Calculate the present value of the following:
(a)  A perpetuity with a face value of $950.34 and a coupon of 4.5% where the
interest rate is 10% and the 1st payment is made at the end of year 1.
(b) A perpetuity with a face value of $950.34 and a coupon of 4.5% where the
interest rate is 10% and the 1st payment is made at the end of year 10.
(c) A 9 year annuity with a face value of $950.34 and a coupon of 4.5% where the
interest rate is 9% and the 1st payment is made at the end of year 1

Let's calculate each part step by step:

### Part (a): Present Value of a Perpetuity with Payments Starting at Year 1

For a perpetuity, the present value (PV) can be calculated using the formula:
\[
PV = \frac{C}{r}
\]
Where:
- $C$ is the annual coupon payment (4.5% of the face value),
- $r$ is the interest rate (10%).

First, we need to calculate the coupon payment:
\[
C = 950.34 \times 0.045 = 42.7653
\]

Now, using the formula for a perpetuity:
\[
PV = \frac{42.7653}{0.10} = 427.653
\]

Thus, the present value for part (a) is **$427.65**.

---

### Part (b): Present Value of a Perpetuity with Payments Starting at Year 10

When payments start at a future date (in this case, year 10), the present value is the present value of a deferred perpetuity.

1. First, calculate the present value at year 9, just before the first payment is made, as a regular perpetuity:
\[
PV_{\text{year 9}} = \frac{C}{r} = \frac{42.7653}{0.10} = 427.653
\]
   
2. Now, discount this value back to today (at year 0), using the present value of a lump sum formula:
\[
PV = PV_{\text{year 9}} \times \frac{1}{(1 + r)^9} = 427.653 \times \frac{1}{(1.10)^9}
\]
\[
PV = 427.653 \times 0.42241 = 180.692
\]

Thus, the present value for part (b) is **$180.69**.

---

### Part (c): Present Value of a 9-Year Annuity with Payments Starting at Year 1

For an annuity, the present value can be calculated using the formula:
\[
PV = C \times \left(1 - \frac{1}{(1 + r)^n}\right) \times \frac{1}{r}
\]
Where:
- $C = 42.7653$ (the annual coupon payment),
- $r = 0.09$ (the interest rate),
- $n = 9$ (the number of years).

First, calculate the factor inside the parentheses:
\[
1 - \frac{1}{(1 + 0.09)^9} = 1 - \frac{1}{(1.09)^9} = 1 - \frac{1}{2.36736} = 1 - 0.42241 = 0.57759
\]

Now, compute the present value:
\[
PV = 42.7653 \times 0.57759 \times \frac{1}{0.09} = 42.7653 \times 6.4176 = 274.547
\]

Thus, the present value for part (c) is **$274.55**.

---

### Summary of Results:
(a) Present Value of the perpetuity (payments starting at year 1): **$427.65**  
(b) Present Value of the perpetuity (payments starting at year 10): **$180.69**  
(c) Present Value of the 9-year annuity (payments starting at year 1): **$274.55**

Would you like more details on any of these calculations?

---

Here are 5 related questions to expand on the concept:
1. How would the present value change if the interest rate increased to 12%?
2. How does the present value of a perpetuity compare to that of an annuity for the same conditions?
3. What happens to the present value if the coupon rate is lower than the interest rate?
4. How is the present value affected by delaying the first payment of a perpetuity even further, say to year 20?
5. How can you calculate the future value of an annuity?

**Tip:** In annuities, the timing of the first payment (immediate or deferred) significantly affects the present value.

Calculate the present value of the following:
(a)  A perpetuity with a face value of $950.34 and a coupon of 4.5% where the interest rate is 10% and the 1st payment is made at the end of year 1.
(b) A perpetuity with a face value of $950.34 and a coupon of 4.5% where the interest rate is 10% and the 1st payment is made at the end of year 10.
(c) A 9 year annuity with a face value of $950.34 and a coupon of 4.5% where the interest rate is 9% and the 1st payment is made at the end of year 1.

This guide explains how to calculate the present value of perpetuities and annuities with different starting points for payments. It uses a face value of $950.34 with a coupon rate of 4.5%, and applies interest rates of 9% and 10% for scenarios where the first payment is deferred to different years. Learn step-by-step calculations using key financial formulas.

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