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The image contains mathematical expressions related to the present value \( V(0) \) of an annuity based on compound interest. The expressions detail how to calculate the present value at different points in time for an annuity deferred by 4 years, with constant payments \( R \) and interest rate \( i \).

Let's break down each formula provided in the image:

1. **Expression (0)**: 
   \[
   V(0) = R \cdot \frac{1 - (1+i)^{-4}}{i}
   \]
   This is the standard present value formula for an annuity with constant payments \( R \), deferred by 4 years. The term \(\frac{1 - (1+i)^{-4}}{i}\) represents the present value factor of 4 payments, discounted back to the present.

2. **Expression (1)**: 
   \[
   V(0) = R \cdot \frac{1 - (1+i)^{-4}}{i} \cdot (1+i)^{-4}
   \]
   This represents the present value, further discounted by an additional \(4\) years (which could correspond to the start of payments being delayed by 4 years).

3. **Expression (2)** (crossed out): 
   This appears to be incorrect based on the notation and was likely canceled out because of a mistake in the discounting factor, where \( (1+i)^{-3} \) was used incorrectly instead of \( (1+i)^{-4} \).

4. **Expression (3)**: 
   \[
   V(0) = R \cdot \frac{1 - (1+i)^{-4}}{i} \cdot (1+i)^4
   \]
   This formula represents the present value discounted by the interest factor \( (1+i)^4 \), which reflects an increase rather than a delay, indicating possibly projecting the annuity forward by 4 years.

These formulas are variations of the present value of an annuity, adjusted for when the payments begin (either deferred or anticipated).

Would you like further explanation on these formulas? Here are 5 related questions to expand your understanding:

1. What is the difference between immediate and deferred annuities in terms of present value?
2. How does the interest rate \( i \) affect the present value of an annuity?
3. Can you explain the role of the discounting factor \( (1+i)^{-n} \) in annuity calculations?
4. How would you modify these formulas for an annuity with increasing payments instead of constant ones?
5. How can these present value formulas be applied in real-world financial scenarios, such as loan amortization?

**Tip**: The term \( (1+i)^{-n} \) in financial mathematics is used to discount future cash flows back to their present value. The larger \( n \) is

In base al tasso annuo di interesse composto i, il valore attuale V(0) di una rendita annua differita di 4 anni, anticipata e costituita da 4 rate costanti di importo R.

This page explains the present value of a deferred annuity under compound interest. It covers the key formulas for annuities with constant payments, deferred by 4 years, and how to apply discounting for present and future value calculations.

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