Appendix B: More Real World Examples


Risk/Reward Ratio

Let’s consider the following example to understand RRR. Investor A wants to create a balanced portfolio using the following input data:

Asset Value


Accepted Risk


Time Period Y


Predict Growth



This is about investing $10000. He is willing to accept 10% of the risk with the expected 20% of the return on a short-term investment of 1Y. He shortlisted the two stocks, AAPL and MSFT, at the current prices $157 (AAPL) and $330 (MSFT) per share. He analyzed both stock trends using historical market data. According to his stock BI analysis, the predicted AAPL share price range is $100-200, whereas the predicted MSFT share price range is $300-400 in a period of 12 months:


Current Price

Stop Loss











The outcome of the RRR calculations is as follows:














Investor A suggests that MSFT is the best investment in terms of RRR because RRR(MSFT)<RRR(AAPL), RRR(AAPL)>1.0 and RRR(MSFT)<<1.0. The dollar outcome of this analysis is

Min Exit Value



Max Exit Value



Exit Loss



Exit Return



Accept Exit Loss



Accept Exit Return



Loss Excess



Return Excess




Conservative investor A comes to the decision of accepting low-risk and low-return MSFT while rejecting high-reward and excessively high-risk AAPL. He may want to increase MSFT returns by extending the investment period at least 3-5 times. On the other hand, assertive investor A may choose to invest in both companies by dividing his investment while extending the investment period for MSFT.        


PEG Ratio versus P/E Ratio

Let’s compare stocks of two big tech companies that have different levels of 2021 growth rates




P/E Ratio



Growth Rate (AGR) %



PEG Ratio




Both P/E and PEG Ratios reveal that the stock AMZN is grossly overvalued, whereas GOOG is reasonably undervalued (considering the earnings expected to be derived in the future from the stocks).  Since AGR(GOOG)>>AGR(AMZN), an ideal stock for investors is GOOG (S&P AA+) and not AMZN (S&P AA) over the long term.  Since PEG(AMZN)>1, it indicates that the stock is overvalued as the growth expectations are very much high.  

Let’ talk about SME A financials. A share is currently trading at $30 and the EPS of the share is $2. The earnings of the company are expected to grow by 20%. The PEG Ratio is given below



Share Price






PEG Ratio



In this case, PEG=0.5<1. It would mean that the stock is undervalued as the growth expectations are very less. The current price of the stock is said to be consistent with its estimated earnings, which in turn means that the stock is priced at par.


Earnings per Share (EPS)

The formula for EPS is

EPS = (Net Income – Preferred Dividend) / Weighted Average No. of Shares Outstanding,


Weighted Average No. of Shares Outstanding = (No. of Outstanding Shares at The Start of The Year + No. of Outstanding Shares at The End of The Year) / 2.

Let’s take the example of the company A that reported a net income of $6 billion during the year 2020, out of which $1 billion has been contributed to the non-controlling interest. The number of shareholders of the common stock at the start of the year was 0.5 million that went up to 1.5 million. The EPS of the company for the year 2020 is



Net income ($ mln)


Net income attributed to non-controlling interest ($ mln)*


No. Outstanding Shares at the Start of the Year (mln)


No. Outstanding Shares at the End of the Year (mln)


Weighted Average No. Shares Outstanding




*It could also be the Preferred Dividend that has to be paid out to the preferred shareholders.

For the sake of apples-to-apples comparison, it is very useful to compare the EPS(A) value with its competitors & industry benchmark ratio EPS(M), where M means Market. For example, we can conclude that the company is doing well if EPS(A)> EPS(M). This is because companies within a sector or industry will experience similar macroeconomic factors such as politics, economy, demographics, etc. So the chances of them performing similarly are very high.

The Bottom Line: stocks with higher EPS attract higher prices. EPS along with the share price can be used to check the rate of return. But be aware that the companies can manipulate the EPS by reducing the number of outstanding shares by buying back their own shares. 


Calmar Ratio

The Calmar Ratio (CR) or the drawdown ratio [1,21] is a risk-adjusted key performance metric for mutual funds, hedge funds and commodity trading. In fact, it measures the return per unit of risk and lets the investor decide whether the given amount of return is worth it at the given level of risk or not. Calmar is short for California Managed Accounts Report and is very similar to MAR ratio. The CR was first published in 1991. It is most similar to the Sterling ratio [22] in its calculation, it takes the average annual compounded rate of return and divides it by the maximum drawdown for that same time period, usually over a period of 3 years [21]. The higher the CR the better with anything over 0.5 or close to 1.0 is good (Amber), CR=3-5 is really good (Green).   

The Calmar ratio formula is

CR = Average Annual Rate of Return / Maximum Drawdown,


Average Annual Rate of Return = Portfolio Return - Risk-Free Return


Maximum Drawdown = ((Highest Value – Lowest Value) / Highest Value) * 100.

Let’s consider a fund started 3 years ago. It reached a value of $2 mln but went as low as $1.5 mln. Over this period the average annual return was 10%. In this example, the CR should help in evaluating whether the fund is worth investing. The outcome is given below:



Highest Value mln $


Lowest Value mln $


Average Annual Return %


Maximum Drawdown %


Calmar Ratio



It appears that the risk-adjusted ratio is 0.4. If the investor has a criterion of a minimum CR of 0.5 [21], then the fund is not worth investing in (Red). Further, we may compare CR to another fund which has CR>0.5 or CR~1.0, and therefore has a higher risk-adjusted return and should be selected over this fund.

The CR is an improvement of both the Sharpe and Sterling Ratios in that it provides an up-to-date appraisal of commodity trading advisor (CTA) performance.


Sterling Ratio

Sterling Ratio (STR) [22] is another helpful risk-adjusted measure that is used to evaluate the performance of hedge funds. The STR measures risk using the average drawdown as follows:

STR=Compounded Annual Return/(Average Maximum Drawdown-Risk-Free Rate (e.g. 10%))


STR=(Annual Return – Annual Risk-Free Rate) / |Average Annual Max Drawdown|  

The STR ignores both the opportunity cost of money and STDEV of monthly results. Instead, it measures volatility of investments with respect to the earnings through the last 3 years. The example below shows the reasonable value of SR>1.0



Annual volatility %


Annual return %


Average Drawdown %


Average Return %


Compounded Return %


Risk-Free Rate %


SR 1


SR 2


(Source: OpenML [25])

The R package tseries (Time Series Analysis and Computational Finance) [23] contains the function sterling(data), where data = a numeric vector or univariate time series corresponding to a portfolio's cumulated returns. This function computes the Sterling ratio of data as follows:



dax <- log(EuStockMarkets[,"DAX"])

ftse <- log(EuStockMarkets[,"FTSE"])



The output SR values are SR(DAX)=4.72 and SR(FTSE)=3.97 (Green), but the investor may want to select the former because SR(DAX)>SR(FTSE).   


Sharpe Ratio

The Sharpe ratio is defined as a portfolio's mean return in excess of the riskless return divided by the portfolio's standard deviation. In finance the Sharpe Ratio represents a measure of the portfolio's risk-adjusted (excess) return. The R package tseries [23] contains the function sharpe(data), where data = a numeric vector or univariate time series corresponding to a portfolio's cumulated returns. This function computes the Sharpe ratio of data as follows:



dax <- log(EuStockMarkets[,"DAX"])

ftse <- log(EuStockMarkets[,"FTSE"])



The output Sharpe ratio values are Sharpe(DAX)=1.00 and Sharpe(FTSE)=0.86 (Green). As with SR, the investor may want to select the former because Sharpe(DAX)>Sharpe(FTSE).   

Growth Rate Metrics and Gordon Model

Let’s consider the following example of a company A scheduled to pay dividends with the calculated annual DGR [47]:


Dividend $


















We can calculate the forward-looking growth rate as the arithmetic average of the historical DGR over (n-1)=4 years

Arithmetic Average = [DGR(2)+…+DGR(n-1)]/(n-1)=(5%+9.52%+1.74%+6.84%)/4=5.78%

or the Compound Annual Growth Rate (CAGR)

CAGR=[((Dividend(n)/Dividend(1))**n)-1]*100%=4.56% with n=5.

This can be compared to the average DGR(M)=4% in the industry in which the company A is operating. In principle, we can just use that rate for DGR(A). The sustainable growth rate is the maximum growth rate that a company can sustain without external financing. The related Sustainable Growth Rate (SGR) can be found using the following formula

SGR = ROE *(1-Dividend Payout Ratio)

Assuming that ROE(A)=15% and the Dividend Payout Ratio = 65%, we have

SGR = 15% *(1-0.65)=5.25%.

This is the maximum rate of growth that a company or social enterprise can sustain without having to finance growth with additional equity or debt. The SGR involves maximizing sales and revenue growth without increasing financial leverage.

Finally, let’s look at a popular and straightforward variant of the dividend discount model (DDM) - the Gordon Growth Model [1,47]


where P is the current stock price, g is the constant growth rate expected for dividends, r is the constant rate of return, and D1 is the value of next year’s dividends.  

As an example, consider a company A whose stock is trading at $110 per share. This company requires an 8% minimum rate of return (r) and will pay a $3 dividend per share next year (D1), which is expected to increase by 5% annually (g).

The intrinsic value (P) of the stock is calculated as follows:


According to the DDM, the shares are currently $10 overvalued in the market.







  1. Let’s look at the Morningstar’s annual fair value estimates across various industry sectors.
    A ratio above 1.00 indicates that the stock’s price is higher than Morningstar’s estimate of its fair value. The further the price/fair value ratio rises above 1.00, the more the median stock is overvalued. A ratio below 1.00 indicates that the stock’s price is lower than our estimate of its fair value. The further it moves below 1.00, the more the median stock is undervalued.
    The Bottom Line: If you’re building your long-term portfolio and you want it to have high growth potential, you’re looking for undervalued stocks such as Energy, Communication Services (Green) and possibly Utilities to be studied further (Amber).

  2. 9. Retirement Savings in Python
    Let’s see how saving at different time period can affect the amount an individual A has in retirement [99]: current age is 55, retires in 5 years, live 10 years in retirement, yearly cost in retirement $30000, and expected returns is 8% per year. How much should A save each year before retirement? The solution script is as follows:
    import pandas as pd
    import numpy as np
    interest = 0.08
    n1 = 5 #Time Horizon 1
    n2 = 10 #Time Horizon 2
    pmt_in_retirement = 30000
    retirement_amount = np.pv(rate = interest,nper=n2,pmt = pmt_in_retirement,when=1)
    #type = 1, withdrawing at the beginning period
    print(retirement_amount * -1)
    So A will need about $217,406 to cover his expenses in retirement. Next we will calculate the amount needed to save today to accumulate $217,406.
    saving = np.pmt(rate = interest, nper = n1, pv = 0, fv = retirment_amount, when=1)
    So A needs to save $34313 each year to have enough money for his retirement.

  3. 10. Retirement Savings in R
    The aforementioned retirement savings Python script can be rewritten in R as follows:
    interest <- 0.08
    n1 = 5 #Time Horizon 1
    n2 = 10 #Time Horizon 2
    pmt_in_retirement <- 30000
    retirment_amount <- pv(pmt = 30000, r = interest, n = n2, fv = 0, type = 1)
    #type = 1, withdrawing at the beginning period
    print(retirment_amount * -1)
    saving = pmt(pv = 0, fv = retirment_amount, r = interest, n = n1, type = 1)

  4. 11. Stock Returns in R
    Let’s install and load the following packages into your R environment:
    We can write a function to get monthly return data for our individual stocks. Our function takes two arguments: ticker , the stock’s symbol, and base_year , the year that we want to start analysing the data [100] :
    monthly_returns <- function(ticker, base_year)
    # Obtain stock price data from Yahoo! Finance
    stock <- getSymbols(ticker, src = "yahoo", auto.assign = FALSE)
    # Remove missing values
    stock <- na.omit(stock)
    # Keep only adjusted closing stock prices
    stock <- stock[, 6]
    # Confine our observations to begin at the base year and end at the last available trading day
    horizon <- paste0(as.character(base_year), "/", as.character(Sys.Date()))
    stock <- stock[horizon]
    # Calculate monthly arithmetic returns
    data <- periodReturn(stock, period = "monthly", type = "arithmetic")
    # Assign to the global environment to be accessible
    assign(ticker, data, envir = .GlobalEnv)

  5. 12. Fair Value Estimation
    Let’s look at the Peter Lynch’s Stock’s Fair Value Calculator with the following formula [111]:

    Peter Lynch’s Fair Value (PLFV) = (Earnings Growth Rate + Dividend Yield) / P/E,

    where you can take an average of the next 3-5 years EPS growth rate, divide the latest annual cash dividend per share by the share price and you will the dividend yield, and divide the stock price by EPS to get the P/E ratio. The value PLFV < 0.5 means that the stock is very over-valued. The value PLFV = 0.5-1 means that the stock is over-valued. The value PLFV = 1-2 means that the stock is trading at a fair valuation. The value PLFV = 2-3 means that the stock is under-valued. The value PLFV >3 means that the stock is very under-valued. The ideal time to buy a stock is when it is very under-valued and the ideal time to sell could be when it is very over-valued.

    Let’s use the Yahoo Finance and Morningstar data to compare the value of high growth stocks such as FB, GOOGL, MSFT and AMZN as it is at 2021, August: PLFV(FB) = 1.02, PLFV(GOOGL) = 0.78, PLFV(MSFT) = 0.43, and PLFV(AMZN) = 0.62. You can see that GOOGL and AMZN are over-valued (amber score, action to monitor), MSFT is very over-valued (red score, action to sell) and FB is fairly-valued (green score, currently no action).

  6. 13. Crucial Liquidity Ratios
    Working out ratios offers a quick and easy way to see whether your business can satisfy its debts [112]. When ratios are less than 1:1, this means you need to find ways to increase liquidity. In fact, creditors and investors prefer to see liquidity ratios closer to 2:1 or 3:1 rather than 1:1, because this indicates that the company has plenty of room to pay its short-term bills and still have working capital to continue operations.

    The current ratio is the first popular liquidity ratio formula. To calculate current ratio, simply divide total assets by total liabilities: Current Ratio (CR) = Current Assets / Current Liabilities.
    Business managers will generally look for a current ratio above 2:1 at a minimum. For example, a start-up company A reported $1000 of current liabilities and only $250 of current assets. This means that CR(A) = 0.25. As you can see, the company has enough current assets to pay off 25% of its current liabilities. This shows that A is highly risky, so that the bank would not approve any loan.
    The second option is the quick ratio (aka the acid test)
    Quick Ratio (QR) = (Cash + Accounts Receivable) / Current Liabilities,
    Cash + Accounts Receivable = Total Current Assets - Inventory - Prepaid Expenses. The acid test ratio measures the liquidity of a company by showing its ability to pay off its current liabilities with quick assets. If a firm has enough quick assets to cover its total current liabilities, the firm will be able to pay off its obligations without having to sell off any long-term or capital assets. A good quick ratio would be 1.5:1, due to its slightly stricter guidelines. An acid ratio of 2 shows that the company has twice as many quick assets than current liabilities. For example, let’s look at the balance sheet of a start-up A: cash = $10k, Accounts Receivable = $5k, Inventory = $5k, Stock Investments = $1k, Prepaid taxes = $0.5k, and current liabilities = $15k. This yields QR(A) = ($10k+$5k+$1k)/$15k = 1.07. This means that A can pay off all of its current liabilities with quick assets and still have some quick assets left over.
    The third liquidity ratio is the cash ratio, which restricts its current assets to cash and marketable securities
    Cash Ratio (CR) = (Cash + Marketable Securities) / Current Liabilities

    CR = (Cash in Hand + Cash in Bank + Bank Deposit + Treasury Bills) / (Current Liabilities + Outstanding Expense + Provision of Tax).

    This makes it even stricter than the quick ratio since only cash and cash equivalents are taken into consideration. For example, company A reported the following balance sheet: cash in hand = $15k, cash in bank = $10k, bank deposit = $86k, treasury bill = $1.2k, current liabilities = $50k, outstanding expense = $12k, and provision of tax = $1.2k. This yields
    CR(A) = ($15k+ $10k+$86k+$1.2k)/ ( $50k+$12k+$1.2k)=1.78.
    As cash ratio is 1.78 which means the company has more cash than they need to pay off current

  7. 14. Gold & Currencies Correlations
    Investors typically buy large quantities of gold when their country is experiencing high levels of inflation [1]. Gold has a profound impact on the value of world currencies. Even though the gold standard has been abandoned, gold as a commodity can act as a substitute for fiat currencies and be used as an effective hedge against inflation. There is no doubt that gold will continue to play an integral role in the foreign exchange markets. Therefore, it is an important metal to follow and analyse for its unique ability to represent the health of both local and international economies.
    People want to invest or buy gold to protect themselves from volatility and uncertainty. The preference for physical assets makes Indian households view gold as a safe haven, an asset to buy when other assets are losing value. Underlining gold's attraction as an asset for good times and bad, most investors would buy gold whether the domestic economy was growing or in recession.
    When inflation rises, the value of currency goes down and therefore people tend to hold money in the form of gold. Therefore, in times when inflation remains high over a longer period, gold becomes a tool to hedge against inflationary conditions. This pushes gold prices higher in the inflationary period.
    According to some industry experts, under normal circumstances, there is a negative relationship between gold and interest rates. Rising yield indicates an expectation of strong economy. Strong economy gives rise to inflation and gold is used as a hedge against inflation. Also, when rates rise, investors flock to fixed-income investments that yield a fixed return unlike gold which does not carry any such return. So, demand takes a back seat with prices remaining flat.
    It is believed by some economists that gold is a highly effective portfolio diversifier due to its low to negative correlation with all major asset classes. Still, as a rule, gold shows no statistically significant correlation with mainstream asset classes. However, some suggests that there is evidence that when equities are under stress, in other words when shares are falling rapidly in value, an inverse correlation can develop between gold and equities.
    Gold usually does well during geopolitical turmoil and the current crisis over Korea's nuclear capability has boosted the prospects of the yellow metal. Crises such as wars, which have a negative impact on prices of most asset classes, have a positive impact on gold prices since the demand for gold goes up as a safe haven for parking funds.
    Under normal circumstances, gold and dollar share an inverse relationship. Since international gold is dollar denominated, any weakness in the dollar pushes up gold prices and vice versa. The inverse relationship is because firstly, a falling dollar increases the value of currencies of other countries. This increases the demand for commodities including gold. It also increases the prices. And secondly, when the US dollar starts to lose its value, investors look for alternative investment sources.
    Finally, it is interesting to note that gold has a positive correlation with AUD/USD [113]. Across the seven seas, Switzerland’s currency, the Swiss franc, also has a strong link with gold. Using the dollar as base currency, the USD/CHF usually climbs when the price of gold slides. Conversely, the pair dips when the price of gold goes up. Unlike the Australian dollar, the reason why the Swiss franc moves along with gold is that more than 25% of Switzerland’s money is backed by gold reserves. Gold has a negative correlation with USD/CHF (since 2009). When gold goes up, USD/CHF goes down. When gold goes down, USD/CHF goes up.

  8. 15. Asset Allocation Portfolios
    We choose to focus on the long term since ultimately that is the key focus for most of our clients. The 60% stock and 40% bond portfolio (60/40 portfolio) that is often thought of as the starting point for basic asset allocation decisions may no longer be sufficient to generate the returns required by many institutions and individuals to meet their long-term financial goals and objectives.
    We need to continually reassess our portfolio positioning to confirm we can meet our long-run goals and objectives:
    1. Re-evaluate Your Investment Policy Statement and Asset Allocation Regularly;
    2. Adjust Spending Rate and the Methodology Used to Calculate It;
    3. Reinvest Excess Returns;
    4. Consider the Mix of Active and Passive Strategies Employed in Portfolios;
    5. Explore the Use of Alternative Investments.
    Here, a clear understanding of risk is critical. Asset allocation decisions may differ depending on the way one chooses to define risk.

  9. 16. The Impact of Fees & Expenses
    Fees and costs associated with investment products and services may seem small, but over time they can have a major impact on your investment portfolio [9]. Let’s look at a portfolio value from investing $100k over 20 years. In 20 years, 1% annual fees reduce portfolio value by nearly $30k, compared to a portfolio with a 0.25% annual fee. In addition, if you were able to invest that $30k, you would have earned an additional $13k. Here is the ABN-AMRO example cost information sheet as at 31-12-2020: guided investing through profile funds 0.62-0.94%, transaction fees in investment funds 0.6-0.8%, ongoing charges 1.16%, and other charges 0.03%.

  10. 17. Multiple Ratings & Credit Standards
    Credit ratings are predominantly provided by three main independent rating agencies, namely; Standard & Poor’s (S&P), Moody’s Investor Services (Moody’s), and Fitch IBCA (Fitch), although there are others. The agencies provide an overview of their detailed rating methodologies on their websites, but in general the analysis will focus on two broad areas:
    Business Risk - evaluation of strengths/weaknesses of the operations of the entity (geographic diversification, sector strengths/weaknesses, etc.); Financial Risk - evaluation of the financial flexibility of the entity (total sales and profitability measures, margins, growth expectations, liquidity, etc.).
    The credit ratings are not buy, sell, or hold recommendations, or a measure of asset value. Nor are they intended to signal the suitability of an investment. They speak to one aspect of an investment decision—credit quality—and, in some cases, may also address what investors can expect to recover in the event of default.
    For example, a corporate bond that is rated ‘AA’ is viewed by the rating agency as having a higher credit quality than a corporate bond with a ‘BBB’ rating. But the ‘AA’ rating isn’t a guarantee that it will not default, only that, in the agency’s opinion, it is less likely to default than the ‘BBB’ bond.
    Most banks now have their own internal risk rating scale that tends to be far more numerically mechanical than those of the agencies and consequently their scales may not be directly comparable. It should be noted that almost all banks use Moody’s KMV to calibrate their rating scales for expected default probability or expected loss.


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