A security pays $800 every 8 years forever. The appropriate discount rate discount rate is 6% (EAR). What is the value of the security if the first payment occurs 2 years from now?

To calculate the value of the security that pays \$800 every 8 years forever with a discount rate of 6%, where the first payment is 2 years from now, we will use the **perpetuity formula** adjusted for a delayed first payment and periodic payments.

### Step-by-Step Solution:

1. **Determine the Present Value (PV) of Perpetuity:**

   The present value of a perpetuity with periodic payments is given by:

   \[
   PV = \frac{C}{r}
   \]
   
   where:
   - $C$ = payment per period (\$800)
   - $r$ = discount rate per period

2. **Convert Annual Effective Discount Rate to 8-Year Rate:**

   Since the payments are every 8 years, we need to find the equivalent 8-year discount rate based on the 6% EAR (Effective Annual Rate).

   The formula to convert the annual rate to an 8-year rate is:

   \[
   r_{8} = (1 + r_{annual})^n - 1
   \]
   
   where $r_{annual} = 6\% = 0.06$ and $n = 8$.

   Let's calculate $r_8$.

3. **Adjust for the Payment Delay:**

   The first payment is 2 years from now, but the perpetuity calculation assumes the first payment is at time zero. To adjust for this, we need to discount the present value we calculated to account for the 2-year delay. The discount factor is based on the annual rate of 6%.

   \[
   \text{Adjusted PV} = PV \times \frac{1}{(1 + 0.06)^2}
   \]

Let’s proceed with the calculations.The value of the security, considering the first payment occurs 2 years from now, is approximately **\$1,198.96**.

Would you like further details on any step? Here are 5 related questions to deepen your understanding:

1. How does delaying the first payment affect the present value of a perpetuity?
2. Why do we convert the annual discount rate to an 8-year rate?
3. What happens to the present value if the payments occur more frequently?
4. How would the calculation change if the discount rate were compounded monthly instead of annually?
5. What is the significance of the Effective Annual Rate (EAR) in this problem?

**Tip:** Understanding how to adjust for delayed payments is crucial for dealing with perpetuities and annuities. Always account for the time shift when payments don't start immediately.

A security pays $800 every 8 years forever. The appropriate discount rate is 6% (EAR). What is the value of the security if the first payment occurs 2 years from now?

Discover how to calculate the present value of a security that pays $800 every 8 years forever, with a 6% discount rate, and the first payment occurring 2 years from now. This solution uses the perpetuity formula adjusted for delayed payments and explains key finance concepts like EAR and discount rates.

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