What is the value today of $500 per month forever, with the first $500 payment occurring four months from today.  Interest rates are 12% p.a. compounded monthly.


Group of answer choices

$2,965.75

$50,000.00

$48,049.02

$48,529.51 

To find the present value (PV) of $500 paid monthly forever (a perpetuity), starting four months from today, with a 12% annual interest rate compounded monthly, we follow these steps:

### Step 1: Adjust the interest rate
Given the interest rate is 12% per annum compounded monthly, the monthly interest rate is:
\[
r = \frac{12\%}{12} = 1\% = 0.01
\]

### Step 2: Perpetuity formula
For a perpetuity that begins after some delay, we first use the basic perpetuity formula:
\[
PV = \frac{C}{r}
\]
Where:
- $C = 500$ is the monthly payment,
- $r = 0.01$ is the monthly interest rate.

Using this formula:
\[
PV_{\text{immediate}} = \frac{500}{0.01} = 50,000
\]
This is the present value **if payments started today**.

### Step 3: Discount for deferred payments
Since the payments start 4 months from today, we need to discount the present value calculated above by 4 months. The discount factor for 4 months at 1% interest per month is:
\[
\text{Discount factor} = \frac{1}{(1 + 0.01)^4} = \frac{1}{1.04060401} \approx 0.9612
\]

### Step 4: Calculate the present value
Now, we apply the discount factor to the perpetuity value:
\[
PV = 50,000 \times 0.9612 \approx 48,060
\]

Since none of the options matches this exactly, the closest is **$48,049.02**.

### Final Answer: 
\[
\boxed{48,049.02}
\]

Do you need further details or clarification on any part of this? Here are some related questions:

1. How would the value change if payments started immediately?
2. What is the difference between perpetuities and annuities?
3. How does compounding frequency affect the present value of cash flows?
4. What if the interest rate was compounded annually instead of monthly?
5. How would the result differ if the interest rate was 10% instead of 12%?

**Tip**: Always adjust the interest rate for the compounding period to match the frequency of payments when calculating present values.

What is the value today of $500 per month forever, with the first $500 payment occurring four months from today? Interest rates are 12% p.a. compounded monthly.

Calculate the present value of a perpetuity with $500 monthly payments starting four months from today, at a 12% annual interest rate compounded monthly. Understand the perpetuity formula and discounting for deferred payments. Ideal for finance students and professionals.

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