1. **Problem Statement:** Britam must pay 30M at time 3 and 50M at time 5. The interest rate is constant at 10%. We analyze how to immunize the portfolio against interest rate changes. 2. **Key Concepts:** - Immunization means structuring assets so the portfolio value meets liabilities despite interest rate changes. - Duration measures sensitivity to interest rate changes. - For small changes, matching duration and present value of assets and liabilities immunizes the portfolio. 3. **(i) Portfolio with 3-year and 5-year zero coupon bonds:** - Assets: 30M 3-year zero coupon bond and 50M 5-year zero coupon bond. - Liabilities: 30M at year 3 and 50M at year 5. 4. **(i)(a) Immunize against small changes in interest rate:** - Since assets exactly match liabilities in amount and timing, the portfolio is already immunized against small interest rate changes. - Duration of zero coupon bond equals its maturity, so durations match liabilities. 5. **(i)(b) Immunize against any changes in interest rate:** - Perfect immunization against any changes requires matching all higher moments or using derivatives. - With only zero coupon bonds matching liabilities exactly, the portfolio is fully immunized against any interest rate changes. 6. **(ii) Immunize using a 10-year 10% coupon bond:** - Let nominal amount be $N$. - Coupon $C=0.10N$ paid annually for 10 years. - Present value of bond at rate $r=0.10$ is: $$PV = \sum_{t=1}^{10} \frac{0.10N}{(1.10)^t} + \frac{N}{(1.10)^{10}}$$ - Duration $D$ of bond is: $$D = \frac{\sum_{t=1}^{10} t \cdot \frac{0.10N}{(1.10)^t} + 10 \cdot \frac{N}{(1.10)^{10}}}{PV}$$ - Liabilities present value: $$PV_L = \frac{30}{(1.10)^3} + \frac{50}{(1.10)^5}$$ - Liability duration: $$D_L = \frac{3 \cdot \frac{30}{(1.10)^3} + 5 \cdot \frac{50}{(1.10)^5}}{PV_L}$$ - To immunize, solve: $$N \cdot PV = PV_L$$ $$D = D_L$$ - This system determines $N$. 7. **(iii) Immunize using 2-year and 7-year zero coupon bonds:** - Let nominal amounts be $x$ (2-year) and $y$ (7-year). - Present value of assets: $$PV_A = \frac{x}{(1.10)^2} + \frac{y}{(1.10)^7}$$ - Duration of assets: $$D_A = \frac{2 \cdot \frac{x}{(1.10)^2} + 7 \cdot \frac{y}{(1.10)^7}}{PV_A}$$ - Set equal to liabilities present value and duration: $$PV_A = PV_L$$ $$D_A = D_L$$ - Solve for $x$ and $y$ to immunize against small interest rate changes. **Final answers:** - (i)(a) Portfolio is already immunized against small changes. - (i)(b) Portfolio is immunized against any changes. - (ii) Buy nominal $N$ of 10-year 10% coupon bond satisfying $N \cdot PV = PV_L$ and $D = D_L$. - (iii) Buy $x$ and $y$ of 2-year and 7-year zero coupon bonds solving $PV_A = PV_L$ and $D_A = D_L$.