Formulas ALM

Definition of yield

\[P(t) = e^{-yt};\;\;y(t) = -\frac{1}{t}ln\left(P(t)\right)\]

Forward rates \[y_F(t) = y(t) + t\cdot y'(t)\]

Annual compounding \[i = e^y-1\]

Average bond yield \[B = B(\bar{y})\overset{!}{=}\sum_{i=1}^ne^{-\bar y t_i c(t_i)}\]

Present Value sensitivity

Present value of the cash flow

\[B(y)=\int_0^\infty e^{-y(t)t}dC(t)\] Derivatives of the PV of the cash flow \[B'(y) = -\int_0^\infty te^{-y(t)t}dC(t),\;B''(y) = -\int_0^\infty t^2e^{-y(t)t}dC(t)\] Taylor approximation of the change in PV due to a shift in the yield curve \[B(y+\Delta\bar y) - B(y) \approx B'(y)\Delta\bar y + \frac{1}{2}B''(y)(\Delta\bar y)^2\] Duration and convexity \[D = D(y) = -B'(y)/B(y)\] \[C = C(y) = B''(y)/B(y)\]

Dollar duration and dollar convexity of the surplus: \[\begin{matrix} DD & = PV_A\cdot D_A - PV_L\cdot D_L \\ DC & = PV_A\cdot C_A - PV_L\cdot C_L \end{matrix} \]

First order matching \[D_A = D_L\] Second order matching \[PV_A\cdot C_A\geq PV_L\cdot C_L\]

Term structure equilibrium models

Vasicek model \[dr(t) = a (b - r(t))dt + \sigma dz(t)\] Cox-Ingersoll-Ross model \[dr(t) = a(b - r(t))dt + \sigma \sqrt{r(t)}dz(t)\]

Affine term structure

\[P(t,T) = A(t,T) e^{-B(t,T)r(t)}\]

Vasicek model \[B(t,T) = \frac{1-e^{-a(t-T)}}{a}\]

\[A(t,T) = e^{\frac{(B(t,T)-(T-t))(a^2b-\frac{1}{2}\sigma^2)}{a^2}}\cdot e^{-\frac{\sigma^2B^2(t,T)}{4a}}\] CIR model

\[A(t,T) = \left(\frac{2\gamma e^{(\gamma+a)(T-t)/2)}}{(\gamma+a)(e^{\gamma(T-t)}-1)+2\gamma}\right)^{2ab/\sigma^2}\]

\[B(t,T) = \frac{2(e^{(\gamma (T-t)} - 1)}{(\gamma + a) (e^{\gamma (T-t)} - 1 ) + 2\gamma}\]

\[\gamma = \sqrt{a^2+2\sigma^2}\]

Mean Variance Analysis

Minimum variance Portfolio\[\underset{\textbf w}{\min}\mathbf w'\Sigma\mathbf w,\;\mathrm{subject\;to\;(only)}\;\mathbf w'\mathbf 1 = 1\]

Optimal Portfolio of Risky Assets \[\underset{\textbf w}{\min}\mathbf w'\Sigma\mathbf w,\;\mathrm{subject\;to\;\mathbf w'\mu=r\;and\;(of\; course)}\;\mathbf w'\mathbf 1 = 1\]

Optimal portfolio with a risk-free asset \[\underset{w_0,\mathbf w}{\min}\mathbf w'\Sigma\mathbf w,\;\textrm{ subject to }\;w_0\mu_0+\mathbf w'\mathbf\mu = r\;\textrm{ and }\;w_0 + \mathbf w'\mathbf 1 = 1\]