Question
Question 7
Suppose you are given a choice of the following two securities:
(a) an annuity that pays $10,000 at the end of each of the next 6 years;
or (b) a perpetuity that pays $10,000 forever, but the first cash payment is 11 years from today.
Which security do you choose if the annual interest rate is 5%?
(a) an annuity that pays $10,000 at the end of each of the next 6 years
(b) a perpetuity that pays $10,000 forever, but the first cash payment is 11 years from today
Suppose you are given a choice of the following two securities:
(a) an annuity that pays $10,000 at the end of each of the next 6 years;
or (b) a perpetuity that pays $10,000 forever, but the first cash payment is 11 years from today.
Which security do you choose if the annual interest rate is 5%?
(a) an annuity that pays $10,000 at the end of each of the next 6 years
(b) a perpetuity that pays $10,000 forever, but the first cash payment is 11 years from today
Answers
GPT-4o mini
To determine which security is more valuable, we need to calculate the present value (PV) of both options given an annual interest rate of 5%.
### (a) Present Value of the Annuity
The annuity pays $10,000 at the end of each year for 6 years. The formula for the present value of an annuity is:
\[
PV = C \times \left(1 - (1 + r)^{-n}\right) / r
\]
Where:
- \( C = 10,000 \) (cash flow per period)
- \( r = 0.05 \) (interest rate)
- \( n = 6 \) (number of periods)
Plugging in the values:
\[
PV = 10,000 \times \left(1 - (1 + 0.05)^{-6}\right) / 0.05
\]
Calculating \( (1 + 0.05)^{-6} \):
\[
(1 + 0.05)^{-6} \approx 0.746215
\]
Now substitute back in:
\[
PV = 10,000 \times \left(1 - 0.746215\right) / 0.05
\]
\[
PV = 10,000 \times 0.253785 / 0.05
\]
\[
PV = 10,000 \times 5.0757 \approx 50,757
\]
### (b) Present Value of the Perpetuity
The perpetuity pays $10,000 beginning in year 11. The present value of a perpetuity that pays \( C \) starting after \( n \) years is given by:
\[
PV = \frac{C}{r} \times (1 + r)^{-n}
\]
For our case:
- \( C = 10,000 \)
- \( r = 0.05 \)
- \( n = 10 \) (the payment starts after 10 years)
Calculating the present value of the perpetuity starting from year 11:
\[
PV = \frac{10,000}{0.05} \times (1 + 0.05)^{-10}
\]
Calculating \( (1 + 0.05)^{-10} \):
\[
(1 + 0.05)^{-10} \approx 0.613913
\]
Now substituting back in:
\[
PV = 200,000 \times 0.613913 \approx 122,782.65
\]
### Conclusion
- Present value of the annuity (a): **$50,757**
- Present value of the perpetuity (b): **$122,782.65**
Given these calculations, you should choose option **(b)**, the perpetuity that pays $10,000 forever starting in 11 years, as it has a higher present value.
### (a) Present Value of the Annuity
The annuity pays $10,000 at the end of each year for 6 years. The formula for the present value of an annuity is:
\[
PV = C \times \left(1 - (1 + r)^{-n}\right) / r
\]
Where:
- \( C = 10,000 \) (cash flow per period)
- \( r = 0.05 \) (interest rate)
- \( n = 6 \) (number of periods)
Plugging in the values:
\[
PV = 10,000 \times \left(1 - (1 + 0.05)^{-6}\right) / 0.05
\]
Calculating \( (1 + 0.05)^{-6} \):
\[
(1 + 0.05)^{-6} \approx 0.746215
\]
Now substitute back in:
\[
PV = 10,000 \times \left(1 - 0.746215\right) / 0.05
\]
\[
PV = 10,000 \times 0.253785 / 0.05
\]
\[
PV = 10,000 \times 5.0757 \approx 50,757
\]
### (b) Present Value of the Perpetuity
The perpetuity pays $10,000 beginning in year 11. The present value of a perpetuity that pays \( C \) starting after \( n \) years is given by:
\[
PV = \frac{C}{r} \times (1 + r)^{-n}
\]
For our case:
- \( C = 10,000 \)
- \( r = 0.05 \)
- \( n = 10 \) (the payment starts after 10 years)
Calculating the present value of the perpetuity starting from year 11:
\[
PV = \frac{10,000}{0.05} \times (1 + 0.05)^{-10}
\]
Calculating \( (1 + 0.05)^{-10} \):
\[
(1 + 0.05)^{-10} \approx 0.613913
\]
Now substituting back in:
\[
PV = 200,000 \times 0.613913 \approx 122,782.65
\]
### Conclusion
- Present value of the annuity (a): **$50,757**
- Present value of the perpetuity (b): **$122,782.65**
Given these calculations, you should choose option **(b)**, the perpetuity that pays $10,000 forever starting in 11 years, as it has a higher present value.