# Price of an equity

An equity has a value of 100 Euros, and pay a dividend of 5 Euros in 6 months. The interest rate of 6 months is 5% and the interest rate for 1 year is 6%. I would like to compute the value of the price of this equity after 1 year ?

• You can compute the forward value, but not the equity value, which is a random variable. – Gordon Feb 3 '16 at 21:21
• You can also simulate. – SmallChess Feb 4 '16 at 0:06
• How to compute it ? – Sino Feb 4 '16 at 20:47

Let $T_1=0.5$ be the dividend payment time, and $T=1$. Moreover, let $r_1=5\,\%$ be the annualized interest rate to $T_1$, $r=6\,\%$ be the interest rate to $T$, and $d=5$ be the dividend payment. Then, the forward value, under the risk-neutral measure with the deterministic interest rate assumption, is given by \begin{align*} F &= E(S_T)\\ &=E\big( E(S_T\mid\mathcal{F}_{T_1})\big)\\ &=E\left( e^{rT} E\left(\frac{S_T}{e^{rT}}\mid\mathcal{F}_{T_1}\right)\right)\\ &=E\left(\frac{e^{rT}}{e^{r_1T_1}} S_{T_1} \right)\\ &=E\left(\frac{e^{rT}}{e^{r_1T_1}} (S_{T_1-} -d)\right)\\ &=\frac{e^{rT}}{e^{r_1T_1}} E(S_{T_1-}) - \frac{e^{rT}}{e^{r_1T_1}}\,d\\ &=S_0\,e^{rT} - \frac{e^{rT}}{e^{r_1T_1}}\,d\\ &=101.01. \end{align*}