Solar Panel Investment Comparison
A household is choosing between two solar financing options. Option A: pay $12,000 upfront and receive a government rebate of $3,000 after one year. Option B: pay nothing upfront, then pay $2,400 per year for 6 years, with the first payment due immediately. The household can invest spare cash at 5% per annum compounded annually. An adviser claims Option B is cheaper because the total payments ($14,400) are only slightly more than Option A's net cost ($9,000) and the money is spread out. The household wants a fair comparison in today's dollars and also wants to know the break-even interest rate at which the two options would be equal in present value.
1. Calculate the present value of each option at 5% per annum and determine which is cheaper in today's dollars.
Read the worked answer
Option A: $12,000 paid now, minus $3,000 received after one year. Present value = 12,000 - 3,000/1.05 = 12,000 - 2,857.14 = $9,142.86.
Option B: an annuity due of $2,400 per year for 6 years at 5%. Present value = 2,400 × [1 + (1 - 1.05^-5)/0.05].
Compute (1 - 1.05^-5)/0.05. 1.05^5 ≈ 1.27628, so 1.05^-5 ≈ 0.78353. Then (1 - 0.78353)/0.05 = 0.21647/0.05 = 4.32948.
So PV = 2,400 × (1 + 4.32948) = 2,400 × 5.32948 ≈ $12,790.75.
Option A's present value ($9,142.86) is much lower than Option B's ($12,790.75). Option A is cheaper by about $3,648 in today's dollars. The adviser's reasoning is flawed because it ignores the time value of money.
2. The adviser now argues that if interest rates were high enough, Option B could become cheaper. Test this claim conceptually and state whether any positive interest rate makes Option B cheaper.
Read the worked answer
As the interest rate rises, the present value of future payments falls. Option B's payments are spread over 6 years, so its PV falls faster than Option A's, which is mostly paid immediately.
As r → ∞, Option B's PV approaches the immediate first payment of $2,400 (later payments discounted to near zero). Option A's PV approaches $12,000 (the rebate becomes worthless).
So at very high rates, Option B's PV ($2,400) is below Option A's ($12,000). Therefore, beyond some break-even rate, Option B does become cheaper.
The adviser's claim is directionally correct, but the rate required is far above typical rates. Setting up the equation and solving numerically would show the break-even rate is very high (well over 30%), so at realistic rates Option A remains cheaper. The claim is theoretically true but practically irrelevant.