1. **Problem Statement:** Calculate the PAYG contribution rate, General Average Premium (GAP), and Scaled Premium for a social insurance scheme using the given data. 2. **Formulas and Concepts:** - PAYG contribution rate at year $t$ is given by: $$\text{PAYG rate}_t = \frac{\text{Total pensions paid at } t}{\text{Total insured wages at } t}$$ - General Average Premium (GAP) is the average contribution rate over the years, assuming no initial reserve. - Scaled Premium is calculated to accumulate a reserve equal to the expenditure of the last year by the end of the period. 3. **Step 1: Calculate PAYG contribution rate for year 0:** - Total pensions paid at year 0: $$50 \times 100,000 = 5,000,000$$ - Total insured wages at year 0: $$1280 \times 156,250 = 200,000,000$$ - PAYG rate at year 0: $$\frac{5,000,000}{200,000,000} = 0.025 = 2.5\%$$ 4. **Step 2: Calculate GAP for years 0 to 5:** - For each year $t$, calculate: $$\text{PAYG rate}_t = \frac{\text{Number of pensioners}_t \times \text{Average pension}_t}{\text{Number of contributors}_t \times \text{Average insured wages}_t}$$ - Calculate each year's PAYG rate: - Year 0: $\frac{50 \times 100,000}{1280 \times 156,250} = 0.025$ - Year 1: $\frac{64 \times 125,000}{1280 \times 195,312.5} = 0.032$ - Year 2: $\frac{75 \times 160,000}{1280 \times 234,375} = 0.04$ - Year 3: $\frac{84 \times 187,500}{1280 \times 273,437.5} = 0.045$ - Year 4: $\frac{100 \times 200,000}{1280 \times 312,500} = 0.05$ - Year 5: $\frac{110 \times 225,000}{1280 \times 351,562.5} = 0.055$ - GAP is the average of these rates: $$\text{GAP} = \frac{0.025 + 0.032 + 0.04 + 0.045 + 0.05 + 0.055}{6} = 0.0412 = 4.12\%$$ 5. **Step 3: Calculate Scaled Premium:** - The Scaled Premium $p$ is constant each year and satisfies: $$\sum_{t=0}^5 \frac{1280 \times p \times \text{Average insured wages}_t}{(1+0.04)^t} = \sum_{t=0}^5 \frac{\text{Number of pensioners}_t \times \text{Average pension}_t}{(1+0.04)^t} + \frac{\text{Reserve at end}}{(1+0.04)^6}$$ - Reserve at end is one time the expenditure of year 5: $$110 \times 225,000 = 24,750,000$$ - Calculate present value of expenditures: $$PV_{expenditures} = \sum_{t=0}^5 \frac{\text{Number of pensioners}_t \times \text{Average pension}_t}{(1.04)^t}$$ $$= \frac{5,000,000}{1} + \frac{8,000,000}{1.04} + \frac{12,000,000}{1.082} + \frac{15,750,000}{1.125} + \frac{20,000,000}{1.170} + \frac{24,750,000}{1.217} = 5,000,000 + 7,692,308 + 11,088,235 + 14,000,000 + 17,094,017 + 20,333,333 = 75,207,893$$ - Calculate present value of reserve: $$PV_{reserve} = \frac{24,750,000}{(1.04)^6} = \frac{24,750,000}{1.265} = 19,560,000$$ - Total present value of outflows: $$75,207,893 + 19,560,000 = 94,767,893$$ - Calculate present value of insured wages: $$PV_{wages} = \sum_{t=0}^5 \frac{1280 \times \text{Average insured wages}_t}{(1.04)^t}$$ $$= \frac{200,000,000}{1} + \frac{250,000,000}{1.04} + \frac{300,000,000}{1.082} + \frac{350,000,000}{1.125} + \frac{400,000,000}{1.170} + \frac{450,000,000}{1.217} = 200,000,000 + 240,384,615 + 277,350,000 + 311,111,111 + 341,880,341 + 369,900,000 = 1,740,626,067$$ - Solve for $p$: $$1280 \times p \times PV_{wages} = 94,767,893 \Rightarrow p = \frac{94,767,893}{1,740,626,067} = 0.0544 = 5.44\%$$ **Final answers:** - PAYG contribution rate at year 0: **2.5%** - General Average Premium (GAP) for years 0-5: **4.12%** - Scaled Premium for years 0-5: **5.44%**