1. **Problem Statement:** We are given cash flows for a project and asked to calculate the Net Present Value (NPV) ignoring risk and considering risk using certainty equivalent coefficients. Also, we need to state two advantages of certainty. 2. **Formulas and Concepts:** - NPV ignoring risk is calculated as $$\text{NPV} = \sum_{t=0}^n \frac{CF_t}{(1+r)^t}$$ where $CF_t$ is cash flow at year $t$ and $r$ is the discount rate. - NPV with risk uses certainty equivalent cash flows: $$\text{NPV} = \sum_{t=0}^n \frac{CE_t}{(1+r)^t}$$ where $CE_t = CF_t \times \alpha_t$ and $\alpha_t$ is the certainty equivalent coefficient. - Risk discount rate is given as 10% or 0.10. 3. **Given Data:** | Year | Cash Flow (000) | $\alpha$ | |-------|-----------------|----------| | 0 | 30000 | 1.00 | | 1 | 12000 | 0.90 | | 2 | 14000 | 0.70 | | 3 | 10000 | 0.50 | | 4 | 6000 | 0.30 | 4. **Step i) NPV ignoring risk:** Calculate present value of each cash flow: - Year 0: $$\frac{30000}{(1+0.10)^0} = 30000$$ - Year 1: $$\frac{12000}{(1.10)^1} = \frac{12000}{1.10} = 10909.09$$ - Year 2: $$\frac{14000}{(1.10)^2} = \frac{14000}{1.21} = 11570.25$$ - Year 3: $$\frac{10000}{(1.10)^3} = \frac{10000}{1.331} = 7513.15$$ - Year 4: $$\frac{6000}{(1.10)^4} = \frac{6000}{1.4641} = 4097.17$$ Sum all present values: $$NPV = 30000 + 10909.09 + 11570.25 + 7513.15 + 4097.17 = 64089.66$$ 5. **Step ii) NPV with risk:** Calculate certainty equivalent cash flows: - Year 0: $$30000 \times 1.00 = 30000$$ - Year 1: $$12000 \times 0.90 = 10800$$ - Year 2: $$14000 \times 0.70 = 9800$$ - Year 3: $$10000 \times 0.50 = 5000$$ - Year 4: $$6000 \times 0.30 = 1800$$ Calculate present value of certainty equivalent cash flows: - Year 0: $$\frac{30000}{1.10^0} = 30000$$ - Year 1: $$\frac{10800}{1.10} = 9818.18$$ - Year 2: $$\frac{9800}{1.21} = 8090.91$$ - Year 3: $$\frac{5000}{1.331} = 3755.64$$ - Year 4: $$\frac{1800}{1.4641} = 1229.93$$ Sum all present values: $$NPV = 30000 + 9818.18 + 8090.91 + 3755.64 + 1229.93 = 52894.66$$ **Comment:** The NPV considering risk is lower than ignoring risk, reflecting the riskiness of future cash flows. 6. **Step iii) Two advantages of certainty equivalent approach:** - It separates risk adjustment from discounting, making risk assessment clearer. - It allows use of a single discount rate, simplifying calculations while incorporating risk.