(1) Internal Rate of Return (IRR)

Definition

IRR is a financial metric used to assess the profitability of an investment project – it is an expected rate of return that makes the Net Present Value (NPV) of the investment equal to zero (just not making a profit or loss).

Pros and Cons

Pros

1. Investment Decision Making: IRR can be used as a decision-making tool to determine whether a project is financially viable. If the IRR of a project exceeds the required rate of return, the project is considered a good investment, and vice versa.

2. Project Comparison: IRR can be used to compare the profitability of multiple projects or investments. The one with the highest IRR would typically be considered the best.

3. Capital Budgeting: IRR is widely used in capital budgeting. It helps organizations to identify the maximum possible return on an investment or the most profitable allocation of resources.

4. Cash Flow Analysis: IRR takes into account the time value of money and provides a more accurate picture of a project’s potential profitability by considering both the magnitude and the timing of expected cash flows.

5. Risk Assessment: While IRR itself doesn’t measure risk, a higher IRR often implies a shorter payback period, reducing the time the investment is at risk. It’s important to note that this isn’t always the case, as some projects might have higher IRRs due to higher risks involved.

Cons

There are some limitations in its application.

1. Ignoring the Scale of Investment: IRR mainly considers the rate of return on investment, not the actual scale of funds. For instance, if two projects have the same IRR, but one project’s investment is 10 times the other’s, there would be a significant difference in actual returns.

2. Ignoring the Project Duration: IRR also does not consider the duration of the project. If there are two projects with the same IRR, but one project has a shorter duration than the other, then the shorter-term project would have a faster cash flow recovery rate and lower risk.

3. Unrealistic Reinvestment Assumption: The calculation of IRR assumes that all cash flows can be reinvested at the IRR rate. However, in reality, it may not be possible to guarantee that all cash flows can be reinvested at such a rate.

4. Existence of Multiple IRRs: In certain cash flow scenarios (such as alternating positive and negative cash flows), there may be multiple IRRs (the equation has multiple solutions), making it complex and challenging to use IRR as a basis for decision-making.

5. Calculation Difficulty: IRR does not have an explicit analytical solution, and usually needs to be solved through numerical methods (such as the bisection method or Newton’s method), which could be quite complex computationally.

Therefore, when using IRR as an investment decision-making tool, these limitations need to be considered, and other evaluation tools such as Net Present Value (NPV) might need to be used in combination, to fully assess the risks and returns of the investment.

Application

Type of Bond	Applicability of IRR (Internal Rate of Return)	Situation or Reason for Applying IRR
Zero-Coupon Bond	Yes	There are no periodic interest payments in a zero-coupon bond, therefore the IRR is equivalent to its yield to maturity.
Fixed-Rate Bond	Yes	Fixed-rate bonds have periodic interest payments, the IRR considers all cash flows throughout the lifecycle of the bond (interest and principal).
Floating Rate Bond	Yes	Although the interest payments of floating-rate bonds may change, the IRR can still be calculated, albeit with some assumptions about future rates.
Hybrid Bonds (A mix of Zero and Fixed Coupon)	Yes	The IRR is also applicable to hybrid bonds, considering all cash flows throughout the bond’s lifecycle.

Formula derivation

This formula of IRR is derived based on the time value of money associated with the bond.

Please refer to this blog for background knowledge on the time value of money –
link # The Time Value of Money (TVM)

The IRR is the discount rate that makes the following equation hold:

$$
NPV = \frac{C_0}{(1+IRR)^0} + \frac{C_1}{(1+IRR)^1} + \frac{C_2}{(1+IRR)^2} + … + \frac{C_T}{(1+IRR)^T}
$$

Where:

• $C_t$ represents the cash flow (income or expenditure) in the $t$th period. $C_0$ is usually the initial investment expenditure (negative value), and $C_1,…,C_T$ are usually future income (positive value).
• $T$ is the total number of investment periods.
• $IRR$ is the Internal Rate of Return we are looking for.

The formula can be simplified to

$$
NPV = \sum_{t=0}^{T} \frac{C_t}{(1+IRR)^t}
$$

Approximate solution of the equation

Suppose we have the following cash flows:

• In the 0th year, we invest 100 units.
• In the 1st year, we receive 60 units.
• In the 2nd year, we receive 60 units.

We need to find an IRR that makes NPV equal to 0. That is, we need to find an IRR that satisfies the following equation:

$$
NPV = \frac{-100}{(1+IRR)^0} + \frac{60}{(1+IRR)^1} + \frac{60}{(1+IRR)^2} = 0
$$

The solution to equations usually involves methods such as the Bisection Method and Newton’s Method.

Please refer to this chapter for details on the solution methods – link # Methods of Equation Solving: Bisection and Newton’s Approach

YTM (Yield to Maturity) vs IRR (Internal Rate of Return)

Similarities

	Yield to Maturity (YTM)	Internal Rate of Return (IRR)
Discounted Cash Flows	The sum of discounted future cash flows equals the current investment or price	The sum of discounted future cash flows equals the current investment or price
Investment Assessment	Used to evaluate the effectiveness of an investment	Used to evaluate the effectiveness of an investment
Interest Rate Calculation	Requires a numerical method to solve an equation involving an unknown interest rate	Requires a numerical method to solve an equation involving an unknown interest rate
Investment Assumptions	Assumes that bond’s interest and principal will be continually paid until the bond matures	Assumes that future cash flows are determined based on the forecast of the investment project

Differences

	Yield to Maturity (YTM)	Internal Rate of Return (IRR)
Field of Application	Mainly used for investment analysis of bonds and other fixed income securities	Commonly used in areas such as capital budgeting and project investment
Certainty of Cash Flows	Future cash flows (such as interest and principal repayments) are generally known and fixed	Future cash flows (such as project income and costs) may be projected and have a certain degree of uncertainty
Regularity of Cash Flows	Typically considers equal, periodic cash flows	May need to handle unequal, non-periodic cash flows
Reinvestment Assumption	Assumes all interest income will be reinvested at the YTM rate	Assumes all positive cash flows will be reinvested at the IRR rate