Futures: Compounding and discounting techniques.
by J Victor on April 17th, 2012We discussed continuous compounding in the last post. Derivatives pricing are done using this method of compounding. The justification is that returns on assets like shares change continuouslyon a minute by minute basis. However, in real life, since interest rates are expressed as an annual percentage, conversion to continuous rate is necessary to make accurate calculations.
We also saw in the last article that interest rates and dividends are the two factors that determine the theoretical pricing of futures. The correctness of calculations therefore depends on how accurately you’ve assumed these two figures.
HOW TO FACTOR IN EXPECTED DIVIDENDS
In the case of dividends, you can only make a best guess by studying the past dividend history and the present financial strength of the company. That’s where it ends. Dividends have to be discounted at appropriate rate to find the present value. The technique used for this is continuous discounting – the reverse process of continuous compounding.
The formula is:
 Present Value of future dividends = Future dividends x E ^{} ^{(R x T)}
For example –
The dividend expected on the X Ltd’s stock 3 months from now is Rs 10,000. What would be the present value of dividends if the risk free continuous compounding interest rate is 16% Per annum?
 Present value = 10,000 x E ^{– (0.16 x .25)}
 = 10,000 x E ^{–(0.4)}
 10,000 x 0.96079 = Rs 9607.90
Note: Just like E ^{(+x), }E^{ (–x) }values are available from the natural log tables.
INTEREST RATES
Risk free interest rates would be expressed as a percentage on an annual basis. The question is How do we convert interest rates expressed on annual or semi annual basis to continuous basis?
The mathematical formula that helps to achieve this task is as follows:
 M x N_{L }(1+ Normal rate/ M)
 Where, M = frequency of compounding and
 N_{L = }Natural logarithm_{ }
Here are two examples:
1. The risk free interest rate is quoted as 18 % P.a. What is the equivalent continuous compounding rate?
 M x N_{L }(1+ Normal rate/ M)
 Where , M= 1( annual compounding), normal rate = 18%
 1 x N_{L }(1+ 0.18/1)_{ }=_{ }1 x 0. 16551 = 0.16551 or 16.551%
2. The risk free interest rate is quoted as 18 % P.a. with half yearly compounding. What is the equivalent continuous compounding rate?
 M x N_{L }(1+ Normal rate/ M)
 Where , M= 2( half yearly compounding), normal rate = 18%
 2 x N_{L }( 1+ 0.18/2) = 2 x 0.08618 = 0.1724 or 17.24%
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2 Responses to “Futures: Compounding and discounting techniques.”
cool
April 26, 2012 at 4:55 pm
in ur example when we calculate present value of expected div of Rs. 10000 using exponential , it comes to Rs. 9607.9. but when we discount the same amount for 3 months @ 4% . we get Rs. 9615.38 .. Please explian the difference
J Victor
April 27, 2012 at 9:18 am
Hi
The difference is because
In present value, we do it assuming that it’s 16% annually compounded. so you take 16/4 for computing present value for 3 months.
E(x) in log tables use continuous discounting.(The reverse of continuous compounding).
when we Compound / discount interest continually, we use the shortest possible interval of time to compound/discount.
so, the difference in results is due to the ‘time interval’ used for discounting.
The technique to convert annual or any other base to continuous basis is also discussed in the same post under the head ‘interest rates’. m
I hope it’s clear.