Time Value of Money :Discrete & Continuous

Time value of money is the concept that money today is worth more than money in the future.

Why do i say so?

Because

• We may invest that given money into government bonds ,stocks or whatever we want and get a premium for that.
• Another reason is inflation.So future money worths less because inflation pushes prices higher.So the buying power of the same amount of money is lower.

Imagine you have $100 today. You have the option to invest it, and over time, it could grow to $110, $120, or more. Conversely, if you receive $100 a year from now, you've missed out on potential earnings. This discrepancy in value over time is what TVM elucidates.

Discrete Approach: Future Value Calculation

One way to compute the future value (FV) of an investment using discrete compounding is through the formula:

FV=PV(1+r)?        

Here, PV represents the present value, r signifies the interest rate per period, and n is the number of periods.

Let's break it down: If you invest $100 (PV) for two years at an annual interest rate of 5% (r), the future value FV would be calculated as FV=100(1+0.05)2=110.25

Continuous Approach: Euler's Formula

Another method, often used in continuous compounding scenarios, involves Euler's formula:

FV = PV * e^(rt)        

Here, e is Euler's number (approximately 2.71828), r remains the interest rate, and t signifies time.

To grasp how we arrive at this formula, let's consider the concept of infinitesimal changes. By observing the change in the investment value x(t+dt)?x(t) over a tiny interval dt, we can model continuous growth. Integrating this expression leads to

x(t)=x(0)* e^(rt)        

which underpins the continuous FV formula.

Now, let's delve into the derivation of the continuous growth equation

dx/dt = rx        

Therefore, the solution to this equation is:

x(t)=A * e^(rt)        

This formula represents continuous compounding, where x(t) is the amount at time t, A is the initial amount, r is the continuous interest rate, and e is Euler's number.

Python code for both discrete and continuous

from math import exp

def future_discrete_value(x, r, n):
    return x*(1+r)**n

def present_discrete_value(x, r, n):
    return x*(1+r)**-n

def future_continuous_value(x, r, t):
    return x*exp(r*t)

def present_continuous_value(x, r, t):
    return x*exp(-r*t)

if __name__ == '__main__':
    # value of investment in dollars
    x = 100
    # define the interest rate (r)
    r = 0.07
    # duration (years)
    n = 3

    print("Future value (discrete model) of x: %s" % future_discrete_value(x, r, n))
    print("Present value (discrete model) of x: %s" % present_discrete_value(x, r, n))
    print("Future value (continuous model) of x: %s" % future_continuous_value(x, r, n))
    print("Present values (continuous model) of x: %s" % present_continuous_value(x, r, n))        

Conclusion

In summary, whether employing discrete or continuous approaches, the time value of money underscores the importance of considering the temporal dimension of cash flows. By understanding these principles, individuals and businesses can make informed financial decisions, maximizing their resources and achieving their goals.