# 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. – HelloWorld Feb 4 '16 at 0:06
• How to compute it ? – Sino Feb 4 '16 at 20:47

## 2 Answers

We consider the forward value, which can be employed to estimate the equity value.

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*}

The relationship between interest rates and equity prices being at best unstable and weak, I'll assume that the level of interest rate is irrelevant here. So the answer to your question (price of the equity in a year) is 95, everything else being equal. Of course it's unlikely that the equity will actually price at 95 in a year due to market movements, but that's a different story.

If you ask for the forward value of the equity, you need to discount that future value with the relevant interest rates.