1. **State the problem:** We want to find the size of a six-year annual ordinary annuity that, combined with investments of 2000000 at year 2, 4000000 at year 3, and 3000000 at year 5, grows to a sum sufficient to pay 1000000 at the end of year 8 and 2000000 perpetually starting from year 10. The interest rate is 12.5% per year. 2. **Identify the variables and formulas:** - Interest rate $r = 0.125$ - Let the annuity payment be $A$ - The annuity lasts 6 years, payments at the end of each year from year 1 to year 6. - Future value (FV) of annuity at year 6 is $$FV_{annuity} = A \times \frac{(1+r)^6 - 1}{r}$$ - Future value of lump sums at year 6: - 2000000 invested at year 2 grows for 4 years: $$2000000 \times (1+r)^4$$ - 4000000 invested at year 3 grows for 3 years: $$4000000 \times (1+r)^3$$ - 3000000 invested at year 5 grows for 1 year: $$3000000 \times (1+r)^1$$ 3. **Calculate the present value (PV) at year 8 of the payments to be made:** - Payment of 1000000 at year 8 is a single payment. - Payment of 2000000 perpetually starting at year 10 means a perpetuity starting at year 10. - PV at year 8 of perpetuity starting year 10 is the PV of payments starting 2 years later: $$PV_{perpetuity,8} = \frac{2000000}{r} \times \frac{1}{(1+r)^2}$$ 4. **Calculate total amount needed at year 8:** $$Total_{8} = 1000000 + PV_{perpetuity,8} = 1000000 + \frac{2000000}{0.125} \times \frac{1}{(1.125)^2}$$ 5. **Calculate the amount at year 6 needed to grow to $Total_8$ at year 8:** $$Amount_{6} = \frac{Total_8}{(1+r)^2}$$ 6. **Sum of annuity FV and lump sums FV at year 6 must equal $Amount_6$:** $$A \times \frac{(1+r)^6 - 1}{r} + 2000000 \times (1+r)^4 + 4000000 \times (1+r)^3 + 3000000 \times (1+r)^1 = Amount_6$$ 7. **Plug in $r=0.125$ and calculate each term:** - $(1+r)^6 = 1.125^6 \approx 2.0122$ - $(1+r)^4 = 1.125^4 \approx 1.6010$ - $(1+r)^3 = 1.125^3 \approx 1.4238$ - $(1+r)^1 = 1.125$ - $(1+r)^2 = 1.125^2 = 1.2656$ Calculate $Total_8$: $$Total_8 = 1000000 + \frac{2000000}{0.125} \times \frac{1}{1.2656} = 1000000 + 16000000 \times 0.7894 = 1000000 + 12630320 = 13630320$$ Calculate $Amount_6$: $$Amount_6 = \frac{13630320}{1.2656} = 10770000$$ Calculate lump sums FV at year 6: $$2000000 \times 1.6010 = 3202000$$ $$4000000 \times 1.4238 = 5695200$$ $$3000000 \times 1.125 = 3375000$$ Sum lump sums = 3202000 + 5695200 + 3375000 = 12272200$$ 8. **Set up equation for $A$:** $$A \times \frac{2.0122 - 1}{0.125} + 12272200 = 10770000$$ Simplify fraction: $$\frac{2.0122 - 1}{0.125} = \frac{1.0122}{0.125} = 8.0976$$ Equation: $$8.0976 A + 12272200 = 10770000$$ 9. **Solve for $A$:** $$8.0976 A = 10770000 - 12272200 = -1502200$$ $$A = \frac{-1502200}{8.0976} = -185600$$ 10. **Interpretation:** Negative annuity payment means the problem setup or assumptions might need review, but mathematically the annuity payment is approximately -185600 (which could mean withdrawals rather than payments).