Everything so far has been in the form of a story. Math helps us conceptualize things in a way that makes it easier to think about more complex situations.
Time Value of Money
We said earlier that we could buy $100,000 of US Treasury bonds and receive $4,000 per year.
- After one year we would receive $100,000 times 4% = $4,000. Including the original $100,000, we now have $104,000.
- If we take this $104,000 and invest it again, we would then receive $104,000 times 4% = $4,160. Including our original $100,000, plus the $4,000 we made last year, we now have $100,000 + $4,000 + $4,160 = $108,160.
How much would we have after 5 years? After 30 years? After 50 years? It would be too cumbersome to figure it out the way we did above. So, we can use math to turn a complex problem into something simple.
The math here is:
- $100,000 times 1.04 = $104,000 for the first year
- $104,000 times 1.04 = $108,160 for the second year
Why are we using 1.04 instead of 4% above? Because 1.04 is equal to 104%. When we invest $100,000 we don’t just get 4% back, we get the 4% plus 100% of our original investment back.
We can short cut things by using exponents:
- $100,000 times 1.04 to the power of 2 = $108,160
If we wanted to know how much we would have after 5 years, we could do:
- $100,000 times 1.04 to the power of 5 = $121,665
After 30 years, it would be:
- $100,000 times 1.04 to the power of 30 = $324,339
We can simplify this by saying:
- Present Value x (1 + rate) to the power of N = Future Value of that principal
- We can abbreviate words and just write, PV x (1 + r)^N = FV
Here Present Value means the $100,000, rate means 4%, N is how many years into the future we’re interested in, and the Future Value is what the principal will be worth in the future.
The nice thing about putting things in the form of math is that we can easily do things with it. For example, we can solve for PV:
- PV = FV / (1 + r)^N
This would be useful if someone told us something like, “I’ll give you $200,000 five years from now in exchange for X dollars today.” You would first try to figure out what the risk associated with this is to decide what interest rate to use. Say you determine this should be 5% return rate. Then we can simply do:
- PV = $200,000 / (1 + 5%)⁵
- PV = $156,705
So, we should be willing to pay this person $156,705 in exchange for $200,000 five years from now, assuming we chose that 5% based on our other available investment opportunities.
We can double check this by going the other way:
- $156,705 x (1 + 5%)⁵ = $200,000
Growing Perpetuity
What happens if the lemonade stand’s profits grew every year? I’ll let you read the Wikipedia on this to learn it.
Dividend discount model
Take a look at the Wikipedia article for this also.
Internal Return 