The IRR formula: Your ally for choosing between bond investments

When faced with the decision to invest in different fixed income assets, we need a reliable metric that goes beyond the simple offered coupon. That’s where the Internal Rate of Return, commonly known as IRR, comes into play—a vital tool for assessing the true profitability of our investments.

Why is IRR more relevant than the bond’s coupon?

Many investors make the mistake of focusing solely on the percentage of the coupon promised by a bond, without considering the price at which they acquire it. This highlights the importance of understanding the IRR formula: it allows us to calculate the effective return considering not only periodic payments but also the gain or loss we will realize at maturity.

Let’s imagine two scenarios. Bond A pays an 8% coupon but trades at €105 (above par), while Bond B distributes a 5% coupon but is traded at €95 (below par). At first glance, A might seem superior. However, when we correctly apply the IRR formula, we discover that B offers a higher return because buying below the nominal value generates a capital gain at maturity.

What exactly is IRR?

The Internal Rate of Return is a percentage interest rate that captures the total profitability of a bond. This figure incorporates two fundamental components:

Coupon flows: These are the payments made by the issuer during the life of the bond, whether annual, semiannual, or quarterly. They can be fixed, variable, or floating (linked to inflation indices).

Revaluation or devaluation of the price: If we buy a bond at €94 that will be redeemed at €100 at maturity, we capture that difference as a gain. Conversely, if we pay €106, we assume a guaranteed loss of €6.

The IRR formula in practice

To calculate IRR, we need to know three variables: the current price of the bond (P), the periodic coupon © expressed as a percentage, and the time until maturity (n) in years.

The mathematical expression we use is:

P = C/(1+IRR)¹ + C/(1+IRR)² + … + C/(1+IRR)ⁿ + N/(1+IRR)ⁿ

Where N represents the nominal value (usually €100 or €1,000).

Although this IRR formula may seem complex, there are online calculators that automate the process by simply inputting the basic data.

Practical cases: Understanding the IRR formula with real examples

Case 1: Purchase below par

We have a bond trading at €94.5, with an annual coupon of 6% and a maturity in 4 years. Applying the IRR formula, we obtain: IRR = 7.62%

Note that the IRR (7.62%) significantly exceeds the coupon (6%). This occurs because the discount at purchase increases the final return.

Case 2: Purchase above par

The same bond now trades at €107.5, maintaining the same coupon (6%) and a 4-year term (4 years). When applying the IRR formula, the result is: IRR = 3.93%

Buying above the nominal value results in an implicit loss that drastically reduces the actual return, bringing it almost to half of the promised coupon.

Fundamental differences: IRR vs TIN vs TAE

It’s crucial not to confuse these metrics, as each measures different aspects:

IRR reflects the total return discounted for cash flows and purchase price. It’s the most comprehensive measure for bonds.

TIN (Nominal Interest Rate) is simply the agreed percentage without including additional costs. For example, in mortgages, you might see a 2% TIN.

TAE (Annual Equivalent Rate) includes all additional expenses: commissions, insurance, etc. In the same mortgage example, the TAE could reach 3.26% when these are incorporated.

The Technical Interest is used in insured products and includes the cost of the underlying life insurance.

Factors that modify the IRR outcome

Without complex calculations, we can anticipate movements in IRR considering these elements:

Higher coupon: Increases IRR proportionally.

Purchase below par: Raises IRR by adding a guaranteed gain at maturity.

Purchase above par: Reduces IRR due to an assured loss.

Special features: Convertible bonds, inflation-linked securities, or other structured products may see their IRR affected as their underlying assets evolve.

Applications in investment selection

When analyzing multiple fixed income opportunities, the IRR formula provides an objective comparison criterion. It helps identify which asset will deliver higher returns after considering all relevant factors.

Suppose we compare a corporate bond from Company A with a government bond from Country B. Although the first offers a 9% coupon and the second only 4%, if the IRR shows that B yields 6.5% while A only reaches 4.8%, the decision should lean toward the second, also considering the risk profile.

Final warning: Return vs credit safety

Although the IRR formula is a powerful tool, it should never be the sole decision criterion. History provides sober examples: during the Greek crisis of 2015, Greek 10-year bonds offered an IRR above 19%, an abnormally high figure reflecting imminent default risk.

Only the intervention of the Eurozone prevented Greece from defaulting on its obligations. Investors chasing that extraordinary IRR without analyzing the issuer’s solvency experienced devastating losses.

Therefore, use the IRR formula as a compass to identify opportunities, but always navigate considering the issuer’s credit quality and the macroeconomic circumstances surrounding the investment.