## Introduction section Examples of calculations

### Examples of calculations

### BTP ITALIA – Calculation examples for the 28 June 2030 BTP Italia (17th issue) in the event of inflation/deflation

#### What would happen if inflation at the end of Semester 1 were 5%?

Assuming inflation at the end of Semester 1 of 5%, the reference daily price index valid for calculation of the first coupon (i.e., on 28 December 2022) would be approximately 114.66.

The Indexation Coefficient (IC) used would thus be equal to IC = 114.66/109.2 = 1.05000.

Considering that the annual coupon rate has been set at 1.6%, with an invested capital of 1.000, the remuneration that the retail investor would receive at the end of Semester 1 would be equal to:

Period | Reference index number | Indexation Coefficient | Coupon (€) | Capital Revaluation (€) | Half-Yearly Remuneration (€) |
---|---|---|---|---|---|

28 June 2022 (Issue date) |
109.2 | 1 | |||

28 December 2022 (End of Semester 1) |
114.66 | 1.05 | 8.40 | 50 | 58.40 |

In detail, the Coupon (C) is calculated according to the following formula:

C = Real Annual Coupon Rate 2 * Nominal Subscribed Capital * IC

namely,

*C* = 1.6% 2 * 1000 * 1.05 = 0.8% * 1050 = 8.40€

The Revalued Capital (RC) is instead calculated as follows:

*RC = Nominal Subscribed Capital* * (*IC* - 1) = 1000 * (1.05 - 1) = 50€

Therefore, the total remuneration of the bond before tax at the end of Semester 1 would be €58.40, which is the sum of the coupon interest (€8.40) and the six-monthly capital appreciation due to inflation (€50).

#### What would happen if, at the end of Semester 2, inflation for the same semester were 2%?

If inflation at the end of Semester 1 were therefore equal to 5% and 2% in the following semester, this would mean that the daily reference price index valid for calculation of the first coupon (i.e., on 28 December 2022) would be 114.66, while the daily reference price valid for calculation of the second coupon (i.e., on 28 June 2023) would be 116.9532.

The Indexation Coefficient (IC) used at the end of Semester 1 would therefore be equal to IC = 114.662/109.2 = 1.05000.

The Indexation Coefficient (IC) used at the end of Semester 2 would instead be equal to IC = 116.9532/114.662 = 1.02000.

Considering that the annual coupon rate has been set at 1.6%, with an invested capital of 1.000, the remuneration the investor would receive at the end of the first two semesters would be equal to:

Period | Reference index number | Indexation Coefficient | Coupon (€) | Capital Revaluation (€) | Half-Yearly Remuneration (€) |
---|---|---|---|---|---|

28 June 2022 (Issue date) |
109.2 | 1 | |||

28 December 2022 (End of Semester 1) |
114.66 | 1.05 | 8.40 | 50 | 58.40 |

28 June 2023 (End of Semester 2) |
116.9532 | 1.02000 | 8.16 | 20 | 28.16 |

Total |
16.56 |
70.00 |
86.56 |

In detail, the annual Coupon (C) would be equal to:

*C* = 1.6% 2 * 1000 * 1.05 + 1.6% 2 * 1000 * 1.02 = 16.56€

The Revalued Capital (RC) at the end of Semester 2 would instead be calculated as follows:

RC = 1000 * (1.05 - 1) + 1000 * (1.02 - 1) = 70€

The total remuneration of the bond before tax at the end of Semester 1 would be €86.56, which is the sum of the coupon interest (€16.56) and the six-monthly capital appreciation due to inflation (€70).

#### What would happen if there were a 2% deflation at the end of Semester 1?

It is useful to remember that deflation occurs when, with reference to a given period, the price index decreases in absolute value, not when it increases less than in previous periods.

Thus, assuming deflation of 2% at the end of Semester 1, this would mean that the daily reference price index valid for calculation of the first coupon (i.e., on 28 December 2022) would have fallen to 107.016.

The Indexation Coefficient (IC) used would be equal to IC = 107.016/109.2 = 0.98. However, being less than 1, the so-called floor mechanism would be applied, bringing the IC back to one. Therefore, the Indexation Coefficient used for the coupon payment would be equal to 1.

Considering that the annual coupon rate has been set at 1.6%, with an invested capital of € 1.000, the gross remuneration that the investor would receive at the end of Semester 1 would be equal to:

Period | Reference index number | Theoretical Indexation Coefficient | Actual Indexation Coefficient | Coupon (€) | Capital Revaluation (€) | Half-Yearly Remuneration (€) |
---|---|---|---|---|---|---|

28 June 2022 (Issue date) |
109.2 | 1 | ||||

28 December 2022 (End of Semester 1) |
107.016 | 0.98 | 1 | 8.00 | 0 | 8.00 |

In detail, the Coupon (C) is equal to:

*C* = 1.6% 2 * 1000 * 1.00 = 0.8% * 1000 = 8.00€

In addition, no capital appreciation is paid. Indeed, the floor mechanism protects the invested capital from deflation, preventing an indexation coefficient below 1 could devalue it.

Therefore, the total remuneration of the bond before tax at the end of the first semester is €8, corresponding to coupon interest only.

#### What would happen if there was a 2% semester deflation at the end of Semester 1 and 3% semester inflation at the end of Semester 2?

Thus, if at the end of Semester 1 there had been a deflation of 2%, this would mean that the daily reference price index at the date of payment of the first coupon (i.e., on 28 December 2022) would have fallen to 107.016.

The Indexation Coefficient used to calculate the gross coupon would therefore be equal to IC = 107.016/109.2 = 0.98. However, being less than 1, the so-called floor mechanism would be applied, bringing the IC back to unity. As mentioned above, the Indexation Coefficient used for the first coupon would be equal to 1.

If, at the end of Semester 2, an inflation rate of 3% were then to occur, the daily reference price index valid for calculation of the second coupon (i.e., on 28 June 2023) would rise to 110.2265.

The Indexation Coefficient used for trading on the secondary market transaction would be higher than 1, as IC = 110.2265/107.016 = 1.03000. However, for the purposes of calculating the coupon and the capital revaluation, the IC would be calculated by taking as a basis the highest index number recorded in the preceding semesters, which in our case is the one applicable on the issue date of the bond (28 June 2022) equal to 109.2. Therefore, the correct value of the Indexation Coefficient in this case would be IC = 110.2265/109.2 = 1.00940.

In other words, this time the investor would only receive back a 1% remuneration for inflation (IC = 1.00940) instead of 3% (IC = 1.03000), since 2% has already been remunerated through the application of the floor (since the principal is not subject to any devaluation in the face of 2% deflation). Otherwise, paying 3% inflation would mean paying twice the inflation share of 2%.

Considering that the annual coupon rate has been set at 1.6%, with an invested capital of 1.000, the gross remuneration the investor would receive at the end of Semester 1 would be equal to:

Period | Reference index number | Theoretical Indexation Coefficient | Actual Indexation Coefficient | Coupon (€) | Capital Revaluation (€) | Half-Yearly Remuneration (€) |
---|---|---|---|---|---|---|

28 June 2022 (Issue date) |
109.2 | 1 | ||||

28 December 2022 (End of Semester 1) |
107.016 | 0.98 | 1 | 8.00 | 0 | 8.00 |

28 June 2023 (End of Semester 2) |
110.2265 | 1.03000 | 1.00940 | 8.08 | 9.4 | 17.48 |

Total |
16.08 |
9.40 |
25.48 |

In detail, the Coupon (C) paid at the end of Semester 1 would be equal to:

*C* = 1.6% 2 * 1000 * 1.00 = 0.8% * 1000 = 8.00€

The Coupon (C) paid at the end of Semester 2 would be equal to:

*C* = 1.6% 2 * 1000 * 1.00940 = 0.8% * 1009.4 = 8.08€

Moreover, while no capital appreciation would be paid at the end of Semester 1, capital appreciation at the end of Semester 2 would be equal to:

*RC* = 1000 * (1.00940 - 1) = 9.40€

Therefore, the total remuneration of the bond before tax at the end of the first two semesters is €25.48, given by the sum of the coupon interest (€16.08) and the capital revaluation due to inflation (€9.40).