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Risk and Return

Risk and reward go hand in hand and have a direct relationship. Return is the reward that a person gets for taking risk.

Measures of Risk

We will first look at the various measures of risk, that is how risk can be measured and a particular investment can be gauged.

The various measures of risk are discussed below:

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  • Variance- It measures how the investment is dispersed. The larger the variance for an expected rate of return, the greater the dispersion of expected returns and the greater the uncertainty, or risk, of the investment. It should be noted that, in perfect certainty, there is no variance of return because there is no deviation from expectations and, therefore, no risk or uncertainty.
  • Standard Deviation- it is the square root of the Variance and even this measure is widely used to measure the risk associated with an investment.

In certain investment cases, a simple variance or standard deviation can give misleading results if the conditions for two or more investment alternatives are not similar. So we use an adjusted measure called as coefficient of variation (CV). It is calculated as:

Standard rate of return/Expected rate of return

Random Variables and Expected Values

Basically, the expected value is the average outcome of the event is drawn randomly for an infinite period. For example, if the coin is tossed randomly the expected outcome is A head or A tail.

Random Variable – Basically it is the variable in which the outcome is not defined. It is also called the variable with a Statistical and probability distribution. For example, we can’t say that if there are clouds in the sky it will definitely rain. The outcome is not definite.

Expected value is the sum of the product of probabilities of each event with their respective payoffs

Suppose we have two expected outcomes with their respective probabilities of occurrence and their payoffs.

Then the expected value will be Probability Event 1*(Payoff incase event 1) + Probability Event 2*(Payoff incase event 2)

In the probability, the basic assumption has to be made. The market should be perfect.

Fair Bet

A fair bet is a bet that cost the expected value of any event which will occur. For Example If a coin is tossed there will be a possible outcome when it will be a head or tail.

Variance and Standard Deviation

Standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. In simple words, it means a quantity expressing by how much the members of a group differ from the mean value for the group.

Variance is the square of a standard deviation of a random variable from its mean.

In the financial term, if we had made an investment then we will get a reward for the investment. Another important concept is the risk. It is denoted by the word “Standard Deviation”.

Calculating Standard deviation – Solved Example

The mean of the event is $15. The following are the three outcomes;

$5 $9 and $13. Find out the standard deviation of the outcome


OutcomeMean ValueMean Deviations squared
$5$15($5 - $15)^2 = (-$10)^2 = $100
$9$15($9 - $15)^2 = (-$6)^2 = $36
$13$15($13 - $15)^2 = (-$2)^2 = $4

Total Variance = $100 + $36 + $4 = $140
Standard Deviation = √Variance
= √$140
= $11.83

Risk Neutrality

In economics and finance, risk neutral refers as neither risk averse or risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indifferent between choices with equal expected payoffs even if one choice is riskier. No high risk or no low risk. The investor is in the position of a both high risk and low-risk situation of its investment.

A risk averse investor is an investor who prefers lower returns with known risks rather than higher returns with unknown risks. In other words, among various investments giving the same return with different level of risks, this investor always prefers the alternative with least interest. It basically means getting on a safer side. Rather going for a huge profit with a huge risk or end at a zero profit, it is better to be at a safer side with a normal return on the investment.

Risk sharing

In the financial market, the risk of a particular investor is shared by the financial market as a whole. Risk sharing means that the premiums and losses of each member of a group of policyholders are allocated within the group based on a predetermined formula. Their aggregate risk absorption capabilities are considerably higher than those of their individual investors. In effect, the financial markets are less risk averse than individual investors.

Interest Rate

Basically, in the general term, the interest rate is an amount charged by a bank or any person who is lending any kind of assets (whether it is cash or any tangible asset) therefore. It is expressed in percentage on an annual basis.

Credit Risk

A credit risk is the risk of default on a debt that may arise from a borrower failing to make required payments. Basically, if the lender is not able to get its principal amount or interest rate timely than it is in a situation of Credit risk. Credit Risk is also called as Default Risk.

If the person is having a neutral risk, then he/she can demand a higher promised rate for their investment in the financial market. The promised rate which he /she will get on their investment will always be less than the actual promised rate.

The difference between the promised rate and expected rate of return is known as Default premium. The investor always used to quote promised rate. Default premium is also called default risk and credit risk.

Promised rate is also called quoted interest rate and the stated interest rate.

Solved numerical problem questions on calculation of expected return, yield and promised interest rate

Question 1: XYZ Ltd Investment proposal with the probability of receiving full payment in one year on a $200 investment of $210 is only 95%, the probability of receiving $100 is 1%, and the probability of receiving absolutely $0 payment is 4%.

1. At the promised interest rate of 5%, what is the expected interest rate?
2. What is interest rate required as a promise to ensure an expected interest rate of 5%?

Solution – At the promised interest rate of 5%, the expected rate of return will be –

Expected payoff – Prob * Cash flow (1) + Prob * Cash flow (2) + Prob * Cash flow (3)

= 95% * $210 + 1% * $100 + 4% * $0%
= $199.5 + $1
= 196.5

Expected return of $196.5 is less than the $210 that the govt promises. We promise to pay a rate of return of 5%

Promised (r) = $210 - $200/$200 = 5%
But XYZ Ltd expected rate of return is =
Expected rate of return = $196.5 - $200/$200 = - 1.75%

For an expected rate of return to be 5% the cash flow would be

= 95% * X + 1% * $100 + 4% * $0 = $210
= 95x = $209
= X = $220

The cash flow for an expected rate of return @5% is $ 220.

The promised rate of return at a cash flow will be =
Promised (r) = $220 - $200 / $200 = 10%

The promised rate will provide us the expected rate of return to be =
Expected Interest Rate – Prob * Rate of Return (1) + Prob * Rate of return (2) + Prob * Rate of Return (3)
= 95% * (+10%) +1% * (-50%) + 1% * (-100%)
= .095 + (- .005) + ( - 0.04)
= 0.05
~ 5%

The difference between the promised rate and the expected rate is the default premium caused due to default risk. If the investor is willing to take a higher risk than he/she receive a higher expected rate of return known as a risk premium.

Credit Ratings and Default Rates

A credit rating is an evaluation of the credit risk of a prospective debtor (an individual, a business, company or a government), predicting their ability to pay back the debt, and an implicit forecast of the likelihood of the debtor defaulting. The two biggest credit rating agencies are Moody’s and Standard & Poor's (S&P).

These ratings are then converted into the default rates. The top rating grades are called investment grade, while the bottom grades are Called speculative grade (or junk grade).s

This credit rating makes a lot of money on behalf of investment banks. In the year 2008, when there was financial crisis it all together falls squarely on the shoulders of the rating agencies, which earned billions providing optimistic ratings for issues explicitly engineered by investment Banks to have high ratings.

Bonds features

Different bonds have different features other than the credit risk. Basically, it can differ in the interest rate which may influence their quoted rates of return. Many corporate bonds allow the issuer to repay the loan early. In this case, if the borrower repays the loan at an early stage than in will be beneficial to the borrower but on the other hand, it will be a loss in the form of interest to the lender.

Credit Default Swaps

A credit default swap is a particular type of swap designed to transfer the credit exposure of fixed income products between two or more parties. In a credit default swap, the buyer of the swap makes payments to the swap’s seller up until the maturity date of a contract. In return, the seller agrees that in the event that the debt issuer defaults or experiences another credit event, the seller will pay the buyer the security’s premium as well as all interest payments that would have been paid between that time and the security’s maturity date.

The Credit Default Swap is also referred to as a credit derivative contract. The credit swaps allow different funds to hold different premiums of a bond. Credit swaps are typically traded in lots of $5 million. This market is over the counter (OTC) – negotiated one to one between two parties.

Uncertainty in Capital Budgeting

In the capital budgeting, the decision situation with reference to risk analysis in capital budgeting decision can be broken into three types,

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  1. Uncertainty
  2. Risk
  3. Certainty

Several types of uncertainties are important to the producer, as he formulates plans and designs course of actions for procuring resources at the present time for a product forthcoming at a future date. The types of uncertainties can be classified as

  1. Price uncertainty
  2. Production uncertainty
  3. Production technology uncertainty
  4. Political uncertainty
  5. Personal and People uncertainty

Present Value with Outcome-Contingent Payoff Tables

Example - XYZ Ltd plans to purchase a building whose future value is uncertain. Next year this investment be either be $60 thousand or $100 thousand. Probability will be (¼ for $60) and (¾ for $100)

Make the payoff table and determine the future value. Assume an expected rate of return is 20% for 1 year.

Solution – Let us assume that the bad outcome is “Rain” and the good outcome is “Sun”

Expected future value of building = Prob * Value Rain + Prob * Value Sun
= ¼* $60 + ¾*$100
= $90

Present value of the building = Future Value/(1+Expected rate )
= $90/(1+20 )
= $75

Alternative method

Present value of the building = Prob * (Value Rain discounted ) + Prob * (Value Sun discounted)
= ¼ * $60 * 1/(1+20%) + ¾ * $100 * 1/(1+20%)
= ¼ * $50 + ¾ * $83.33
= $75

Note that in the NPV formula,

  • Known future cash flows are replaced by expected discounted cash flows, and
  • Known appropriate rates of return are replaced by appropriate expected rates of return.

The State-Contingent Rates of Return

The state-contingent rates of return can also be probability-weighted to arrive at the average (expected) rate of return.

Considering the same example, we have to calculate the rate of return in the two situations and also calculate the expected rate of return.

If it is Sunny
In Sun = $100 - $75/$75 ~ +33%
If it is Rainy
In Rain = $60 - $75/$75 ~ -20%
Therefore the expected rate of return is
= Prob * Rain rate of return + Prob * Sun rate of return
= ¼ * (-20%) + ¾ (+33%)
= 20%

The probability state-weighted rates of return add up to the expected overall rate of return.

Therefore, the proper price of the building today $75 using an expected rate of return @20%.

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Splitting Uncertain Project Payoffs into Debt and Equity

We make the state payoff table to understand cash flow rights. The most important concept in finance is debt and equity. Majorly in the corporate world, all the companies financed their project with both debt and equity.

The debt is basically the amount of money which has to be repaid by the company. A duty or obligation to pay money, deliver goods or render service under an express or implied agreement. A person who owes is called the debtor. Equity is nothing but the ownership in the business. It is the owner capital in the business. The difference between the value of assets and the value of the liabilities. The levered equity is the amount left with the owner after the payment of debt in the business. The debt should never be more than the equity. Basically, the ratio is 2:1.


The amount which we use to take from the market or a person for our own use or for the company growth is called the loan. In the case of a loan the person who has given the amount will get a fixed rate of interest as per the term and condition. In the loan, there are basically two projects in term of above example of purchase of building –

  1. Mortgage Lending: - It is basically the act of lending the amount of money to the buyer of houses and other property.
  2. Residual Building Ownership: - It is basically the levered equity. It is the amount left with the investor after payment of all its debts.

In the case of a loan, the liability of the owner or investor is limited to an extend of the payment of loan amount and interest only.

Example – Taking the same building example let us compute the present value of the mortgage lending.

EventProbabilityValueDiscount Factor

Solution - The creditor receives the property worth $60 if it rains, or the full promised amount (to be determined) if the sun shines.

Lending amount $70 today
Promised payoff
Loan value = Prob * Rain Loan PV + Prob * Sun loan PV.
$70 = ¼ * ($60/(1+20 ) ) + ¾ * ( Promise/(1+20 ))
Promise = ((1+20%)*$70 -1/4*$60)/(3/4)
Promise Payoff = $92
In repayment, Paid by the borrower only if the sun shines.
Promised Rate of return r = ($92-$70/$70)
= + 31.4%
If it rains than the lender rate of return will be –
( ($60-$70)/$70 )
= - 14.3%

Therefore, the lender expected rate of return is

= Prob * Rain rate of return + Prob * Sun rate of return
= ¼ * (-14.3%) + ¾ (+31.4%)
= 20%

The stated rate of return is 31.4% (and it is not an exorbitant rate!), but the expected rate of return is 20%. After all, in our risk-neutral perfect market, anyone investing for one year expects to earn an expected rate of return of 20%.

The Levered Equity

The difference between the debt and equity is called levered equity. After payment of all the debt, the amount which is left is called levered equity.

Let us consider the following example of the building
If the sun shines –
Payoff = Value of the building – promised value
= $100 - $92 = $8 If the rain comes – Payoff = Value of the building – promised value = $60 - $92 = $0

Payoff Table

Event FactorProbValueDiscount

Expected future levered building ownership payoff

= ¼*$0 + ¾*$8
= $6

Present value of levered building ownership

PV = ¼ * ($0/(1+20 ) ) + ¾ * ( $8/(1+20))
= $5
The rate of return –
If Sun – r = ($8-$5)/$5
r = +60%
If Rain – r = ($0-$5)/$5
R = -100%
Expected rate of return of levered equity ownership =
= Prob * Rain rate of return + Prob * Sun rate of return
= ¼ * (-100%) + ¾ (+60%)
= 20%

Payoff table is fundamental tools to help in projects and financial claims settlement. The table shows everything – the state – contingent payoffs, expected payoffs, net present value and expected rate of return.

In the risk neutral world, all that matter is the expected rate of return. So, in the real world which investment is riskiest?

  • Full Ownership
  • Loan Ownership
  • Levered Ownership

Solution – The loan ownership is the least risky, followed by the full ownership and the riskiest one is the levered ownership.

Understanding debt and equity are as important to corporations as it is to building owners. After all, stocks in corporations are basically levered ownership claims that provide money only after the corporation has paid back its liabilities. The building example has given you the skills to compute state-contingent, promised, and expected payoffs, as well as state-contingent, promised, and expected rates of return. These are the necessary tools to work with debt and equity.

Leverage both Financial and Operational Leverage

In finance, leverage is any technique involving the use of borrowed funds in the purchase of assets, with an expectation that the after-tax income from the asset and asset price appreciation will exceed the borrowing cost. Leverage is a general concept rather than an accounting term, you should think of it in even broader terms. The idea of leverage is always that a smaller equity investment can control the firm and is more sensitive to firm value changes.

Basically, leverage is of Two types

  1. Operating leverage
  2. Financial leverage

Operating Leverage

It refers to the percentage of fixed costs that a company has. Operating leverage is the ratio of fixed cost to variable cost

Degree of operating Leverage = Contribution/Earning before interest and tax

Financial Leverage

It refers to the amount of debt in the capital structure of the business firm. Financial leverage refers to how the firm will pay for it or how the operation will be financed.

Degree of Financial Leverage = Earning before interest and tax/Earning before Tax

Risk and Return

In the perfect market under risk aversion, the project can influence one another from an “overall risk”. If the one project return is high than the other project return is always low. It is the basic concept that with the increase in the amount of risk the value of the return increases. There is a direct relationship between the risk and return.

A risk averse investor is an investor who prefers lower returns with known risks rather than higher returns with unknown risks. In other words, among various investments giving the same return with different level of risks, this investor always prefers the alternative with least interest.

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