VIPSolutionsAn employee's compensation includes an annuity with 1 0 annual payments that pays $ 6 5 , 0 0 0 at retirement, with each subsequent payment growing by 7 % . The firm ’ s policy is to pre - fund such annuities one year before retirement. At an interest rate of 1 0 % , how much would the firm need to invest one year before retirement? What is the sum of the annuity payments? Please round your answer to the nearest dollar. Answer Save your answer What is the net present value of the annuity payments? Please round your answer to the nearest dollar.
Question:
An employee's compensation includes an annuity with 1 0 annual payments that pays $ 6 5 , 0 0 0 at retirement, with each subsequent payment growing by 7 % . The firm ’ s policy is to pre - fund such annuities one year before retirement. At an interest rate of 1 0 % , how much would the firm need to invest one year before retirement? What is the sum of the annuity payments? Please round your answer to the nearest dollar. Answer Save your answer What is the net present value of the annuity payments? Please round your answer to the nearest dollar.
This is AI generated Response.
Asked by: Test User 100150
Created at: 2025-05-14 08:59:46
Asked by: Test User 100150
Created at: 2025-05-14 08:59:46
Final Answer:
Rendered
Raw Text
Did you like this answer?
Let's break down the problem step by step:
**Given:**
- Number of annual payments (\(n\)) = **10**
- First payment (\(P_1\)) = **$65,000** (at retirement)
- Each payment grows by **7%** (\(g = 0.07\))
- Discount rate (\(r\)) = **10%** (\(0.10\))
- The firm pre-funds **one year before retirement**
---
### 1. **Sum of the Annuity Payments**
This is just the sum of a 10-term geometric series:
\[
S = P_1 + P_2 + ... + P_{10}
\]
Where \(P_k = P_1 \cdot (1+g)^{k-1}\)
So,
\[
S = P_1 \cdot \frac{1-(1+g)^{n}}{1-(1+g)}
\]
\[
S = 65,000 \cdot \frac{1-(1.07)^{10}}{1-1.07}
\]
Calculate \(1.07^{10}\):
\[
1.07^{10} = e^{10 \cdot \ln(1.07)} \approx e^{0.677} \approx 1.9672
\]
So,
\[
S = 65,000 \cdot \frac{1-1.9672}{1-1.07}
\]
\[
S = 65,000 \cdot \frac{-0.9672}{-0.07}
\]
\[
S = 65,000 \cdot 13.817
\]
\[
S \approx \$898,105
\]
**Rounded answer:**
\[
\boxed{\$898,105}
\]
---
### 2. **Net Present Value (PV) of the Annuity at Retirement**
This is the present value of a growing annuity:
\[
PV_0 = P_1 \cdot \frac{1 - \left(\frac{1+g}{1+r}\right)^n}{r-g}
\]
Plug in the numbers:
- \(P_1 = 65,000\)
- \(g = 0.07\)
- \(r = 0.10\)
- \(n = 10\)
First,
\[
\frac{1+g}{1+r} = \frac{1.07}{1.10} = 0.9727
\]
\[
0.9727^{10} = e^{10 \cdot \ln(0.9727)} \approx e^{-0.2778} \approx 0.7575
\]
\[
PV_0 = 65,000 \cdot \frac{1 - 0.7575}{0.10 - 0.07}
\]
\[
PV_0 = 65,000 \cdot \frac{0.2425}{0.03}
\]
\[
PV_0 = 65,000 \cdot 8.083
\]
\[
PV_0 \approx \$525,395
\]
**Rounded answer:**
\[
\boxed{\$525,395}
\]
---
### 3. **Amount to Invest One Year Before Retirement**
This is the PV one year before retirement, i.e., the PV *discounted back one year* at 10%:
\[
PV_{-1} = \frac{PV_0}{1.10}
\]
\[
PV_{-1} = \frac{525,395}{1.10}
\]
\[
PV_{-1} \approx \$477,632
\]
**Rounded answer:**
\[
\boxed{\$477,632}
\]
---
## **Summary Table**
| Quantity | Value |
|--------------------------------------------|------------|
| Sum of annuity payments | $898,105 |
| Net present value at retirement | $525,395 |
| Investment 1 year before retirement | $477,632 |
---
Let me know if you need the formulas or steps explained in more detail!
📢 Instructions:
- Ask anything relatd to above question only.
- For new Question goto :