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	10000
Accepted Risk	0.1
Time Period Y	1
Predict Growth	0.2

 

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:

Stock	Current Price	Stop Loss	Target
AAPL	157	100	200
MSFT	330	300	400

 

The outcome of the RRR calculations is as follows:

Metrics	AAPL	MSFT
Risk	0.36	0.09
Reward	0.27	0.21
RRR	1.33	0.43

 

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	6369.43	9090.91
Max Exit Value	12738.85	12121.21
Exit Loss	3630.57	909.09
Exit Return	2738.85	2121.21
Accept Exit Loss	1000	1000
Accept Exit Return	2000	2000
Loss Excess	-1738.85	90.91
Return Excess	738.85	121.21

 

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

Metrics	AMZN	GOOG
P/E Ratio	68.56	27.39
Growth Rate (AGR) %	27	41.03
PEG Ratio	2.54	0.67

 

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

Metrics	Value
Share Price	15
EPS	3
AGR %	10
PEG Ratio	0.5

 

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,

where

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

Metrics	Value
Net income ($ mln)	6
Net income attributed to non-controlling interest ($ mln)*	1
No. Outstanding Shares at the Start of the Year (mln)	0.5
No. Outstanding Shares at the End of the Year (mln)	1.5
Weighted Average No. Shares Outstanding	1
EPS $	5

*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,

where

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

and

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:

Metrics	Value
Highest Value mln $	2
Lowest Value mln $	1.5
Average Annual Return %	10
Maximum Drawdown %	25
Calmar Ratio	0.4

 

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%))

or

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

Metrics	Value
Annual volatility %	15
Annual return %	6
Average Drawdown %	-8.3
Average Return %	10.3
Compounded Return %	25.1
Risk-Free Rate %	0
SR 1	1.37
SR 2	1.24

(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:

library(tseries)

data(EuStockMarkets)

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

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

sterling(dax)

sterling(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:

library(tseries)

data(EuStockMarkets)

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

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

sharpe(dax)

sharpe(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]:

Year	Dividend $	DGR %
1	1	0
2	1.05	5
3	1.15	9.52
4	1.17	1.74
5	1.27	6.84

 

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]

P=D1/(r-g),

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:

P=$3/(0.08-0.05)=$100

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