# Risk mapping a zero-coupon bond portfolio

I'm trying to understand example 2.6 taken from McNeil and Embrechts "Quantitative Risk Management". The example consists of obtaining the risk mapping of a portfolio of $$d$$ zero-coupon bonds.

• The zero-coupon bond price at time $$s$$ and with maturity $$T$$ is $$p(s, T)= e^{-(T-s)y(s, T)}$$ where $$y(s, T)$$ is the yield curve

• The loss $$L_{t+1}$$ over a time horizon $$\Delta$$ is given by the change in portfolio value given by: $$L_{t+1}=-\left( V_{t+1}-V_t \right)$$

• The portfolio value is $$V_t = \sum_{i=1}^d \lambda_i \ p(t\Delta, T_i)$$, where $$\lambda_i$$ is the number of shares of the $$i$$th bond. Replacing for the bond price we have: $$V_t = \sum_{i=1}^d \lambda_i e^{-(T_i-t\Delta)y(t\Delta, T_i)}$$

My attempted solution:

To find the loss distribution, I take $$V_t$$ and write it in terms of the risk factor $$Z_t=y(t\Delta,\ T_i)$$, so basically: $$$$L_{t+1}=-(V_{t+1}-V_t) = -\sum_{i=1}^d \lambda_i \left[e^{-(T_i-t\Delta) Z_t} -e^{-(T_i-(t+1)\Delta)(Z_t+X_{t+1})}\right] \ \ \ \ \ \ \ \ \ \ \text{(1)}$$$$ where $$X_{t+1}=Z_{t+1}-Z_t$$ is the change of the risk factors.

Factoring $$e^{-(T_i-t\Delta) Z_t} = p(t\Delta, T_i)$$ from equation (1) gives me: $$$$-\sum_{i=1}^d\lambda_i \ p(t\Delta, T_i) \left[ 1-e^{-(T_i-t\Delta) X_{t+1}} e^{\Delta (Z_t +X_{t+1})} \right] \ \ \ \ \ \ \ \ \ \ \text{(2)}$$$$

My question:

This is where I get stuck. The textbook says that taking derivatives of the loss function (what I suppose is equation (2) unless I've made a mistake) and using the first order approximation of the loss function: $$L_{t+1}^{\Delta} = \left[ f_t(t, Z_t) + \sum_{i=1}^d f_{z_i}(t, Z_t)X_{t+1, i} \right]$$

Where $$f(t, Z_t)$$ is the portfolio value function. They get the following answer:

$$L_{t+1}^{\Delta} = -\sum_{i=1}^d \lambda_i \ p(t\Delta, T_i) \left[ y(t\Delta, T_i)\Delta - (T_i-t\Delta) X_{t+1, i} \right]$$

From the loss function that I have found, I haven't been able to get the answer. I'd be really grateful for any insight, explanation, or pointing to any mistake I've made. Thanks for reading this!