You are considering selling your 1⁄2 of the business to your business partner who promised to pay you $200,000 every year for 10 years. You also have an option to sell your part for a cash price of $1.8M to another buyer. You would much rather sell it to your partner, so they offered to change annual payments to match this other offer. Calculate the new annual payment value if the interest rate is 5%.

Question

BlackTom AI · Accepted Answer

Question restatement: You own 1/2 of a business and your partner has promised to pay you $200,000 every year for 10 years. There is also a cash offer of $1.8M for your 1/2 stake. To align with the cash offer, you want to adjust the annual payments so that the present value at 5% of 10 equal annual payments matches $1.8M. Now let's evaluate each answer choice.

Option A: $249,999
- To check whether this could be the adjusted annual payment, calculate the present value of an ordinary annuity of 10 payments of $249,999 at 5%. The PV factor for 10 years at 5% is (1 − (1+0.05)^(−10)) / 0.05 ≈ 7.7217. Multiplying gives PV ≈ 249,999 × 7.7217 ≈ 1,930,400, which is higher than $1.8M. This suggests the payment is too large to equate to the $1.8M cash offer.

Option B: $233,108
- Using the same PV calculation with P = 233,108: PV ≈ 233,108 × 7.7217 ≈ 1,800,000 (approximately). This aligns with the cash price of $1.8M, meaning 10 yearly payments of around $233k would have a present value equal to the cash offer at a 5% discount rate.

Option C: $249,118
- Similar to Option A, this amount would yield a PV ≈ 249,118 × 7.7217 ≈ 1,925,000, which is clearly above $1.8M. The pattern shows again this is too high to match the cash offer.

Option D: $250.004
- With a payment of 250.004, the PV ≈ 250.004 × 7.7217 ≈ 1,930,420, far above $1.8M. This is not consistent with the cash offer either.

Key concept used: The present value of an ordinary annuity with n payments is PV = P × [(1 − (1 + i)^(−n)) / i], where P is the annual payment, i is the interest rate per period, and n is the number of periods. Here, i = 0.05 and n = 10. The goal is to set PV = $1.8M and solve for P. The computed factor for the annuity is roughly 7.7217, so P ≈ 1,800,000 / 7.7217 ≈ 233,108.

Summary of reasoning: Options that produce a present value higher than $1.8M (Options A, C, and D) imply payments that are too large to equate to the $1.8M cash offer. Only the payment around $233,108 (Option B) yields a present value that matches the cash price when discounted at 5% over 10 years. The calculations rely on the standard annuity PV formula and the 5% discount rate.

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