Solar Panel Investment: Financial Mathematics
A family is considering installing solar panels. The upfront cost is $8,500. The panels are expected to reduce their electricity bill by $1,800 per year. However, the electricity price is expected to increase by 3% per year, so the savings will also increase by 3% annually. The family can invest their money elsewhere at a fixed interest rate of 4% per year compounded annually. They plan to stay in the house for 10 years. After 10 years, the panels will have a salvage value of $1,000. The family wants to know if the solar panels are a good investment compared to investing the $8,500 at 4% compounded annually. Ignore tax and maintenance costs.
1. Calculate the total savings from the solar panels over 10 years, assuming the first year's savings is $1,800 and it increases by 3% each subsequent year. Then calculate the future value of the $8,500 if invested at 4% compounded annually for 10 years. Compare the two amounts (including the salvage value) and advise whether the solar panels are a better investment.
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Total savings from panels: This is a geometric series with first term a = 1800, ratio r = 1.03, n = 10. Sum = a(r^n - 1)/(r - 1) = 1800(1.03^10 - 1)/0.03. 1.03^10 ≈ 1.343916. So sum ≈ 1800(0.343916)/0.03 = 1800 × 11.46387 ≈ $20,635. (More precisely: 1800 × 11.463879 = $20,635.0). Add salvage value: $20,635 + $1,000 = $21,635. Future value of investment: FV = 8500(1.04)^10. 1.04^10 ≈ 1.480244. FV ≈ 8500 × 1.480244 = $12,582.07. Comparison: $21,635 > $12,582, so the solar panels yield more. However, note that the savings are spread over time and not reinvested; a more rigorous comparison would discount the savings at 4%. But based on simple future value, the panels are better. Advise that the panels are a better investment if the assumptions hold.
2. Calculate the payback period (in years and months) for the solar panels, assuming the annual savings increase by 3% each year. Use the first year savings of $1,800.
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We need cumulative savings to reach $8,500. Year 1: $1,800 (cumulative $1,800) Year 2: 1800×1.03 = $1,854 (cum $3,654) Year 3: 1854×1.03 = $1,909.62 (cum $5,563.62) Year 4: 1909.62×1.03 = $1,966.91 (cum $7,530.53) Year 5: 1966.91×1.03 = $2,025.92 (cum $9,556.45) Payback occurs during year 5. Amount needed at start of year 5: 8500 - 7530.53 = $969.47. Savings in year 5 = $2,025.92. Fraction of year = 969.47 / 2025.92 ≈ 0.4785 years. 0.4785 × 12 ≈ 5.74 months, so about 5.7 months. Thus payback period ≈ 4 years and 5.7 months, or about 4 years 6 months.