Force of Interest

    Force of interest is a natural extension of compound interest. It is shown that a constant force of interest can be thought of as compound interest, where the interest is compounded continuously. Examples are done using both constant and varying forces of interest, and it is proven that force of interest and force of discount are equal.

Definition (Force of Interest) For an investment whose accumulated value at time is given by the function the force of interest at time is the rate of change of divided by the value of that is,

    

Example (Force of Interest) The accumulation function for compound interest is where is the nominal rate of interest compounded times per investment period. Taking a logarithm of both sides of the equation and using logarithmic differentiation to find we have







The left side of this equation, is the force of interest by definition, so we have the relationship



First note that this is a constant funtion of since is a known quantity, so we will just denote it Second, note that

Example (Constant Force of Interest) (i) Find the accumulated amount of invested at a constant force of interest for 2 years. Using the formula and the above relationship , we have
    (ii)  Find the nominal rate of interest compounded quarterly that is equivalent to a force of interest of 4.5%. We see that and so
    (iii) A constant force of interest is sometimes called interest compounded continuously. To see where this terminology arises, consider the behavior of the function as the number of compounding periods, increases without bound. As is usually the case with exponential functions, the logarithm of this function is easier to analyze, therefore we consider and find the limit as



The right-hand side is an indeterminate form of the form which seems to indicate that if we can rewrite it, we can use l'Hopital's rule to find the limit. Rewriting the right-hand side, we have



Now applying l'Hopital's rule, we take the derivative of the numerator and denominator, remembering that we are considering the expression as a function of





            

Therefore,
    In terms of the accumulation function we see that which is the accumulated amount of an investment at an interest rate compounded continuously for periods.  

Proposition (Force of Interest) The accumulation function for an investment at a force of interest after years is given by



    Proof By the definition of force of interest, It is easily shown that so that



Therefore,



Finally, rearranging the last equation gives the relationship as desired.

Example (Variable Force of Interest)
(i) Find the accumulated value of 1,200 accumulated at a force of interest for 10 years. We have













(ii) Find the annual effective rate of interest equivalent to a force of interest over a 5-year period and a 10-year period. We see from above (ignoring the principal amount of 1200) that the accumulated value of 1 after years would be Therefore we set up the equations





and



        

Note that the effective rate is increasing as increases; it will approach 100% as approaches

Definition (Force of Discount) For an investment whose present value at time is given by the force of discount is defined analagously to the force of interest as the ratio



    The negative sign is needed in the definition to make a positive quantity; since is an increasing function of is a decreasing function of and therefore has a negative derivative.

Proposition (Force of Discount) The force of discount is equal to the force of interest.

    Proof Letting denote the force of discount and denote the force of interest, we have







Therefore,

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Force Of Interest
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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