Discount forward price for a continuous index to an in between time point

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h21

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Hi all, I have a question about discount continuous index forward to the current point
long a 90 days index forward mature in 90 days. The index is currently at 1145. The continuously compounded dividend yield is 1.75 percent. The discrete risk-free rate is 4.25 percent. It is now 28 days since the purchase The index value is at 1225. the question want us to alculate the value of the forward contract 28 days into the contract.
The answer is
St = $1,225
T = 90/365 = 0.2466
t = 28/365 = 0.0767
T − t = 0.1699
r = 0.0425
rc = ln(1 + 0.0425) = 0.0416——my issue is with this, i understand we are discounting the forward price to the time point t, but why do we take the In of the 1+dividend rate? instead of just use e−0.025(0.1699)
δc = 0.0175
Vt(0,T) = ($1,225 × e−0.0175(0.1699)) − (1151.83e−0.0416(0.1699)) = $77.65

Thanks
 
I don’t see that you’re taking the natural log of the dividend rate; their formula has e^(−0.0175(0.1699)), and 1.75% is the (continuous) dividend rate.
 
S2000magician wrote:
I don’t see that you’re taking the natural log of the dividend rate; their formula has e^(−0.0175(0.1699)), and 1.75% is the (continuous) dividend rate.
Click to expand...
That is the eoc answer i am confused too
 
I’m not sure I understand your question. Are you asking why someone would use a continuously compounded dividend yield?
From the curriculum,
“An alternative way to incorporate dividends is to express them as a fixed percentage of the stock price. The more common version of this formulation is to assume that the stock, portfolio, or index pays dividends continuously at a rate of δc. By specifying the dividends in this manner, we are allowing the dividends to be uncertain and completely determined by the stock price at the time the dividends are being paid.”
Also,
“This specification involving a continuous dividend yield is commonly used when the underlying is a portfolio or stock index. If a single stock in the portfolio pays a dividend, then the portfolio or index can be viewed as paying a dividend. Given the diversity of dividend policies and ex-dividend dates, such an assumption is usually considered a reasonable approximation for stock portfolios or stock indices, but the assumption is not as appropriate for individual stocks. No general agreement exists on the most appropriate approach, and you must become comfortable with all of them.”
For consistency, since we’re using a continuously compounded dividend yield in the question (which was given at .0175), you also have to use a continuously compounded risk free rate, thus taking the natural log of the discrete risk free rate (ln 1.0425).
 
I dont understand why we are taking natural log of interest rate, for example , practice of question 14 on reading 47, see the first answer,
Consider the following: The US risk-free rate is 6 percent, the Swiss risk-free rate is 4 percent, and the spot exchange rate between the United States and Switzerland is $0.6667.
  1. Calculate the continuously compounded US and Swiss risk-free rates.
answer is
rfc = ln(1.04) = 0.0392
rc = ln(1.06) = 0.0583
Wait, I think I just figured out, the risk free rate quoted is not continuous compounding rate, but how would I know if it is discreet rate or natural log transformed already????
 
ah, thanks
so i still dont get the intuition behind why we use foureign rate at denominator, domestic rate at numerator, how should i memorize this?
 
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