What is the difference in value between an ordinary 10-year annuity of
£200 per year and a perpetuity of £200 per year, if the annual interest rate
is 3%?

To calculate the difference in value between an ordinary 10-year annuity of £200 per year and a perpetuity of £200 per year with an annual interest rate of 3%, we use the following formulas:

### 1. **Present Value (PV) of an Ordinary Annuity**
The present value of an ordinary annuity can be calculated using the formula:

\[
PV_{\text{annuity}} = P \times \left(1 - (1 + r)^{-n}\right) \div r
\]

Where:
- \(P\) = annual payment (£200)
- \(r\) = interest rate (0.03 or 3%)
- \(n\) = number of years (10 years)

\[
PV_{\text{annuity}} = 200 \times \left(1 - (1 + 0.03)^{-10}\right) \div 0.03
\]

\[
PV_{\text{annuity}} = 200 \times \left(1 - (1.03)^{-10}\right) \div 0.03
\]

\[
PV_{\text{annuity}} = 200 \times \left(1 - 0.74409\right) \div 0.03
\]

\[
PV_{\text{annuity}} = 200 \times 0.25591 \div 0.03
\]

\[
PV_{\text{annuity}} = 200 \times 8.5303
\]

\[
PV_{\text{annuity}} = 1706.06
\]

### 2. **Present Value of a Perpetuity**
The present value of a perpetuity can be calculated using the formula:

\[
PV_{\text{perpetuity}} = \frac{P}{r}
\]

\[
PV_{\text{perpetuity}} = \frac{200}{0.03}
\]

\[
PV_{\text{perpetuity}} = 6666.67
\]

### 3. **Difference in Value**
Now, subtract the present value of the annuity from the present value of the perpetuity to find the difference in value:

\[
\text{Difference} = PV_{\text{perpetuity}} - PV_{\text{annuity}}
\]

\[
\text{Difference} = 6666.67 - 1706.06
\]

\[
\text{Difference} = 4960.61
\]

### Conclusion:
The difference in value between the perpetuity and the 10-year annuity is **£4960.61**.

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Would you like further clarification on this, or do you have any questions?

Here are 5 related questions to explore:
1. How would the difference change if the interest rate was 5% instead of 3%?
2. What is the future value of the same 10-year annuity?
3. How does the present value of an annuity due differ from an ordinary annuity?
4. Can you calculate the value of a growing perpetuity with a growth rate of 1%?
5. How do you calculate the present value of an annuity that pays quarterly?

**Tip:** The present value of a perpetuity is highly sensitive to the interest rate, as small changes in the rate can significantly affect its value.

What is the difference in value between an ordinary 10-year annuity of £200 per year and a perpetuity of £200 per year, if the annual interest rate is 3%?

Explore the difference in value between a 10-year annuity of £200 per year and a perpetuity of £200 per year at a 3% interest rate. This step-by-step solution covers the present value of annuities and perpetuities, and the impact of interest rates on financial instruments. Ideal for college-level finance and math students.

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