu], 2],Subscript[\[Nu],3],Subscript[\[Nu], 4]}]

Output:

$4&space;g_{\nu&space;_1,\nu&space;_2}$

$4&space;g_{\nu&space;_1,\nu&space;_4}&space;g_{\nu&space;_2,\nu&space;_3}-4&space;g_{\nu&space;_1,\nu&space;_3}&space;g_{\nu&space;_2,\nu&space;_4}+4&space;g_{\nu&space;_1,\nu&space;_2}&space;g_{\nu&space;_3,\nu&space;_4}$

However, when I try n = 6 or higher it just keeps running. I'm guess there's something about Module that I'm missing or have misunderstood. I've tried replacing Module with Block and the problem still persists.

• You might be interested in InternalDiracGammaMatrix[]: Table[InternalDiracGammaMatrix[k], {k, 5}]. – J. M.'s torpor Jan 15 '17 at 12:32
• Or you could study the source code of DiracTrace – Rolf Mertig Jan 15 '17 at 18:58

Since the formula has a sum ($\sum$), it's best to use Sum:

TrGamma[nu1_, nu2_] := 4 Subscript[g, nu1, nu2]

TrGamma[indices___] :=
Sum[
(-1)^k Subscript[g, {indices}[[1]], {indices}[[k]]] *
TrGamma @@ Delete[{indices}, {{1}, {k}}],
{k, 2, Length[{indices}]}]

• Tried it and it works! – Feynmaniac Jan 15 '17 at 19:37
• @Feynmaniac Ok, don't forget to upvote this and other answers if you found them helpful. Mark it as accepted if it answers your question. – QuantumDot Jan 15 '17 at 21:32
• Right. Sorry, I'm new here. – Feynmaniac Jan 15 '17 at 21:46

Are you just trying to understand how to get recursion working in Mathematica, or are you mainly interested in having some useful functions for dealing with gamma matrices? If the latter, there is a package GAMMA by Ulf Gran which will handle all your gamma matrix needs, in any dimension and in any signature:

https://arxiv.org/abs/hep-th/0105086

• I'm interested in both, actually. I'll check out the package. – Feynmaniac Jan 15 '17 at 19:37
• @Feynmaniac Depending on what you want to do, you may find FeynCalc quite useful. – QuantumDot Jan 15 '17 at 21:34

format: plain


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# Tag Info

1

In ElGamal encryption like you said for unknown $m_1$ and known $m_2$ we have: $m_1: C_1=g^k\ mod\ p,\ D_1=m_1.g^{a.k}$ $m_2: C_2=g^k\ mod\ p,\ D_2=m_2.g^{a.k}$ $k$ is the same in both encryptions, by dividing $D_1$ and $D_2$ we have: ${D_1\over D_2}={m_1\over m_2} => D_1.m_2 = D_2.m_1$ $m_2$ , $D_1$ and $D_2$ and are known so now we can calculate \$...

1

A nonce on its own does not prevent replay attacks. It is just a number, it doesn't do anything, it can't give any guarantees. You could define a protocol with a nonce, that has no cryptographic functions at all - and it's fairly obvious, that is not secure in any sense. It all comes down to how you use the nonce in a protocol and what kind of security ...

5

Given no knowledge of the MAC key and that M1 != M2, it is not possible to compute MAC(K, M2) from MAC(K, M1). More generally, it isn't possible to determine the MAC of any message you haven't already seen. (Again, without the key.) The definition of a nonce is a number which is only used once with a given key. Bob ensures that nonces are only used once ...

0

Messages 4 and 5 are meant to protect the recipient B from a replay attack. It is a basic challenge response mechanism. Otherwise an attacker could just replay previous messages and have the very same interaction with B again. That is bad, if e.g. the message is just "increase some counter", or another message that has a lasting effect or changes a state. ...

Top 50 recent answers are included


text:
## Random Friday night thoughts about myself

I’m flamboyant. That’s who I am. That’s my style. There’s no two ways about it. I can’t be conservative or risk-averse. That’s not who i am.

And because being flamboyant is who I am, I necessarily take risk in everything I do. This means that occasionally the risks don’t pay off – if they pay off all the time they’re not a risk.

In the past I’ve taken the wrong kind of lessons from risks not paying off. That I should not have taken those risks. That I should have taken more calculated risks. That I should have hedged better.

Irrespective of how calculated your risks are, they will not pay off some of the time. The calculation is basically to put a better handle on this probability, and the impact of the risk not paying off. Hedging achieves the same thing.

For example, my motorcycle trip to Rajasthan in 2012 was a calculated risk, hedged by full body riding gear. I had a pretty bad accident – the motorcycle was travelling at 85 kmph when I hit a cow and got thrown off the bike, but the gear meant I escaped with just a hairline fracture in my last metacarpal – I rode on and finished the trip.

Back to real life – what happened was that between approx 2006-09 a number of risks didn’t pay off. Nowadays I like to think of it as a coincidence. Or maybe it was a “hot hand” of the wrong kind – after the initial set of failed risks, I became less confident and less calculating about my risks, and more of them did not pay off.

This is my view now, of course, looking back. Back then I thought I was finished. I started beating myself for every single (what turned out to be, in hindsight) bad decision. And that made me take worse decisions.

A year of medication (2012), which included the aforementioned motorcycle trip, a new career and a lot of time off, helped me get rid of some of these logical fallacies. I started accepting that risks sometimes don’t pay off. And the solution to that is NOT to take less risk.

However, that thought (that every single risk thay didn’t pay off was a bad decision on my past) has been permanently seeded in my brain – whether I like it or not (I don’t like it). And so whenever something goes bad – basically a risk I consciously took not paying off – I instinctively look for a bad decision that I personally made to lay the blame on. And that, putting it simply, never makes me happy. And this is something I need to overcome.

As I said at the beginning of the post, cutting risk simply isn’t my style. And as I internalise that this is how I inherently am, I need to accept that some of my decisions will inherently turn out to have bad outcomes. And in a way, that is part of my strategy.

This blogpost is essentially a note to myself – to document this realisation on my risk profile and to make sure that I have something to refer to the next time a risky decision I take doesn’t pay off (well that happens every single day – this is for the big ones).

The next time I shoot off my mouth without thinking it’s part of my strategy.

The next time I resist the urge to contain myself and blurt out what I’m thinking it’s part of my strategy.

The next time I unwittingly harm myself because of a bad decision I make it’s just part of my strategy.

To close – there was a time when Inzamam-ul-Haq took someone’s advice and lost weight and found that he just couldn’t bat. In a weird way his belly was positively correlated with his batting. Similarly the odd bad decision I take is positively correlated with how I operate naturally.

And I need to learn to live with it.

## Ronald Coase, Scott Adams and Intrapersonal Vertical Integration

I have a new HR policy. I call it “intrapersonal vertical integration”. Read on.

I

Back in the 193os, economist Ronald Coase wrote an article on “the nature of the firm” (the link is to Wikipedia, not to the actual paper). It was a description of why people form companies and partnerships and so on, rather than all being gig workers negotiating each piece of work.

The key concept here was one of transaction costs – if everyone were to be a freelancer, like I was between 2012 and 2020 (both included), then for every little piece of work there would need to be a piece of negotiation.

“Can you build this dashboard for me?”
“Yes. That would be $10000” “No, I’ll only pay$2000”
“9000”
“3000 final”
“get lost”

During my long period of freelancing, I internalised this, and came up with a “minimum order value” – a reasonable amount which could account for transaction costs like the above (just as I write this, I’m changing videos on Youtube for my wife, and she’s asking me to put 30 second videos. And I’m refusing saying “too much transaction cost. I need my hands for something else (blogging)” ).

This worked out fine for the projects that I actually got, but transaction costs meant that a lot of the smaller deals never worked out. I lost out on potential revenue from those, and my potential clients lost out on work getting done.

So, instead, if I were to be part of a company, like I am now, transaction costs are far lower. Yes, we might negotiate on exact specifications, or deadlines, but price was a single negotiation at the time I joined the firm. And so a lot more work gets done – better for me and better for the company. And this is why companies exist. It might sound obvious, but Coase put it in a nice and elegant theoretical framework.

II

I’ve written about this several times on my blog – Scott Adams’s theory that there are two ways in which you can be really successful.

1. Become the best at one specific thing.
2. Become very good (top 25%) at two or more things.

This is advice that I have taken seriously, and I’ve followed the second path. Being the best at one specific thing is too hard, and too random as well – “the best” is a sort of a zero sum game. Instead, being very good in a few things is easier to do, and as I’d said in one of my other posts on this, being very good in uncorrelated things is a clear winner.

I will leave this here and come back later on in the post, like how Dasharatha gave some part of the mango to Sumitra (second in line), and then decided to come back to her later on in the distribution.

III

I came up with this random theory the other day on the purpose of product managers. This theory is really random and ill-formed, and I haven’t bothered discussing it with any real product managers.

The need for product managers comes from software engineers’ insistence on specific “system requirement specifications”.

I learnt software engineering in a formal course back in 2002. Back then, the default workflow for software engineering was the so-called “waterfall model”. It was a linear sequential thing where the first part of the process goes in clearly defining system requirement specifications. Then there would be an unambiguous “design document”. And only then would coding begin.

In that same decade (2000s), “agile” programming became a thing. This meant fast iterations and continuous improvements. Software would be built layer by layer. However, software engineers had traditionally worked only with precise specifications, and “ambiguous business rules” would throw them off. And so the role of the product manager was created – who would manage the software product in a way that they would interface with ambiguous business on one side, and precise software engineers on the other.

Their role was to turn ambiguity to certainty, and get work done. They would never be hands on – instead their job would be to give precise instructions to people who would be hands on.

I have never worked as either a software engineer or a product manager, but I don’t think I’d enjoy either job. On the one hand, I don’t like being given precise instructions, and instead prefer ambiguity. On the other, if I were to give precise instructions, I would rather use C++ or Python to give those instructions than English or Kannada. In other words, if I were to be precise in my communication, I would rather talk to a computer than to another human.

It possibly has to do with my work history. I spent a little over two years as a quant at a top tier investment bank. As part of the job, I was asked to write production code. I used to protest, saying writing C++ code wasn’t the best use of my time or effort. “But think about the effort involved in explaining your model to someone else”, the higher ups in the company would tell me. “Wouldn’t it be far easier to just code it yourself?”

IV

Coase reasoned that transaction costs are the reason why we need a firm. We don’t need frequent negotiations and transaction costs, so if people were to get together in the form of a firm, they could coordinate much better and get a lot more work done, with more value accruing to every party involve.

However, I don’t think Coase went far enough. Just putting people in one firm only eliminates one level of transaction costs – of negotiating conditions and prices. Even when you are in the same firm, coordinating with colleagues implies communication, and unless precise, the communication links can end up being the weak links in how much the firm can achieve.

Henry Ford’s genius was to recognise the assembly line (a literal conveyor belt) as a precise form of communication. The workers in his factories were pretty much automatons, doing their precise job, in the knowledge that everyone else was doing their own. The assembly line made communication simpler, and that allowed greater specialisation to unlock value in the firm – to the extent that each worker could get at least five dollars a day and the firm would still be profitable.

It doesn’t work so neatly in what can be classified as “knowledge industries”. Like with the product manager and the software engineer, there is a communication layer which, if it fails, can bring down the entire process.

And there are other transaction costs implied in this communication – let’s say you are building stuff that I need to build on to make the final product. Every time I think you need to build something slightly different, it involves a process of communication and negotiation. It involves the product manager to write a new section in the document. And when working on complex problems, this can increase the complexity multifold.

So we are back to Scott Adams (finally). Building on what I’d said before – you need to be “very good” at two or more things, and it helps if these things are uncorrelated (in terms of being able to add unique value). However, it is EVEN MORE USEFUL if the supposedly uncorrelated skills you have can be stacked, in a form of vertical integration.

In other words, if you are good at several things that are uncorrelated, where the output of one thing can be the input into another, you are a clear winner.

Adams, for example, is good at understanding business, he is funny and he can draw. The combination of the first two means that he can write funny business stories, and that he can also draw means he has created a masterpiece in the form of Dilbert.

Don’t get me wrong – you can have a genius storyteller and a genius artist come together to make great art (Goscinny and Uderzo, for example). However, it takes a lot of luck for a Goscinny to find his Uderzo, or vice versa. I haven’t read much Asterix but what I’m old by friends is that the quality dropped after Uderzo was forced to be his own Goscinny (after the latter died).

At a completely different level – I have possibly uncorrelated skills in understanding business and getting insight out of data. One dovetails into the other and so I THINK I’m doing well in business intelligence. If I were only good at business, and needed to keep asking someone to churn the data on each iteration, my output would be far far slower and poorer.

So I extend this idea into “intrapersonal vertical integration”. If you are good at two or more things, and one can lead into another, you have a truly special set of skills and can be really successful.

Putting it another way – in knowledge jobs, communication can be so expensive that if you can vertically integrate yourself across multiple jobs, you can add significant value even if you are not the best at each of the individual skills.

Finish

In knowledge work, communication is the weakest link, so the fewer levels of communication you have, the better and faster you can do your job. Even if you get the best for every level in your chain, the strength (or lack of it) of communication between them can mean that they produce suboptimal output.

Instead if you can get people who are just good at two or more things in the chain (rather than being the best at any one), you can add significantly better value.

Putting it another way, yes, I’m batting for bits-and-pieces players rather than genuine batsmen or bowlers. However, the difference between what I’m saying and cricket is that in cricket batting and bowling are not vertically integrated. If they were, bits and pieces players would work far far better.

The Downside

I’ve written about this before. While being good at uncorrelated things that dovetail into one another can be a great winning strategy, liquidity can be your enemy. That you are unique means that there aren’t too many like you. And so organisations may not want to bet too much on you – since you will be hard to replace. And decide to take the slack in communication and get specialists for each position instead.

PS:

I have written a book on transaction costs and liquidity. As it happens, today it is on display at the Bangalore Literature Festival.

## Risk and data

A while back a group of <a large number of scientists> wrote an open letter to the Prime Minister demanding greater data sharing with them. I must say that the letter is written in academic language and the effort to understand it was too much, but in the interest of fairness I’ll put a screenshot that was posted on twitter here.

I don’t know about this clinical and academic data. However, the holding back of one kind of data, in my opinion, has massively (and negatively) impacted people’s mental health and risk calculations.

This is data on mortality and risk. The kind of questions that I expect government data to have answered was:

1. If I get covid-19 (now in the second wave), what is the likelihood that I will die?
2. If my oxygen level drops to 90 (>= 94 is “normal”), what is the likelihood that I will die?
3. If I go to hospital, what is the likelihood I will die?
4. If I go to ICU what is the likelihood I will die?
5. What is the likelihood of a teenager who contracts the virus (and is otherwise in good health) dying of the virus?

And so on. Simple risk-based questions whose answers can help people calibrate their lives and take calculated enough risks to get on with it without putting themselves and their loved ones at risk.

Instead, what we find from official sources are nothing but aggregates. Total numbers of people infected, dead, recovered and so on. And it is impossible to infer answers to the “risk questions” based no that.

And who fill in the gaps? Media of course.

I must have discussed “spectacularness bias” on this blog several times before. Basically the idea is that for something to be news, it needs to carry information. And an event carries information if it occurs despite having a low prior probability (or not occurring despite a high prior probability). As I put it in my lectures, “‘dog bites man’ is not news. ‘man bits dog’ is news”.

So when we rely on media reports to fill in our gaps in our risk systems, we end up taking all the wrong kinds of lessons. We learn that one seventeen year old boy died of covid despite being otherwise healthy. In the absence of other information, we assume that teenagers are under grave risk from the disease.

Similarly, cases of children looking for ICU beds get forwarded far more than cases of old people looking for ICU beds. In the absence of risk information, we assume that the situation must be grave among children.

Old people dying from covid goes unreported (unless the person was famous in some way or the other), since the information content in that is low. Young people dying gets amplified.

Based on all the reports that we see in the papers and other media (including social media), we get an entirely warped sense of what the risk profile of the disease is. And panic. When we panic, our health gets worse.

Oh, and I haven’t even spoken about bad risk reporting in the media. I saw a report in the Times of India this morning (unable to find a link to it) that said that “young are facing higher mortality in this wave”. Basically the story said that people under 60 account for a far higher proportion of deaths in the second wave than in the first.

Now there are two problems with that story.

1. A large proportion of over 60s in India are vaccinated, so mortality is likely to be lower in this cohort.
2. What we need is the likelihood of a person under 60 dying upon contracting covid. NOT the proportion of deaths accounted for by under 60s. This is the classic “averaging along the wrong axis” that they unleash upon you in the first test of any statistics course.

Anyway, so what kind of data would have helped?

1. Age profile of people testing positive, preferably state wise (any finer will be noise)
2. Age profile of people dying of covid-19, again state wise

I’m sure the government collects this data. Just that they’re not used to releasing this kind fo data, so we’re not getting it. And so we have to rely on the media and its spectacularness bias to get our information. And so we panic.

PS: By no means am I stating that covid-19 is not a risk. All I am stating is that the information we have been given doesn’t help us make good risk decisions

## Uncertain Rewards

A couple of months back, I read Nir Eyal’s Hooked. I didn’t particularly get hooked to the book – it’s one of those books that should have been a blogpost (or maybe a longform article). However, as part of the “Hooked model” that forms the core of the book, the author talks about the importance of “uncertain rewards”.

The basic idea is that it is easier to get addicted to something when the rewards from it are uncertain. If the rewards are certain, then irrespective of how large they are, there is a chance that you might get bored of them. Uncertainty, on the other hand, makes you curious. It provides you “information” each time you “play the game”. And you in the quest for new information (remember that entropy is information?), you keep playing. And you get hooked.

This plays out in various ways. Alcohol and drugs, for example, sometimes offer “good trips”, and sometimes “bad trips”. The memory of the good trips is the reason why you keep at it, even if you occasionally have bad trips. The uncertain rewards hook you.

It’s the same with social media. This weekend, so far, I’ve had a largely good experience on Twitter. However, last weekend on the platform was a disaster. I’d gotten quickly depressed and stopped. So why did I get back on to twitter this weekend when last weekend was bad? Because of an earlier weekend when it had provided a set of good conversations.

Even last weekend, when I started having a “bad trip” on Twitter, I kept at it, thinking the longer I play the better the chances of having a good trip. Ultimately I just ruined my weekend.

Uncertain rewards are also why, (especially) when we are young, we tolerate abusive romantic partners. Partners who treat you well all the time are boring. And there is no excitement. Abusive partners, on the other hand, treat you like a king/queen at times, and like shit at other times. The extent of the highs and lows means that you get hooked to them. It possibly takes a certain degree of abuse for you to realise that a “steady partner who treats you well” makes for a better long term partner.

Is there a solution to this? I don’t think so. As we learn in either thermodynamics or information theory, entropy or randomness is equal to information. And because we have evolved to learn and get more information, we crave entropy. And so we crave the experiences that give us a lot of entropy, even it that means the occasional bad trip.

Finally, I realise that uncertain rewards are also the reason why religion is addictive. One conversation I used to have a lot with my late mother was when I would say, “why do you keep praying when your prayers weren’t answered the last time?”. And she would quote another time when her prayers WERE answered. It is this uncertain reward of answers to prayers (which, in my opinion, is sheer randomness) that keeps religion “interesting”. And makes it addictive.

## Confusing with complications

I’m reading this awesome article by Srinivas Bhogle (with Rajeeva Karandikar) on election forecasting. To be fair, not much of the article is new to me – it’s just a far more readable version of Karandikar’s seminal presentation on the topic made at IIT Kanpur all those years back.

However, as with all good retellings, this story also has some nice tidbits. This one has to do with “index of opposition unity”. The voice here is Bhogle’s:

It is easy to understand why the IOU becomes so critical in such situations. But, and here’s the rub, the exact mathematical formula connecting IOU to the seat count prediction is not easy to find. I searched through the big and small print of The Verdict by Dorab Sopariwala and Prannoy Roy, but the formula remained elusive.

Rajeeva suggests that it was likely based on simple heuristics: something like ‘if the IOU is less than 25%, give the first-placed party 75% of the seats.’ It may also have involved intelligent tweaking based on current survey data, historical data, informal feedback, expert opinion, gut feeling, and so on.

I first came across the IOU in Prannoy Roy and Dorab Sopariwala’s book. The way they had presented in the book, it seemed like it is a “major concept”. It seems, like I did, Bhogle also looked through the book trying to find a precise formula, and failed to do so.

And then Karandikar’s insight above is crucial – that the IOU may not be a precise mathematical formula, but just an intelligent set of heuristics, involving intelligent tweaking.

Sometimes putting a fancy name (or, even better, an acronym) on something can help lend credibility to the concept. For example, IOU is something that has been championed by Roy and Sopariwala for years, and they have done so to a level where it has become a self-fulfilling prophecy, and a respected scientist for Bhogle has gone searching for its formula!

Also, sometimes, telling people that you “used an intelligent heuristic” to come up with a conclusion can lead you to be taken less seriously. Put on a fancy name (even if it is something that you have yourself come up with), and the game changes. You suddenly start to be taken more seriously, like Ganesha assumed when he started sending fan mail under the name “YG Rao”.

And like they say in The Usual Suspects, sometimes the greatest trick that the devil ever pulled was to convince you that he exists. It is the same with “concepts” such as IOU – you THINK they must be sound because they come with a fancy name, when all that they apeear to represent is a set of fancy heuristics.

I must say this is excellent marketing.

## ISAs and Power Laws

There are a number of professions where incomes are distributed according to a power law. The most successful people in the professions corner a very large share of the income that people in the profession make, and unless you reach that very high level of success, you might even struggle to make a living wage.

Professions of this nature include the arts (movies, music, drama, standup comedy, painting, sculpture, etc.), sports, writing and entrepreneurship. The thing with such professions is that it needs some degree of “socialism” – if people are left to their own devices, then the 99% confidence payoffs will mean that few people will enter the profession, and when fewer people enter the profession, the overall quality of the profession goes down.

So what is required in this case is some sort of a safety net – people who are reasonably competent at the profession get paid a sort of regular basic income (could either be one-time, periodic or output-based) by “investors” in exchange for a cut of the upside. And this, for a talented but struggling beginner, is usually a good deal – they are assured a basic income to pursue what they love and think they are good at, and anything they have to pay in return is only probabilistic – contingent upon a heavy degree of success.

And in order for this kind of safety net to work, it is important that the investment be of the nature of “equity” rather than “debt” – the extreme power law nature of these professions is that only a small proportion of the people who get the safety net will be able to pay back, and those that are able to pay back will be able to pay disproportionately large amounts.

Entrepreneurship and film acting have sort of done well in terms of providing these safety net. Entrepreneurs get venture capital investment, which allows them to fund their business and take (nominal) salaries, while working on the thing they hope to make it big in. The venture capitalists make money even when a small proportion of their investments don’t fail.

The model in acting is a little different- studios hire actors on long term contracts at negotiated salaries. These salaries give actors the safety net to continue in the profession. And in case the actors become popular, the studios cash out essentially by “encashing the option” of using the actor at the pre-negotiated rate for the duration of the contract.

There are other examples of these safety nets as well – artist studios pay their artists a basic wage, in exchange for a cut on the sale of their paintings. However, the model is not as popular as it seems.

For sportspersons, for example, apart from things like the Ranji Trophy increasing match fees in a big way in the late noughties, this kind of a safety net has been absent. The studio model in acting hasn’t held on. Writers get advances but that doesn’t represent much of a “living wage”.

The good news is that this is changing. Investment in athletes in exchange for a cut of future earnings is gaining traction. And now we have this deal ($): Taxes will cut into his new 14-year agreement with the Padres, of course. But Tatis also must pay off a previous obligation, a deal he made during the 2017-18 offseason, when he was turning 19 years old and preparing for his first full season at Double A. It was then that Tatis entered into a contract with Big League Advance (BLA), a company that offers select minor leaguers upfront payments in exchange for a percentage of their future earnings in Major League Baseball. Neither Tatis nor BLA has revealed the exact percentage he owes the company. The company’s president and CEO, former major-league pitcher Michael Schwimer, told The Athletic in April 2018 that BLA uses a proprietary algorithm to value every player in the minors. Players who receive offers can accept a base-level payout in return for 1 percent of their earnings, with the chance to receive greater incremental payouts and pay back a maximum of 10 percent. If a player never reaches the majors, he keeps the cash advance, with no obligation to pay it back. This is an awesome thing. For a struggling potential sportsperson, a minor investment (in exchange for equity) can provide a huge boost in their chances of making it – hiring coaches, for example, or eating better food, or living more comfortably. While the media attention will go to the small proportion of investments that do pay off (like how tech media gives disproportionate coverage, and quite rightly so, to startups that do well), arrangements like this mean that more people will play the sport, and the overall standard in the sport will improve. We need to see if such arrangements start making a mark in the rest of the arts and writing as well. Oh, and much has been made of income sharing agreements for professional colleges and “tuition centres”. I’m not sure that is the right model there – the thing is that if you are studying to be a software engineer, your payoffs don’t follow a power law. Yes, if you are successful, you make a few orders of magnitude more money than the less successful ones, but even an average software engineer can expect to make a fairly decent income. From that perspective, selling equity in your future earnings to get paid to study engineering is not a great idea, and can lead to adverse selection on the part of the candidates (the better ones will prefer to get funding through debt, which their average salaries can help pay off). In that sense I prefer what the likes of MountBlue are doing, where the “training fees” get paid off by simply working for the company for a certain period of time. ## Monetising volatility I’m catching up on old newsletters now – a combination of job and taking my email off what is now my daughter’s iPad means I have a considerable backlog – and I found this gem in Matt Levine’s newsletter from two weeks back ($; Bloomberg).

“it comes from monetizing volatility, that great yet under-appreciated resource.”

He is talking about equity derivatives, and says that this is “not such a good explanation”. While it may not be such a good explanation when it comes to equity derivatives itself, I think it has tremendous potential outside of finance.

I’m reminded of the first time I was working in the logistics industry (back in 2007). I had what I had thought was a stellar idea, which was basically based on monetising volatility, but given that I was in a company full of logistics and technology and operations research people, and no other derivatives people, I had a hard time convincing anyone of that idea.

My way of “monetising volatility” was rather simple – charge people cancellation fees. In the part of the logistics industry I was working in back then, this was (surprisingly, to me) a particularly novel idea. So how does cancellation fees equate to monetising volatility?

Again it’s due to “unbundling”. Let’s say you purchase a train ticket using advance reservation. You are basically buying two things – the OPTION to travel on that particular day using that particular train, sitting on that particular seat, and the cost of the travel itself.

The genius of the airline industry following the deregulation in the US in the 1980s was that these two costs could be separated. The genius was that charging separately for the travel itself and the option to travel, you can offer the travel itself at a much lower price. Think of the cancellation charge as as the “option premium” for exercising the option to travel.

And you can come up with options with different strike prices, and depending upon the strike price, the value of the option itself changes. Since it is the option to travel, it is like a call option, and so higher the strike price (the price you pay for the travel itself), the lower the price of the option.

This way, you can come up with a repertoire of strike-option combinations – the more you’re willing to pay for cancellation (option premium), the lower the price of the travel itself will be. This is why, for example, the cheapest airline tickets are those that come with close to zero refund on cancellation (though I’ve argued that bringing refunds all the way to zero is not a good idea).

Since there is uncertainty in whether you can travel at all (there are zillions of reasons why you might want to “cancel tickets”), this is basically about monetising this uncertainty or (in finance terms) “monetising volatility”. Rather than the old (regulated) world where cancellation fees were low and travel charges were high (option itself was not monetised), monetising the options (which is basically a price on volatility) meant that airlines could make more money, AND customers could travel cheaper.

It’s like money was being created out of thin air. And that was because we monetised volatility.

I had the same idea for another part of the business, but unfortunately we couldn’t monetise that. My idea was simple – if you charge cancellation fees, our demand will become more predictable (since people won’t chumma book), and this means we will be able to offer a discount. And offering a discount would mean more people would buy this more predictable demand, and in the immortal jargon of Silicon Valley, “a flywheel would be set in motion”.

The idea didn’t fly. Maybe I was too junior. Maybe people were suspicious of my brief background in banking. Maybe most people around me had “too much domain knowledge”. So the idea of charging for cancellation in an industry that traditionally didn’t charge for cancellation didn’t fly at all.

Anyway all of that is history.

Now that I’m back in the industry, it remains to be seen if I can come up with such “brilliant” ideas again.

## Uncertainty and Anxiety

A lot of parenting books talk about the value of consistency in parenting – when you are consistent with your approach with something, the theory goes, the child knows what to expect, and so is less anxious about what will happen.

It is not just about children – when something is more deterministic, you can “take it for granted” more. And that means less anxiety about it.

From another realm, prices of options always have “positive vega” – the higher the market volatility, the more the price of the option. Thinking about it another way, the more the uncertainty, the more people are willing to pay to hedge against it. In other words, higher uncertainty means more anxiety.

However, sometimes the equation can get flipped. Let us take the case of water supply in my apartment. We have both a tap water connection and a borewell, so historically, water supply has been fairly consistent. For the longest time, we didn’t bother thinking about the pressure of water in the taps.

And then one day in the beginning of this year the water suddenly stopped. We had an inkling of it that morning as the water in the taps inexplicably slowed down, and so stored a couple of buckets until it ground to a complete halt later that day.

It turned out that our water pump, which is way deep inside the earth (near the water table) was broken, so it took a day to fix.

Following that, we have become more cognisant of the water pressure in the pipes. If the water pressure goes down for a bit, the memory of the day when the motor conked is fresh, and we start worrying that the water will suddenly stop. I’ve panicked at least a couple of times wondering if the water will stop.

However, after this happened a few times over the last few months I’m more comfortable. I now know that fluctuation of water pressure in the tap is variable. When I’m showering at the same time as my downstairs neighbour (I’m guessing), the water pressure will be lower. Sometimes the level of water in the tank is just above the level required for the pump to switch on. Then again the pressure is lower. And so forth.

In other words, observing a moderate level of uncertainty has actually made me more comfortable now and reduced my anxiety – within some limits, I know that some fluctuation is “normal”.  This uncertainty is more than what I observed earlier, so in other words, increased (perceived) uncertainty has actually reduced anxiety.

One way I think of it is in terms of hidden risks – when you see moderate fluctuations, you know that fluctuations exist and that you don’t need to get stressed around them. So your anxiety is lower. However, if you’ve gone a very long time with no fluctuation at all, then you are concerned that there are hidden risks that you have not experienced yet.

So when the water pressure in the taps has been completely consistent, then any deviation is a very strong (Bayesian) sign that something is wrong. And that increases anxiety.

## Shooting, investing and the hot hand

A couple of years back I got introduced to “Stumbling and Mumbling“, a blog written by Chris Dillow, who was described to me as a “Marxist investment banker”. I don’t agree with a lot of the stuff in his blog, but it is all very thoughtful.

He appears to be an Arsenal fan, and in his latest post, he talks about “what we can learn from football“. In that, he writes:

These might seem harmless mistakes when confined to talking about football. But they have analogues in expensive mistakes. The hot-hand fallacy leads investors to pile into unit trusts with good recent performance (pdf) – which costs them money as the performance proves unsustainable. Over-reaction leads them to buy stocks at the top of the market and sell at the bottom. Failing to see that low probabilities compound to give us a high one helps explain why so many projects run over time and budget. And so on.

Now, the hot hand fallacy has been a hard problem in statistics for a few years now. Essentially, the intuitive belief in basketball is that someone who has scored a few baskets is more likely to be successful in his next basket (basically, the player is on a “hot hand”).

It all started with a seminal paper by Amos Tversky et al in the 1980s, that used (the then limited) data to show that the hot hand is a fallacy. Then, more recently, Miller and Sanjurjo took another look at the problem and, with far better data at hand, found that the hot hand is actually NOT a fallacy.

There is a nice podcast on The Art of Manliness, where Ben Cohen, who has written a book about hot hands, spoke about the research around it. In any case, there are very valid reasons as to why hot hands exist.

Yet, Dillow is right – while hot hands might exist in something like basketball shooting, it doesn’t in something like investing. This has to do with how much “control” the person in question has. Let me switch fields completely now and quote a paragraph from Venkatesh Guru Rao‘s “The Art Of Gig” newsletter:

As an example, take conducting a workshop versus executing a trade based on some information. A significant part of the returns from a workshop depend on the workshop itself being good or bad. For a trade on the other hand, the returns are good or bad depending on how the world actually behaves. You might have set up a technically perfect trade, but lose because the world does something else. Or you might have set up a sloppy trade, but the world does something that makes it a winning move anyway.

This is from the latest edition, which is paid. Don’t worry if you aren’t a subscriber. The above paragraph I’ve quoted is sufficient for the purpose of this blogpost.

If you are in the business of offering workshops, or shooting baskets, the outcome of the next workshop or basket depends largely upon your own skill. There is randomness, yes, but this randomness is not very large, and the impact of your own effort on the result is large.

In case of investing, however, the effect of the randomness is very large. As VGR writes, “For a trade on the other hand, the returns are good or bad depending on how the world actually behaves”.

So if you are in a hot hand when it comes to investing, it means that “the world behaved in a way that was consistent with your trade” several times in a row. And that the world has behaved according to your trade several times in a row makes it no more likely that the world will behave according to your trade next time.

If, on the other hand, you are on a hot hand in shooting baskets or delivering lectures, then it is likely that this hot hand is because you are performing well. And because you are performing well, the likelihood of you performing well on the next turn is also higher. And so the hot hand theory holds.

So yes, hot hands work, but only in the context “with a high R Square”, where the impact of the doer’s performance is large compared to the outcome. In high randomness regimes, such as gambling or trading, the hot hand doesn’t matter.

## What is the Case Fatality Rate of Covid-19 in India?

The economist in me will give a very simple answer to that question – it depends. It depends on how long you think people will take from onset of the disease to die.

The modeller in me extended the argument that the economist in me made, and built a rather complicated model. This involved smoothing, assumptions on probability distributions, long mathematical derivations and (for good measure) regressions.. And out of all that came this graph, with the assumption that the average person who dies of covid-19 dies 20 days after the thing is detected.

Yes, there is a wide variation across the country. Given that the disease is the same and the treatment for most people diseased is pretty much the same (lots of rest, lots of water, etc), it is weird that the case fatality rate varies by so much across Indian states. There is only one explanation – assuming that deaths can’t be faked or miscounted (covid deaths attributed to other reasons or vice versa), the problem is in the “denominator” – the number of confirmed cases.

What the variation here tells us is that in states towards the top of this graph, we are likely not detecting most of the positive cases (serious cases will get themselves tested anyway, and get hospitalised, and perhaps die. It’s the less serious cases that can “slip”). Taking a state low down below in this graph as a “good tester” (say Andhra Pradesh), we can try and estimate what the extent of under-detection of cases in each state is.

Based on state-wise case tallies as of now (might be some error since some states might have reported today’s number and some mgiht not have), here are my predictions on how many actual number of confirmed cases there are per state, based on our calculations of case fatality rate.

Yeah, Maharashtra alone should have crossed a million caess based on the number of people who have died there!

Now let’s get to the maths. It’s messy. First we look at the number of confirmed cases per day and number of deaths per day per state (data from here). Then we smooth the data and take 7-day trailing moving averages. This is to get rid of any reporting pile-ups.

Now comes the probability assumption – we assume that a proportion $p$ of all the confirmed cases will die. We assume an average number of days ($N$) to death for people who are supposed to die (let’s call them Romeos?). They all won’t pop off exactly $N$ days after we detect their infection. Let’s say a proportion $\lambda$ dies each day. Of everyone who is infected, supposed to die and not yet dead, a proportion $\lambda$ will die each day.

My maths has become rather rusty over the years but a derivation I made shows that $\lambda = \frac{1}{N}$. So if people are supposed to die in an average of 20 days, $\frac{1}{20}$ will die today, $\frac{19}{20}\frac{1}{20}$ will die tomorrow. And so on.

So people who die today could be people who were detected with the infection yesterday, or the day before, or the day before day before (isn’t it weird that English doesn’t a word for this?) or … Now, based on how many cases were detected on each day, and our assumption of $p$ (let’s assume a value first. We can derive it back later), we can know how many people who were found sick $k$ days back are going to die today. Do this for all $k$, and you can model how many people will die today.

The equation will look something like this. Assume $d_t$ is the number of people who die on day $t$ and $n_t$ is the number of cases confirmed on day $t$. We get

$d_t = p (\lambda n_{t-1} + (1-\lambda) \lambda n_{t-2} + (1-\lambda)^2 \lambda n_{t-3} + ... )$

Now, all these $n$s are known. $d_t$ is known. $\lambda$ comes from our assumption of how long people will, on average, take to die once their infection has been detected. So in the above equation, everything except $p$ is known.

And we have this data for multiple days. We know the left hand side. We know the value in brackets on the right hand side. All we need to do is to find $p$, which I did using a simple regression.

And I did this for each state – take the number of confirmed cases on each day, the number of deaths on each day and your assumption on average number of days after detection that a person dies. And you can calculate $p$, which is the case fatality rate. The true proportion of cases that are resulting in deaths.

This produced the first graph that I’ve presented above, for the assumption that a person, should he die, dies on an average 20 days after the infection is detected.

So what is India’s case fatality rate? While the first graph says it’s 5.8%, the variations by state suggest that it’s a mild case detection issue, so the true case fatality rate is likely far lower. From doing my daily updates on Twitter, I’ve come to trust Andhra Pradesh as a state that is testing well, so if we assume they’ve found all their active cases, we use that as a base and arrive at the second graph in terms of the true number of cases in each state.

PS: It’s common to just divide the number of deaths so far by number of cases so far, but that is an inaccurate measure, since it doesn’t take into account the vintage of cases. Dividing deaths by number of cases as of a fixed point of time in the past is also inaccurate since it doesn’t take into account randomness (on when a Romeo might die).

Anyway, here is my code, for what it’s worth.

deathRate <- function(covid, avgDays) {
covid %>%
mutate(Date=as.Date(Date, '%d-%b-%y')) %>%
gather(State, Number, -Date, -Status) %>%
arrange(State, Date) ->
cov1

# Need to smooth everything by 7 days
cov1 %>%
arrange(State, Date) %>%
group_by(State) %>%
mutate(
TotalConfirmed=cumsum(Confirmed),
TotalDeceased=cumsum(Deceased),
ConfirmedMA=(TotalConfirmed-lag(TotalConfirmed, 7))/7,
DeceasedMA=(TotalDeceased-lag(TotalDeceased, 7))/ 7
) %>%
ungroup() %>%
filter(!is.na(ConfirmedMA)) %>%
select(State, Date, Deceased=DeceasedMA, Confirmed=ConfirmedMA) ->
cov2

cov2 %>%
select(DeathDate=Date, State, Deceased) %>%
inner_join(
cov2 %>%
select(ConfirmDate=Date, State, Confirmed) %>%
crossing(Delay=1:100) %>%
mutate(DeathDate=ConfirmDate+Delay),
by = c("DeathDate", "State")
) %>%
filter(DeathDate > ConfirmDate) %>%
arrange(State, desc(DeathDate), desc(ConfirmDate)) %>%
mutate(
Lambda=1/avgDays,
}


text:
# Whiff Rates Adjusting for Location

#### Introduction

In last week’s post, we began an exploration of whiff rates on sliders. We looked at pitcher whiff rates for pitches thrown in and out of the zone. In the 2019 season, we saw that Max Scherzer and Kyle Gibson had similar whiff rates for balls out of the zone, but Scherzer had a much higher whiff rate than Gibson in the zone. That motivated the fitting of a model where we represented the probability of a miss as a smooth function of the pitch location. By adding random pitcher effects to the model, we showed that some pitchers such as Max Scherzer have abilities to induce whiffs on slider beyond what are predicted by the pitch locations.

That work motivated some new questions:

• Since random effects estimates are a bit abstract, is there a more basic way to demonstrate these “beyond location” abilities to induce whiffs on sliders?
• How does this work carry over to other pitch types such as four-seamers or change-ups?
• Can we identify remarkable pitchers who are unusually good (and unusually weak) in inducing whiffs on swung pitches?

In this post, I illustrate the use of standardized scores to measure a pitcher’s whiff rate performance beyond one would predict on the basis of the pitch locations.

#### The Method

To begin, using 2019 Statcast data, I collected all of the pitches thrown of a particular type, say Slider, and a particular pitching side, say Right. I focused on the pitches where the batter swings and defined a variable Miss which is equal to 1 if the batter whiffed and 0 otherwise. I fitted the generalized additive model

$\log \frac{p}{1-p} = s(plate\_x, plate\_x)$

where p is the probability of a Miss, (plate_x, plate_z) gives the coordinates of the pitch location, and s() represents a smooth function. Using this model, one can predict the probability of a whiff for any pitch location.

For each pitcher, I computed the observed number of swing misses M. In addition, I computed an Expected number of misses E for that pitcher by adding up the whiff probabilities on the pitch locations where the batter swings. To assess the difference between M and E, I compute a Z-score

$Z = \frac{M - E}{\sqrt{E}}$

Z-scores are attractive since they adjust the increase in number of misses, M – E, for the sample size and have a clear interpretation. If the miss probabilities are primarily explained by the pitch locations, we anticipate that most of the Z-scores will fall between -2 and 2. If a pitcher has a “large-positive” Z-score (much larger than 2), this positive residual indicates that he has an “extra” ability to induce a whiff on this particular pitch. On the other hand, a “large-negative” Z-score means that this pitcher does worse in whiffing batters than one would predict based on his pitch locations. To implement this calculation, I have a special R function pitch_z_score() that has three inputs — the Statcast dataset, the pitch type, and the pitching arm. The output includes a data frame containing the number of pitches, the number of swings, the number of whiffs, and the Z-scores for all pitchers that season who threw that particular pitch. By graphing the Z-scores against the number of pitches, we can see general patterns and the remarkable values (high and low) that deviate from the general pattern.

#### Z-Scores for Four Pitch Types

For each pitching arm, I decided to look at four pitch types — sliders (Statcast code SL), four-seamers (FF), two-seamers (FT), and change-ups (CH) — and compute the Z-scores for all pitchers who threw that particular pitch in the 2019 season. Below I am plotting the number of pitches against the Z-scores for each pitch type. The red dotted lines correspond to Z-score values of -2.5 and 2.5 — points either above or below the red lines correspond to pitchers with extreme whiff rates. The points on the right-hand side of each display correspond to starters who throw a large number of pitches of that type. What do we take away from this graph for right-handed pitchers?

• Most of the points fall within the (-2.5, 2.5) band which indicates that pitch location explains the whiff rates for many of the pitchers. All of the Z-scores for Sliders (SL) fall between -3 and 3.
• But there are a number of Z-scores above the “2.5 line” indicating that there are pitchers whose whiff rate is higher than one would predict based on the pitch locations. This is most obvious for change-ups (CH) and four-seam fastballs (FF).

Here is the graph for the 2019 southpaws. What really stands out are the large Z-scores for pitchers throwing the four-seamer.

#### Remarkable Pitchers

Who are these extreme pitchers with large Z-scores? For the 8 pitchers with Z-scores exceeding 5, the following tables display, for each pitcher, the number of swings, number of misses and the Z-scores, first for right-handers and then for left-handers. Note that six of these pitchers have outstanding four-seamers and two have remarkable change-ups. Josh Harder had the largest Z-score among pitchers across these four pitch types. Of the 566 four-seamers thrown by Hader, 214 are missed (38%). Based on the location model, we expect the miss count to be 129, and the corresponding Z-score would be

$Z = \frac{214 - 129}{\sqrt{129}} = 7.5$

What is special is that Hader’s whiff rate of 38% is remarkably high after adjusting for the pitch locations. There was a 538.com article a few years back that referred to Josh Hader’s fastball as baseball’s most mysterious pitch.

#### Z-Scores are Measurements of Pitcher Ability

To demonstrate that these Z-scores are meaningful measurements of pitcher ability, we run this exercise for two consecutive seasons (here 2018 and 2019) and see if the Z-scores from one season are associated with the Z-scores for the second season. I’ve done this by focusing on right-handed pitchers who throw four-seamers. The scatterplot below demonstrates that pitchers who are above-average for one season are very likely to be above-average in the second season.