# Investors' required rate of return on bonds explained with formula and example

Investor’s required rate of return is the

“Minimum rate of return (profit) that is required to attract an investor to invest or keep holding the investment (e.g. bonds, shares)."

## Explanation

An investor will only purchase a security if he considers worthwhile (profitable) to do so. How does he realize the investment in given security is worthwhile? The answer is in calculating required rate of return and then comparing this 1) required rate of return with 2) the rate of return offered by the security (e.g. bonds, debentures, common stock).

For example a bond in the market has yield to maturity of 10% whereas the required rate of return of the investor is 11%. He will definitely not purchase the bond or if he is willing to purchase, he will require the bond’s price to be further discounted in order to match his required rate of return.

## How to calculate required rate of return?

Calculation of required rate of return demands considering the individual elements of the return an investor is expecting to receive. Individual elements of return are

• Opportunity cost of risk free investment: investor wants to be compensated for delayed consumption. E.g. if he invests 100 dollars today in the security, he definitely is delaying his ability to consume these 100 dollars today. In other words he can purchase some useful items today but he has decided to delay them by purchasing the security therefore he wanted to be compensated for such delay. This is normal return for delaying consumption which has no risk at all.

• Risk premium: the other thing, an investor wants to be compensated for is assuming the risk associated with the security i.e. the more risky the security the more return investor expects to receive. This is perfectly logical as why one should assume risk when there are no rewards (gains) for doing so.

## Formula

Based on two components of return (reward) i.e. opportunity cost & risk premium (as discussed above), we can write a formula.

 Required rate of return  = Opportunity cost of delayed consumption + Reward of assuming risk associated with the security

 K       = k risk free + k risk premium

 K       = k risk free + β (K market – k risk free )

Where
K = is the required rate of return of the investor.
k risk free = is the market rate of short term government treasury bills.
K market = is the historic rate of return of market on the particular security under consideration.
β = is the measure of relationship of returns on security under consideration and returns of the market.

## Logical explanation of the formula

#### How to calculate k risk free?

k risk free is equal to short term government Treasury bill’s return of rate. Short term T Bills are risk free because there is

• No credit risk (i.e. it is rarely possible that government will default on payments) and
• No interest risk (since these are short term from 1 month to three months the fluctuation of interest will not affect the cash flows associated with these T bills.)

Since rate of return on T bills is readily available in the markets we can easily calculate this part of the formula.

#### How to calculate k risk premium?

k risk premium is reward for assuming risks. Before calculating the risk premium, two facts should be known.

• In practice only market risk associated with securities is relevant. All other risks can be diversified. Therefore the focus should be on market risk of the security.
• Beta is the term used to denote the risk of a particular security with respect to markets overall risk. More about beta is here.

Using this information we can write a formula for risk premium:

 k risk premium      = β (K market + k risk free )