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Relationship Between Risk and Rate of Return
READING
9 mins read
In our previous chapter, we took a deep dive into the concept of 'risk' in the investment world. We explored the various types of risks – from the market swings that can sway all investments to the more specific risks like credit and liquidity issues. Each type of risk, we learned, demands its own strategy and understanding for effective navigation.
If you remember, we also saw how risk was tied to returns (and also losses). It’s natural to ponder over the question – how exactly are risk and return related? How does one affect the other? Let’s answer those questions in this chapter.
This chapter is all about understanding how the risks you take (or avoid) are intrinsically linked to the potential returns you might gain. Think of it as the balance between the thrill of sailing in uncharted waters (high risk) and the tranquillity of cruising along a calm coastline (lower risk).
What is the Risk Return Relationship?
Picture this – you are all set to go for an adventurous trek. Now, track back a bit and try to think how exactly you decide on the destination for your trek. It would seem reasonable to say that you chose the path and the destination based on your comfort with challenges and potential rewards. You did some riskreturn ratio analysis, to put it in simple terms, and came to some conclusions.
Likewise, when you’re starting your investment journey, it’s important to understand the relationship between risk and return in order to be able to choose the best path and the most reasonable destination.
Risk Return Relationship Simplified
Risk and returns can be simply thought of as two interlinked parameters or knobs of your investment journey. The higher the peaks you want to conquer (returns), the more rugged the paths (risks) you might have to navigate. Various risks, such as those specific to a project or industry, can impact your investment journey. Returns, on the other hand, are your rewards for embarking on this journey, often expressed as a percentage gain or loss on your investment.
Imagine walking on a tricky trail. Risk could be many things – it could be the probability of stumbling on a tricky path (losing part of your investment) or the unevenness of the terrain (return volatility). Highrisk investments are akin to treacherous paths where a wrong step could mean a significant setback. The volatility of these investments is often measured by standard deviation, a statistic that captures how much your investment journey might deviate from the average path.
Furthering our trekking journey, returns can be seen as the scenic views you get to enjoy after the hike. They can be 'nominal', including the inflationary effect, or 'real', which discounts the inflation. For instance, if you start with an investment of ₹100 and it grows to ₹120, your nominal return is 20%. But after adjusting for a 3% inflation, your real return, which reflects the actual value of your scenic view, is 17%.
So then the question becomes, how do you protect yourself from risk while optimising for returns? Well, it isn’t that straightforward, but let’s stay on our trek. On the trek, you’d carry different equipment for varied terrains, right? If you expect rain, you’d carry an umbrella, a raincoat, and so on. If parts of your hike are dark and not lit, you’d carry a torch. You get the idea!
By spreading your investments across different types (say, equities, bonds, real estate), you reduce the impact of rough patches in one area on your entire portfolio. In a diversified portfolio, positive and negative impacts across different investments tend to balance out, smoothing your journey.
So, if we have to put it simply, the riskreturn relationship in investing is about how you balance your appetite for high scenic points against walking through riskier paths. It's about making choices that align with how much risk you're comfortable taking for potential rewards. This principle guides your investment decisions, helping you pick the right mix of assets that align with your financial goals and risk tolerance.
So far, so good. But how exactly do you calculate the socalled risk and return values? Well, there are different ways to answer this question – but let’s start with the basics first!
Risk And Return Calculation
Alright, let's dive into the world of risk and return calculations – think of it as a chef figuring out the perfect recipe for a delicious dish. You're the chef, and your ingredients are your investments. How do you mix them to get the tastiest (most profitable) and safest (least risky) dish?
Understanding and Quantifying Return
1. Absolute Return
Absolute Return measures the growth or decline in your investment without considering the time period. It's expressed as a percentage of the investment's value. For example, if you invested ₹1,00,000 in an asset, and its current market value is ₹1,40,000, the absolute return is 40%. This method is typically used for periods of less than a year.
Absolute Return=[(Ending ValueBeginning Value) / (Beginning Value)] × 100
So, if the investment starts at ₹1,00,000 and grows to ₹1,40,000,
Absolute Return = [(1,40,000  1,00,000) / 1,00,000)]*100
This comes to 40%, as mentioned earlier!
2. Annualised Return
Annualised Return spreads the total return of an investment over a number of years. It includes the effect of compounding, which can significantly impact the return over longer periods. For example, you might compare 15% annualised returns over 5 years versus 85% absolute returns. Due to compounding, 15% annualised returns actually yield a total effective return of 101.14% over 5 years.
Annualised Return = (Ending Value  Beginning Value) (1/Number of Years)1
So, if an investment grows from ₹1,00,000 to ₹1,50,000 over 3 years, you know how to calculate its annualised return!
3. Total Return
Total Return considers all forms of inflows and appreciation from an investment. For stocks and mutual funds, it includes capital gains and dividends. For instance, if you invest in a mutual fund with a net asset value (NAV) of ₹20, and after a year, it grows to ₹22, plus you receive a dividend of ₹2 per unit, your total return is 20%. Get the same answer using the formula below:
Total Return = [(Capital Gain+Income) / (Initial Investment)] × 100
4. PointtoPoint Return
This calculates the annualised returns between two specific dates. For instance, if a mutual fund's NAV was ₹50 on January 1, 2020 and ₹55 on December 31, 2020, the pointtopoint return for that year is 10%. Here’s the formula to get the pointtopoint returns.
Point to Point Return = [(Ending ValueStarting Value) / (Starting Value)] × 100
5. Compounded Annual Growth Rate (CAGR)
CAGR is used for longer periods to give compounded annual returns, especially for mutual funds. It considers the start and end value of the investment over a period, normalising all highs and lows. CAGR does not account for multiple cash flows or periodic investments.
CAGR = (Ending Value  Beginning Value) (1/Number of Years)1
This formula might look similar to the one for annualised returns, and you’re right in that observation – because they are doing the same thing in principle. However, keep in mind that the CAGR is often presented using only the beginning and ending values, whereas the annualised total return is typically calculated using the returns from several years
6. Extended Internal Rate of Return (XIRR)
XIRR is suitable when you have multiple cash flows, like in the case of a Systematic Investment Plan (SIP). It calculates the CAGR of each cash flow and combines them to give an overall CAGR. Since XIRR is primarily used for irregular cash flows, it involves complex calculations and is often done using financial software or spreadsheets.
While these basics are good to keep in mind, always remember to factor in investment fees, commissions, or other charges while doing these calculations, as they can affect the overall return. Also, consider comparing your investment's performance against similar investments for a fair assessment.
Measuring the Risk
Let’s now see how exactly we can measure risk, if at all. Please note that these terms use concepts and formulas from statistics. So, if you don’t understand the formulas and examples perfectly, that’s fine – as long as you get the gist of what the risk is all about.
Below are the different types of risks without using jargons:

Alpha and Beta:
Such are values used in finance to calculate the performance and volatility of investments compared to the market. 
RSquared:
This is the correlation between the performance of an investment and a benchmark. 
Standard Deviation:
This is a statistical measure of the dispersion of investment returns. Consider this the recognition of the variability of returns from the year to year. 
Sharpe Ratio:
This ratio measures the riskadjusted return of an investment.
Now, let's look at the formulas and examples for each:

Alpha Calculation:
 Formula: Alpha = Actual Return of Portfolio  [RiskFree Rate + Beta * (Market Return  RiskFree Rate)]
 Example: If a portfolio's actual return is 16%, the riskfree rate is 4%, the market return is 11%, and the beta is 1.3, the alpha would be calculated as 16%  [4% + 1.3 * (11%  4%)] = 2.9%.

Beta Calculation:
 Formula: Beta is typically provided as a value in financial reports or investment analysis tools.
 Example: If a portfolio has a beta of 1.2, it means the portfolio is 20% more volatile than the market.

RSquared Calculation:
 Formula: RSquared is generally calculated using statistical software as it involves complex regression analysis.
 Example: An RSquared value of 95% means 95% of the portfolio's movements are explained by movements in its benchmark index.

Standard Deviation Calculation:
 Formula: Standard deviation is calculated as the square root of the variance of investment returns. More simply, consider the probability of each year's return occurring, then calculate the squared deviation of each year's return from the average return, and finally, average these values. The square root of this average gives us the standard deviation.
 Example: If investment returns are 10%, 12%, 8%, and 11% over four years, the standard deviation would be calculated based on the variance from the mean return. This would give you some sort of an idea of how much your figure varies from the mean figure.

Sharpe Ratio Calculation:
 Formula: Sharpe Ratio = (Mean Portfolio Return  RiskFree Rate) / Standard Deviation of Portfolio Return.
 Example: If the mean return is 10%, the riskfree rate is 3%, and the standard deviation is 5%, the Sharpe Ratio would be (10%  3%) / 5% = 1.4.
Now that we’ve understood returns and risks let’s see how to perform the return calculation of a portfolio.
Return Calculation of a Portfolio
The concept of return calculation for a portfolio involves understanding the relationship between the expected returns from individual securities and their proportion in the portfolio. Essentially, the expected return of a portfolio (ε(Rp)) is the weighted average of the expected returns of each security within it. This includes factors like the proportion of funds invested in each security (Wa and Wb for securities A and B, respectively) and their individual expected returns (Ra and Rb). The calculation is guided by the formula:
ε(Rp)=Wa×Ra+Wb×Rb
For a practical application of this concept, let's consider an example portfolio. Suppose Ms Ridhi's portfolio consists of securities with varying proportions and returns. The portfolio's overall expected return would be the sum of the products of each security's proportion of investment and its return.
Contrastingly, the risk of a portfolio is not merely the average of the risks of individual securities. It also takes into account the correlation between the securities, which measures the relationship between the returns from one security with another. This covariance affects the overall risk of the portfolio and plays a key role in determining the gains from diversification.
To simplify this process, let’s consider two securities – A and B.
 Portfolio Composition: Let us assume that you have Security A and Security B.
 Security A: ₹1,00,000 investment, Expected Return: 8%
 Security B: ₹50,000 investment, Expected Return: 12%

Weighted Average Return Calculation:
 The total value of the portfolio is ₹1,50,000; ₹100,000 from A and ₹50,000 from B.
 A portion of Security A is ₹1,00,000/₹1,50,000 = 2/3, and a portion of Security B is ₹50,000/₹1,50,000= 1/3.
 Weighted Average Return = (2/3 × 8%) + (1/3 × 12%) = 5.33% + 4% = 9.33%.

Risk Calculation (using variance or standard deviation):
 Assume standard deviations: Security A = 5%, Security B = 10%.
 Assuming the securities are perfectly uncorrelated, the risk of the portfolio is obtained from the following formula for the standard deviation of a twoasset portfolio.

Applying the Formula:
 Portfolio Standard Deviation = √((Wa^2 * σa^2) + (Wb^2 * σb^2)), where Wa and Wb are weights of A and B, and σa and σb are standard deviations of A and B.
 Substituting the values, we get: Portfolio Standard Deviation = sqrt(((2/3)^2 * 5%^2 + (1/3)^2 * 10%^2)).
 This gives us the overall risk measure of the portfolio.

Analysis and Comparison:
 Security A has a lower return and lower risk, while Security B has a higher return and higher risk.
 The overall portfolio return is somewhere between these two, demonstrating that diversification impacts both returns and risks.
In this chapter, we have examined the intricate nature of risk and return on investment. This riskreturn tradeoff is one of the most important aspects of investment decisionmaking. We’ve observed how different types of investments can have a variety of risks and the importance of assessing one’s own risk profile.
In the next chapter, ‘Types of Investments’, we will discuss types of investments, including fixedincome, equity, and debt investments. Our discussion will revolve around their peculiarities, their risks and returns, and their role in the various investment strategies. This will enable the creation of a comprehensive appreciation of the diverse investment environment and enable a proper choice of investments.