Thursday, 14 May 2020

Black- Scholes Model for Non Math Users

There are two kinds of people on this planet. Mathematicians who may constitute 0.001% (or even lesser than that!) of the total population and the other lesser mortals. So if you are one of those lesser mortals (like me!) and you want to use the Black Scholes model then, this article is for you. This article is based on Prof. Damodaran's real options pricing theory.

To understand Black Scholes without delving too much into calculus, one needs to the understand two concepts - a) replication portfolio and b) arbitrage.

To value or price an option we replicate the cash flows from the option by using other financial instruments as described below.

Call = Borrowing + Buying D of the Underlying Asset

Put = Selling Short D on Underlying Asset + Lending

To replicate the cash flows on a call option without actually buying the option, we need to borrow some money (put in some personal equity) and buy D units of underlying asset. At the end of the time period if the stock price goes up you can sell the stock and repay the borrowed amount (+ interest) and the balance is your profit. Similarly with a put option, you sell short on the underlying asset at the beginning of the period and lend the amount at an interest rate i. At the end of the time period t, if the price of the underlying asset decreases and the principal + interest you received from the borrower can be used to square of your short position and you keep the difference as your profit.

The cash flows on the option and the replication portfolio are the same and in case there is a difference there is scope for arbitrage. Arbitrageurs chip in to ensure that there is an equilibrium in these prices.

In a two period binomial model, we can easily determine the price of the options by working backwards as shown in the figure below. The number of shares that you need to buy in case of a call option is given by D, B is the borrowed amount, K is the strike price, t is the number of time periods, r is the interest rate per period.

By equating the cash flows on the replication portfolio and the call, we are implicitly assuming zero arbitrage and thus we can arrive at the price of the option today. (working backwards from the cash flows at the end of the period)

In the real world, asset prices change continuously and the Black Scholes model is just an extension of the two period model in to the real world in which asset prices change continuously and are normally distributed.

Remember those pay off diagrams for calls and puts. A call option derives value only when the underlying asset is at a price above the strike price. Similarly a put option derives its value only when the price is below the strike price of the option.

Payoff on Options

Once you have an understanding of these basic ideas it is not really difficult to use the Black Scholes model which is presented below.

The Black Scholes Formula for Calls and Puts


C is the price of the call option

P is the price of the put option

S is the Stock Price

K is the strike price

y is the annual dividend yield - dividend paid/current market price of the underlying stock

t is the time period

r is the interest rate per time period

? and sigma for standard deviation

When we move from a two period model to a continuously compounding model, the value of money is assumed to compound at e units per unit of time, e is the euler's number which is equal to 2.7182818284590452353602874713527 (and more ...). So in the equations above determining call and put prices where we use K*e^(-r*t), we are just determining the present value of the strike price as we do not have to forgo or receive the strike price till the end of the period.

A call derives its value only when it is in the money, that is only when K < S, so to arrive at the price/value of the call we find the difference between present value of the stock price and strike price.

N(d1) is the option delta gives you the responsiveness in the value of the option to a change in the value of the underlying asset.

N(d2) is the risk neutral probability of the option being in the money.

In effect to arrive at the value/price of a call option we are deducting the strike price of the option multiplied by the probability of the option being in the money from the present value of the stock price multiplied by option delta. In case of the put option, we are doing the opposite we are subtracting the present value of the stock price multiplied by option delta from the strike price of the option multiplied by the probability of the option being in the money.

d1 and d2 are the mathematical derivations using calculus that you can leave for Manjul Bhargava or Brian Greene !

Template to use Black Scholes Model

Given above is the link to a spread sheet template for using the Black Scholes model. You just have to input the variables in the yellow fields to arrive at the option price.

The template has been tried on some listed stock options on CBOE and the option prices are quite close to the actual bid ask quotes. Hope you find it useful!

(Source: Prof Damodaran Real Options)

Friday, 1 May 2020

Using Regression Analysis to Screen for Stocks (Relative Valuation Approach)

1. Data Source: The data sets for this article have been sourced from (open to everyone) It looks like updates its data every day at 1930 hrs. As the earnings season progresses more numbers will get updated but most of the earnings numbers should belong to 2019. (GIGO - the quality of outputs is subject to the quality of inputs)

2. Consistency and Timing: Multiples approach to valuation is subject to whims and fancies of the analysts. For example, PE ratio which is the most widely used multiple across the global financial markets can have many variations within the rules of the game. Although price is the current market price most of the time, earnings can be previous years earnings or trailing twelve month earnings or forward earnings. An analyst who is bearish on the stock may use past earnings while an analyst who is bullish on the stock may use forward earnings. Even the price used can be the current price, weekly or monthly average or a moving average. To facilitate a fair comparison across companies consistency and timing in the definition of multiples is critical. Both the numerator and the denominator used in the multiple should match each other in consistency and timing. Example, Price to Earnings is a consistent multiple but Price to EBITDA is not a consistent multiple as it scales an equity variable to a firm level variable.

3. Descriptive Stats: Price multiples are skewed distributions as shown in the charts below and subject to selection bias. Negative multiples are not included in the sample as they are not useful in making any meaningful conclusions.

As you can see in the summary stats above, median multiple is a much better representative measure of these data sets than average multiple. These average multiples are affected by very large observations in the right tail. 

 4. Analysis and Application: There are some 5000 stocks listed on BSE and it is very difficult for ordinary unsophisticated retail investors to screen these stocks.

To be fair, it is not a great time to run this regression as many stocks may look under priced due to the impact of Covid-19. (most of the earnings numbers are of 2019 and prices are current prices) Nevertheless the results are presented below for your perusal.

Based on inputs provided by Prof. Damodaran's Relative Valuation a multiple regression has been run on Indian stock multiples and presented in the spreadsheets below.

Price to Earnings Multiple Regression:

Price to Book Value Multiple Regression:

EV to EBITDA Multiple Regression:

(It is not really difficult to run a regression on an excel sheet and predict multiples and readers may give it a shot as this can be a good starting point to make more informed investment decisions !)

A summary of regression results is presented in the table below. By substituting these variables into the equation we can find a predicted multiple. In effect, we are estimating a multiple which is the dependent variable based on its relationship with the independent variables. 

 The results obtained by using these equations can be very different from the results obtained using discounted cash flow (intrinsic value) techniques. Nevertheless, we have a starting point which we can use to investigate further into the fundamentals of under priced companies to discover undervalued companies.Similar analysis can be performed across sectors or global markets.

(Reference and Acknowledgements Prof. Damodaran)

Wednesday, 29 April 2020

Resource Allocation and Utilisation: Corporate Profitability meets Economic Objectives

One of the key issues that the field of economics addresses is the optimum allocation and utilization of resources. How do we choose between competing objectives when our resources are limited ? How do we transform scarce resources in to productive outputs and at what cost? In spite of all the advancements of mankind in most fields, these fundamental questions of economics still persist.

In a free market economy, the burden of the choice between competing projects/undertakings falls on individuals and businesses. At least once in a while, it is necessary to assess if our collective decision making is navigating us in the right direction. So how do we know if our micro economic decision making enhances broader welfare ?

One of the key indicators of efficiency in resource allocation and utilization is corporate profitability. In this context, take a look at these graphs on the basis of data provided by Prof. Damodaran. (Reference and Acknowledgements: Prof. Damodaran's blog post)

On a global scale, 52.05% of all companies have generated returns (measured in terms of return on invested capital) that are lower than their cost of capital in the year 2019. 15.32% of all companies generated returns that are within 2% of their cost of capital and only 32.63% of all companies were able to generate good returns in excess of their cost of capital. If you are crying foul that this is only one year's data (2019),the last decade does not present a great picture either. Growth does not have any value when the cost of capital exceeds return on capital.

Purists would argue that this definition of  returns and costs is very narrow as it does not include environmental costs, tail risk behavior and other externalities. If we include these costs as well then the picture may depreciate further.

Rather than fretting over methodology or data, every company needs to reassess its profits and costs. At least, in the longer run it should be possible to improve the return performance of existing businesses, exit non-profitable businesses and move into more productive sectors.

Activist investors have a role to play here. By taking positions on corporate boards they can steer the management towards the right path.

All hail Lord Keynes, government intervention is required not just during recessionary periods but almost always to nudge private investments in the right direction. Governments cannot limit themselves to infrastructure investments. Productive investments in different sectors is as important as divestment of unproductive businesses. Healthy competition between public and private sector is the need of the hour.

Wednesday, 15 April 2020

Implied Equity Risk Premium for S & P BSE Sensex

Based on Prof. Damodaran's implied equity risk premium for the US market, an equity risk premium for the Indian equity market has been calculated in the spread sheet below. For the calculation of historical equity risk premiums actual dividend yields, nominal growth rate and the yield on 10-year benchmark bond for the respective years have been used.

Implied ERP for S & P BSE Sensex

The methodology for the calculation of ERP is available on Prof. Damodaran's blog. 

Implied equity risk premiums are akin to bond yields for bonds, they are inversely proportional to stock prices or index level.

Current implied equity risk premium is 1.13% which is actually lower than previous estimates  since earnings growth rate (nominal GDP growth rate) has been slashed to 6% for the current year due to covid-19, 10.5% for the subsequent years before reaching a steady state growth rate of 6% in the long term. The average implied ERP for the period 2010 to 2019 is 2.23% vs the current ERP of 1.13%.
If the current estimate of 1.13% is correct then there is scope for further correction in the Indian equity markets based on the reduced growth estimates.

The implied ERP  is subject to your assumptions of growth and dividend payout.

 Reading Equity Risk Premiums 

In this chart, Prof. superimposed equity risk premiums and BBB bond spreads in the same graph. There are multiple takeaways from this chart for practitioners of finance and economics and  the rest of the article is devoted to describing some of these key insights provided by Prof. 

Equity Risk Premiums can be used to gauge overheating in the stock markets. For example, in 1999
equity risk premiums touched a low of 2% . This was at the peak of dot com bust and the expected returns on equity were very low which hinted at unreasonable price levels. So whenever ERPs reach historically low levels, investors need to get cautious and reduce risk taking.

Also notice that equity risk premiums and bond spreads on BBB bonds intersected in 1999 and 2008.
Under normal circumstances, the required rate of return on equity must be higher than the return on bonds. But when these curves intersect, chances are that either the equity prices are inflated (1999) or bond markets are in panic (2008).

Also notice the decline in bond spreads between 2002 and 2007, this period has been dubbed by market participants as 'Greenspan put' which resulted in the sub prime crisis.

So the bottom line here is that equity risk premiums and bond default spreads need to stay in balance with appropriate reward to risk ratio. If that is not the case, then there could be an asset price correction looming round the corner.

Data Source: S & P Indices, Yahoo Finance, Google Search

Investors' discretion is advised.

Tuesday, 14 April 2020

Microsoft: Key Financial Ratios and Metrics


Microsoft is betting big on its cloud revenues but it faces increased competition from Amazon, Google, Sales Force, IBM, Oracle among others. If we combine Microsoft's Server Products & Cloud Services  segment and Office Products & Cloud Services  segment into a new segment called Cloud and Related Services, it accounts for 51.17% of the revenues for the year ended in June 2019. The Cloud related services segment grew at a CAGR of 14.47% in the last three years. Windows is the second largest contributor to Microsoft's revenues after cloud, accounting for 16.21% of the total revenues for the year ended in June 2019.

The 3-year segment CAGRs are shown in the figure below. Based on these figures, it is assumed that Microsoft is going to transform from a growth company to a mature company in the next five to ten years. (unless they pull off another high growth segment!)

Having said that, it is important to note that Microsoft invests heavily in research and development, acquisitions and patents. For the year ended on June 30, 2019 Microsoft invested 13.41% - close to
$ 17 billions of its revenues in research and development.


Microsoft enjoys high margins on its products and services. Between June 2015 and June 2019 Microsoft's operating margin increased from 19.21% to 34.12%. The Net Profit Margin increased from 13.03% to 31.18% during the same period.

When compared to 'FAANG' companies Microsoft has higher higher net profit margins.


Return on Invested Capital has been calculated as EBIT*(1-t)/(Total Assets - Cash and Short Te In)

Microsoft's return on invested capital has increased from 15.15% in June 2015 to 23.68% in June 2019.

Microsoft's EPS and DPS grew at a CAGR of 33.85% and 10.44% respectively in the past four years.


Microsoft has $134. 231 billions in cash and short term investments. Therefore, there are no solvency or liquidity problems.

As of Dec 31, 2019 Microsoft has $ 78.36 billions of debt outstanding on its balance sheet. The proportion of total debt (both short term and long term) out of invested capital (Total Assets -  Cash)
is shown in the chart below.


As of 30 June 2019, Microsoft has unearned revenues of $ 123.67 billions which is 98.27% of its total revenues. Microsoft earns majority of its revenues through client subscriptions and therefore activity ratios are not suitable. Instead, Microsoft's unearned revenues by segment and financial year have been shown in the figure below.

So the conclusion is that Microsoft is moving from growth stage to mature stage of its life cycle with high profit margins and piles of cash.

Data Source: Reuters Finance, Microsoft Investors Centre

Monday, 13 April 2020

Microsoft:Revenues and Product Mix

There are very few companies on the face of this planet that have impacted our daily life and Microsoft is one of them. Today it is hard to imagine any workplace or home anywhere in the world without Microsoft products or services. Over the decades Microsoft has transformed it self from an operating system software provider to a commercial cloud solutions giant.
Microsoft reports its financials from July to June each year unlike most companies which report financials by calendar year. Microsoft's revenues for YE June 2019 clocked $ 125.843 billions growing at a CAGR of 11.35%.

Microsoft revenues for the trailing twelve months ending on Dec 2019 added up to $ 134.249 billions which are classified into three segments as shown below.

As per Microsoft's Annual Report 2019 the following is the description of the composition of their segment-wise revenues:

Productivity and Business Processes
Productivity and Business Processes segment consists of products and services in the portfolio of productivity, communication, and information services, spanning a variety of devices and platforms. This segment primarily comprises:
     Office Commercial, including Office 365 subscriptions and Office licensed on-premises, comprising Office, Exchange, SharePoint, Microsoft Teams, Office 365 Security and Compliance, and Skype for Business, and related Client Access Licenses (“CALs”).
    Office Consumer, including Office 365 subscriptions and Office licensed on-premises, and Office Consumer Services, including Skype,, and OneDrive.
     LinkedIn, including Talent Solutions, Marketing Solutions, and Premium Subscriptions.
    Dynamics business solutions, including Dynamics 365, a set of cloud-based applications across ERP and CRM, Dynamics ERP on-premises, and Dynamics CRM on-premises.
Intelligent Cloud
Intelligent Cloud segment consists of public, private, and hybrid server products and cloud services that can power modern business. This segment primarily comprises:
     Server products and cloud services, including Microsoft SQL Server, Windows Server, Visual Studio, System Center, and related CALs, GitHub, and Azure.
     Enterprise Services, including Premier Support Services and Microsoft Consulting Services.
More Personal Computing
The More Personal Computing segment consists of products and services geared towards harmonizing the interests of end users, developers, and IT professionals across all devices. This segment primarily comprises:
   Windows, including Windows OEM licensing and other non-volume licensing of the Windows operating system; Windows Commercial, comprising volume licensing of the Windows operating system, Windows cloud services, and other Windows commercial offerings; patent licensing; Windows Internet of Things (“IoT”); and MSN advertising.

     Devices, including Microsoft Surface, PC accessories, and other intelligent devices.
     Gaming, including Xbox hardware and Xbox software and services, comprising Xbox Live transactions, subscriptions, cloud services, and advertising (“Xbox Live”), video games, and third-party video game royalties.

Apart from these products Microsoft also invests in Research and Development to build new products and services and intellectual property. For the year ending in June 2019 Microsoft invested close to $ 17 billions on research and development expenses.

 As of YE 30 June 2019, Microsoft total revenues were $125.843 billions and segment wise revenues are broken down in the pie below. More than 50% of Microsoft's revenues are coming from cloud related services and Windows accounts for only 16.21% of total revenues.

Microsoft's Server Products and Cloud Services is the fastest growing segment @ 19.61% CAGR in the past three years while office products and services grew at a rate of 10 %.

As per Microsoft's annual report 2019:

No sales to an individual customer or country other than the United States accounted for more than 10% of revenue for fiscal years 2019, 2018, or 2017. 

At a conference organized by Microsoft in Mumbai recently, Satya Nadella, stressed on the importance of building tech capacity for every organization in the digital era. Microsoft offers to collaborate with its customers in digital transformation with solution areas including:

  • Applications and Infrastructure
  • Data & AI
  • Business Applications
  • Modern Work Place
  • Modern Life
  • Gaming
Microsoft is well and truly on its way to achieve its mission 'to empower every person and every organization on the planet to achieve more.'