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Equity Forward Value with an Interim Dividend and Term Rates

Article Quant Q&A · Author: Sino
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Summary

The document considers the one-year forward value of an equity that pays a known dividend after six months, with different interest rates for the six-month and one-year terms. Under deterministic rates and risk-neutral valuation, the answer carries the initial share value forward and subtracts the dividend adjusted across the relevant rate periods. It reports a forward value of 101.01 using the stated inputs.

A second response distinguishes this forward calculation from predicting the share's actual market price in a year. That price is uncertain and cannot be inferred from the dividend and interest rates alone; market movements also matter. The calculation is conditional on the stated assumptions, including a known dividend and deterministic rates. It is therefore a pricing relationship for a forward under a simplified model, not a forecast of the realized equity price.

Key ideas

  • A known interim dividend reduces the value carried into a later forward horizon.
  • The forward calculation uses interest rates corresponding to the dividend and contract maturities.
  • Risk-neutral forward value is distinct from a forecast of the future spot price.
  • The stated result relies on deterministic rates and the assumed dividend payment.

Tags

EquitiesDerivatives pricingFixed income
Full text
# Price of an equity


# 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 ?

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/23115

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*}

## Answer by assylias (score 1)

https://quant.stackexchange.com/a/23094

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.