I originally wrote this as a twitter thread, but I thought it might be useful to send it out on substack as well. The style and flow might be a little different for that reason, but you will get the key points and I’m sure you can forgive me for the lack of literary flair you have become accustomed too.
1/ Pricing Deterministic Payoffs (Part 2).
(Note: We are still building a foundation but this will be the last thread before we get into the more useful stuff like Bonds.)
2/ I wanted to give you another example of deterministic payoff before we move into bonds. This example will look unrealistic because we aren’t taking into account randomness. But for the sake of the example lets just assume everything is determined and we know it all.
3/ Lets say you wanted to lease a Dutch Brothers for 10 years (for those of you not in the states it’s a very successful drive through coffee chain,) and let’s assume your annual profit is 2 mil.
3a/ The question now becomes, how much is the fair price of the lease if we wanted to pay for everything today.
4/ To clarify, we are leasing for 10 years. We are assuming we will have an annual profit of 2 mil every year for all ten years (unrealistic), and we are estimating that if investing in anything other than the coffee shop we would receive r=10%.
5/ Massive assumptions we couldn’t possibly know, but indulge me. I’m going to blow through these next steps quickly and you will see why in a second.
Step 1: First we need our discount rate. Which we talked about last week.
6/ Step 2: Use discount rate in the formula below and you end up with a “fair value” for your Dutch Bro’s.
~12M
7/ Here’s the punchline, we don’t use that formula anymore because it excludes randomness. We now use ideas from real options theory, which are used for pricing random payouts. (So why waste time talking about it.)
8/ Ok, back to the matter at hand.
Question: Assume we now take a loan with a deterministic interest rate paid in equal amounts X over m periods. What will our payment on the loan be?
9/ In the tweets above (5)(6),we used this discount rate formula to calculate PV (present value). Now we can simply invert that formula to find X (payments). In other words, with the formula below we were pretending to know X to find PV. But now we know PV and need to find X.
10/ Here's how to calculate payments. Lets say I want to take on a 400k mortgage to buy a house. The PV of that loan is 400k which we are denoting as $V here.
V= 400k (loan)
r = Rate
n= compounding times a year
m= period or number of years.
X= Payment
11/
30 year loan 400k
8% (lolz) 12 times compounding in a year (months)
m= 30 years * 12 months = 360
r= .08//12=.0067
V=400k
Plug that into formula above and you get $2,946/mo.
Done.
12/ Finally, lets scratch the surface of the useful (although I think being able to calculate your mortgage is interesting, I can appreciate not everyone does.)
13/ What is IRR (Internal rate of return)?
The internal rate of return (IRR) is a fun metric used in financial analysis to estimate the profitability of potential investments. – Investopedia.
(I added the “fun”).
14/ You can see how this is going to be useful right? If you are deciding on where to park 10k or 100k or even 1M, you will want to know your IRR. At the very least it will let you know if your investments are outpacing inflation.
15/ The formula below should look familiar by now. Imagine you have a sequence of payments x(0), x(1), x(2), x(m). Some years will be positive, some years will be negative. If that’s the case, we define the IRR for the investment as the number r for which the sum is 0.
16/ In other words, if x(0) is the fair price for x(1), x(2) etc, then total value should be 0, and the fair rate (r) should be a rate that makes it such.
So you simply enter the PV of the payments that you will receive, then solve for r that makes it equal to 0.
r = IRR.
17/ Why did we cover that? Because this is exactly how we are going to define a Bond Yield.
You pay up initially for the bond, then you receive coupons, and finally the payment at maturity. The yield will be defined exactly as the IRR of a bond.
Next week…Bonds.