From Theoretical to Real Profitability: Understanding the IRR Formula

▶ Introduction: Why You Need to Know the IRR

When faced with the decision of where to invest our money, especially in debt instruments like bonds, a fundamental question arises: what is my actual profit? The answer is not always obvious by just looking at the coupon announced by the issuer. The Internal Rate of Return, commonly known as IRR, is precisely the tool that solves this enigma, allowing us to see beyond superficial numbers.

▶ The IRR Unpacked: Definition and Real Meaning

The IRR is an interest rate expressed as a percentage that allows us to make objective comparisons between multiple investment options. When deciding whether to allocate our capital to asset X or asset Y, we need a clear and measurable criterion. This is where the IRR comes into play.

In the context of debt securities, the IRR reveals the effective return we will obtain if we hold the bond from today until its maturity date. This return comes from two different sources:

1. Periodic coupons: Represent the interest payments the issuer delivers to us regularly (annually, semiannually, or quarterly). These can be fixed, variable, or even floating when indexed to indicators like inflation. There are also special securities called zero-coupon bonds that do not generate intermediate payments.

2. Gain or loss from convergence to the nominal value: During the bond’s life, its price fluctuates in the secondary market. Depending on whether we acquire it at a lower, equal, or higher price than the nominal, we will realize additional gains or losses at maturity, when the issuer always returns exactly the nominal value plus the last coupon.

▶ How Bonds Work: The Necessary Context

To deeply understand the importance of the IRR, it is useful to examine the operation mechanism of a standard bond (one with a defined term and constant coupon payments).

The basic scheme is: buy at an initial price (which can differ from the nominal), receive the agreed coupons periodically, and at maturity, the issuer reimburses the nominal plus the final coupon.

Let’s take a five-year bond as an example: its price fluctuates constantly during those five years due to changes in market interest rates, modifications in the issuer’s creditworthiness, and other factors. This volatility is precisely what creates opportunities.

The Importance of When We Buy

Here comes the counterintuitive part: it is most advantageous to buy a bond when its market price is low (below €100, assuming €100 as the reference nominal). Why? Simple: on the maturity date, regardless of the purchase price, they will return the full nominal.

If we buy at €94 and at maturity receive €100, we gain €6. If we buy at €107 and at maturity receive €100, we lose €7. This difference is not trivial, and the IRR captures it perfectly.

The Three Purchase Scenarios

• At par: The purchase price exactly matches the nominal (€1,000 for a €1,000 nominal)
• Above par: We pay more than the nominal (€1,086 for a €1,000 nominal)
• Below par: We acquire for less than the nominal (€975 for a €1,000 nominal)

▶ Differentiating Between IRR, TIN, APR, and Technical Interest

It is essential to clarify these different measures, as confusion can lead to incorrect decisions.

The IRR, as already established, is the net return we obtain from a bond considering all its cash flows discounted at the current market price.

The Nominal Interest Rate (TIN) is simply the interest rate agreed upon directly, without considering additional costs. It is the most “pure” form of interest.

The Annual Percentage Rate (TAE) incorporates expenses not initially apparent, such as management fees, insurance, or other costs. For example, a mortgage may have a TIN of 2% but a TAE of 3.26% because it includes the opening fee and other charges. The Bank of Spain promotes the use of the TAE precisely because it allows for homogeneous comparison of financing offers.

Technical Interest is frequently used in insurance products, including inherent costs like life insurance. A savings insurance could offer a technical interest of 1.50% when the purely nominal interest is 0.85%.

▶ Practical Applications: What Is the IRR Really Used For

When we use the IRR in investment analysis, it allows us to evaluate the feasibility of different options and select the most attractive one, whether for higher potential returns or better risk-return ratio.

In the fixed income universe, the IRR helps identify investment opportunities in bonds that truly deserve our attention.

Consider a scenario:

• Bond A: coupon of 8%, but IRR of 3.67%
• Bond B: coupon of 5%, but IRR of 4.22%

If we only looked at the coupon, we would choose the first. However, the IRR shows that the second asset is genuinely more profitable. Why can this happen? Usually because Bond A is traded at a very high price (above par), and that premium paid today is lost at maturity, significantly eroding our final profitability. If we pay €105 for something that will return €100 at maturity, that €5 loss is unavoidable.

▶ The IRR Formula: How to Calculate It

To mathematically obtain the IRR in fixed-income bonds, we use the following structure: where P is the current price, C represents each coupon, n is the number of periods until maturity, and N is the bond’s nominal value.

Practical Example 1: Suppose a bond trades at €94.5. It pays an annual coupon of 6% and matures in 4 years. What will be its IRR?

Applying the formula: IRR = 7.62%

Notice how the IRR (7.62%) exceeds the coupon (6%) thanks to the favorable low purchase price. We are buying at a discount, which amplifies our effective return.

Practical Example 2: Now the same bond, but trading at €107.5:

IRR = 3.93%

Here we see the opposite effect: although the nominal coupon remains 6%, the IRR drops sharply to 3.93% because we pay a premium that penalizes us significantly at maturity.

For those who do not want to perform these calculations manually (which involve solving high-degree equations), there are numerous online calculators that greatly speed up the process.

▶ Variables That Shape the IRR Outcome

Without delving into complex calculations, we can anticipate the IRR’s behavior by knowing certain key factors:

The Coupon: A direct relationship. Higher coupon equals higher IRR; lower coupon, lower IRR.

The Purchase Price: Equally crucial. Buying below par increases IRR; buying above par decreases it.

Special Features of the Instrument: Some bonds have attributes that make them sensitive to external factors. A convertible bond’s IRR may change depending on the evolution of the underlying stock. Inflation-linked bonds will vary their IRR as this economic metric fluctuates.

▶ Final Consideration: IRR Is Not Everything

Although the IRR is an invaluable tool, it should not be the sole decision criterion. The credit quality of the issuer is equally important.

During the Greek crisis of Grexit, Greek sovereign bonds traded with IRRs above 19% for ten years. Does that seem attractive? Theoretically yes, but that extraordinary yield reflected the huge default risk. Only the intervention of the Eurozone prevented Greece from defaulting, which would have meant total losses for Greek bond investors.

Therefore, IRR should always be considered together with a rigorous analysis of the issuer’s solvency and stability. A brilliant IRR formula can hide behind a credit risk bomb.