What is Continuous Discounting?
Continuous discounting is a way to figure out how much money you will have in the future. It looks at money coming in, like your paycheck. This money is called a cash flow.
Discounting means lowering the value of that future money. Why? Because money in the future will be worth less than money now. A dollar today can buy more than a dollar next year.
Most of the time, we discount money every month or year. But with continuous discounting, we do it all the time, every single second. It’s like a non-stop discounting machine.
The Math Behind Continuous Discounting
We use a unique number called “e” to offer continuous discounts. It’s about 2.71828. This number is unique because it helps us figure out continuous things like discounting.
We also need to know the “cost of capital.” This is like the cost of borrowing money or the amount you could earn if you invested your cash elsewhere. Let’s call the cost of capital “r”.
To discount continuously for one year, we use this math:
e^(-r)
That ^ symbol means “to the power of”. And the little minus sign in front of the r means we’re lowering the value, which is what discounting does.
An Example
Let’s say you’re expecting $1000 in one year. And let’s say your cost of capital, r, is 10% or 0.10.
To continuously discount that $1000, we do:
1000 * e^(-0.10) = $904.84
So, by this math, that future $1000 is really only worth about $905 to you right now. That’s because you’re lowering its value to account for the cost of waiting a whole year to get it.
Why Use Continuous Discounting?
You might be wondering, why bother with this complex continuous discounting? Why not just do regular, periodic discounting every month or year like most people?
Well, some folks think continuous discounting is more accurate. It’s always working, second by second, to keep the value of future money up to date. There’s no lag time.
It’s especially useful for big financial decisions that involve a lot of money over a long time. The non-stop nature of continuous discounting can give a more precise picture.
Downsides of Continuous Discounting
Now, continuous discounting isn’t perfect. It can be confusing because it uses that special “e” number and requires more math.
Plus, in real life, money usually comes in chunks on a regular schedule, like a monthly paycheck. It doesn’t really flow continuously.
So for day-to-day stuff, regular periodic discounting is often enough. The difference in the final numbers between periodic and continuous discounting is usually pretty small for short time frames.
Continuous Compounding
There’s another idea that’s closely related to continuous discounting. It’s called continuous compounding.
Remember how discounting lowers the value of future money? Well, compounding does the opposite. It increases the value of money over time, like interest in a savings account.
Just like discounting, compounding can be done periodically (like monthly or yearly) or continuously. Continuous compounding uses that same special “e” number and similar math.
The cool thing is, continuous discounting and continuous compounding are like opposites that cancel each other out. If you have $1000 that’s continuously compounded at a 10% rate for a year, you’d end up with about $1105.17. But if you then continuously discounted that amount by the same 10% rate, you’d be right back at your original $1000.
Putting It All Together
Let’s sum up what we’ve learned:
- Continuous discounting is a way to lower the value of future money on a non-stop basis.
- It uses a special number “e” and the cost of capital “r” in its math.
- It’s the opposite of continuous compounding, which increases the value of money over time.
- It can be more accurate than periodic discounting, especially for long-term, big money decisions.
- But it’s also more complex and not always necessary for day-to-day financial stuff.
