Going through forward pricing with people who are not natural users of the exponential function (like myself to be honest). The easiest way of explaining it is through no arbitrage and pointing out that no riskless profits can (should) be made. So I used a simple example of a $100 asset with three months maturity and an 8% (3m) interest rate. With no dividend, the forward price is $102, since you can always buy the asset and borrow money. Now I introduce a dividend yield of 4% (3m). If I do a simple cash in/cash out approach, then surely the forward price is 101? The yield is worth $1 in the future. However, using the regular exp(r-d)t approach, and converting the nominal yields to their continuously compounded equivalents (7.921 and 3.994) then I generate a different forward price of 100.986. I'm guessing the latter is correct as regards Hull etc but the first one seems more intuitive and makes sense from a cash in/cash out basis. Is there some difference between yield and discrete cash flows?I can also see (I think) that mathematically there has to be a difference - subtracting exponents is clearly different from subtracting rates. I am obviously doing something incredibly stupid and I'm embarrassed to ask for help among such clever people - but can someone spare me a few seconds to put me out of my misery and show just how stupid I am?

your confusion is entirely due to the fact that its not clear what the compounding frequency of your rate and div yield are. In the first example you are assuming that its simple interest so you pay 8% * 3/12 * 100 and receive 4% * 3/12 * 100 which gives you a new payment of 1. However, you then use the same rates and compounding them continuously - this is not correct. the equivalent continuous rates have to be lower - for the interest rate the continuous compounded equivalent is 7.921%.hth

Dont think this helps me - as mentioned above I adjusted the two yields to the continuously companion equivalents - 7.921 & 3.994. Using exp(r-d)t I get a fwd of 100.986. But using simple interest of 8 and 4 I get 101.

You are right and I apologise. the answer is that if you agree that the forward price is 101 and continuos rates are 7.921% then your equivalent continuous div yield is 3.9801%. the reason this lower than 3.994% is because you are earning interest on the dividend on an ongoing basis.

I'm confused. If I am told (by somebody) that the dividend yield is actually 4% on a simple market basis, are you saying that you cannot simply convert this to a continuously compounded equivalent of 3.994 and calculate the forward price using the exponential equation? This is different from the forward foreign-exchange calculation where I can always convert the rate in each currency from its nominal to its continuously compounded equivalent. I'm also a little bit lost as to why there is a difference in the literature of between present valuing discrete dividends and using the exp(r-d)t approach as surely they are conceptually equal?

if you can find forward prices for equities then you can figure out the implied continuous yield. its a bit of chicken and egg. In fx zero coupon bonds or depo rates can be observerd so its not an issue. In equities dividends are discrete (eve in indices) so you should be calculating your forward as followsF = (S - pvdiv) * exp(rT)hth or has not confused you even more.

I was trying to go the other way around - calculating the forward price given a dividend yield. I wanted to check it using the exp(r-d)t approach, and assumed I could simply convert the nominal yield to a continuous basis (as I can using Fx forwards). I guess the important thing in terms of this exercise simply showing that the forward price is indeed 101, and ignoring the exp(r-d)t approach completely.

if the equity pays a continuous dividend of q per annum then the present value of this is given byD = q*S/r*(1-exp(-rT))where r is the continuous rate and S is the current price of the equity. In your example if the continuous rate is 4% per annum and rates is 7.921% then the PV of this 0.99016 for 3 months.

I had also worked out that the PV of the dividend is 0.99016, but that was simply by taking the future value of one (0,25 x 100 x 0.04!) and discounting that using simple interest. No exponential elements were involved, and I can't actually get your equation above to work. (Shouldn't it be ((S*(exp(qT))/(exp(-rT)) - simply the future value of the dividend (calculated using continuous compounding) discounted back by the interest rate? in any event, I still seem to have the same problem that the age-old exp(r-d)t approach doesn't seem to yield the correct forward value when I convert a dividend yield based on ordinary rates into the continuously compounded equivalent.

well that will work for short periods but wont for longer periods - for example as t->infinity ?