Money Lending EV

[assume_R]

This is not a casino game, per se, but rather a question on EV calculation. If the mods want to move this thread to another section, feel free.

Let's say I have a friend, and I am willing to loan him $100, in which I'll take 14% yearly interest (calculate yearly for simplicity). I am guessing that the chance of him losing that $100 and defaulting is 2% (in which I get no payments).

How do I calculate my EV and Variance (for RoR calculations)?

I have EV for each year as: 98% * (+14%) + 2% * (-100%) = +11.72% EV. Right?

What about variance? Let's assume a normal distribution. How do I calculate the variance, given a default rate of 2%? Thanks in advance.

 

[assume_R]

Got it

Nevermind - I think I figured it out.

Mean = 11%
Var = (+14% - 11%)^2 * 98% + (-100% - 11%)^2 * 2% = .025524
Std = 16.0%

 

[MangoJ]

This is not a casino game, per se, but rather a question on EV calculation. If the mods want to move this thread to another section, feel free.

Let's say I have a friend, and I am willing to loan him $100, in which I'll take 14% yearly interest (calculate yearly for simplicity). I am guessing that the chance of him losing that $100 and defaulting is 2% (in which I get no payments).

How do I calculate my EV and Variance (for RoR calculations)?

I have EV for each year as: 98% * (+14%) + 2% * (-100%) = +11.72% EV. Right?

What about variance? Let's assume a normal distribution. How do I calculate the variance, given a default rate of 2%? Thanks in advance.

EV is correct, it is ~$12 per year. However you do NOT have a normal distribution, it is a simple discrete distribution (assuming your friend either pays your interest, or busts).

For variance (and this is crucial, for a yearly interest payment):
Var = 98% * ($14)² + 2% * (-$100)² - EV² =~ ($16)²

Variance is ($16)² per year.

After N years, variance is N*($16)², while EV is N*$12. If N would be a large number, you could use a normal distribution for your RoR calculus, like you do with blackjack hands. If N is small (say, smaller than 100 years), you should better do exact calculations.

 

[assume_R]

For variance (and this is crucial, for a yearly interest payment):
Var = 98% * ($14)² + 2% * (-$100)² - EV² =~ ($16)²

Variance is ($16)² per year.

After N years, variance is N*($16)², while EV is N*$12. If N would be a large number, you could use a normal distribution for your RoR calculus, like you do with blackjack hands. If N is small (say, smaller than 100 years), you should better do exact calculations.

Yup, you got the same as me (I just had it as a standard deviation of 16%).

Now, could you elucidate what you mean by "exact calculations"? How do I do exactly calculations? Also, this doesn't assume any compounding of interest... which would be useful to include.

 

[MangoJ]

Now, could you elucidate what you mean by "exact calculations"? How do I do exactly calculations? Also, this doesn't assume any compounding of interest... which would be useful to include.

Let's assume we model your friend by a (weighted) coin flip. He either busts within a year (p=2%) or he survives and compounds the interest. We further assumes that busting each year is statistically independent (as a coin flip).

The dept (your bankroll B(N) ) after N years with interest P and initial bankroll B0 will be B(N) = B0 * (1+P)^N.

You double your bankroll after N2 years, which is (1+P)^N2 = 2, or N2 = log(2)/log(1+P).

The probability of your friend busting within N years at least once (which means you bust also) is 1 - (1-p)^N.
For N=N2 this is exactly your RoR, the probability of you busting before doubling your bankroll:

RoR = 1 - (1-p)^N2 = 1 - (1-p)^(log(2) / log(1+P))

For your numbers, P=14% and p=2%, RoR is 10%. (N2 = 5.3 years)


What I meant with "exact calculations": For low number of years you cannot use a normal distribution calculation - unless you have a lot of friends you lend your money to.

 

[assume_R]

Let's assume we model your friend by a (weighted) coin flip. He either busts within a year (p=2%) or he survives and compounds the interest. We further assumes that busting each year is statistically independent (as a coin flip).

The dept (your bankroll B(N) ) after N years with interest P and initial bankroll B0 will be B(N) = B0 * (1+P)^N.

You double your bankroll after N2 years, which is (1+P)^N2 = 2, or N2 = log(2)/log(1+P).

The probability of your friend busting within N years at least once (which means you bust also) is 1 - (1-p)^N.
For N=N2 this is exactly your RoR, the probability of you busting before doubling your bankroll:

RoR = 1 - (1-p)^N2 = 1 - (1-p)^(log(2) / log(1+P))

For your numbers, P=14% and p=2%, RoR is 10%. (N2 = 5.3 years)


What I meant with "exact calculations": For low number of years you cannot use a normal distribution calculation - unless you have a lot of friends you lend your money to.

Thanks, Mango. Some questions:

1. Why did you use the probability of busting before doubling? Isn't traditional RoR probability of busting ever? I'm all for the chance of busting before doubling, but I don't know where you got that from.

2. If I had 100 friends, and loaned them each $1 (or 1000 friends, each $0.10), with the EV and probability of busting I gave, could I use the traditional RoR equation assuming a normal distribution as below? Or would there be another distribution?

RoR_Norm = ((1 - $12/$16) / (1 - $12/$16))^($100/$16) = .00052%

And constrained by 5.3 years (as typed into excel):
RoR_5.3 = NORM.DIST((-100-12*5.3)/sqrt(16^2*5),0,1,TRUE)+EXP(-2*12*5.3*100/(16^2*5))*NORM.DIST((-100+12*5.3)/sqrt(16^2*5),0,1,TRUE) = .000987%

 

[MangoJ]

Why did you use the probability of busting before doubling? Isn't traditional RoR probability of busting ever? I'm all for the chance of busting before doubling, but I don't know where you got that from.

You are right, there are typically two different RoR. One is the probability of bust before doubling, the other is probability of livetime bust.
If you let the interest compound, you are actually playing a progression system (without stop win). For such a progression your lifetime RoR would be 100%, because your friend will bust some year almost surely. To get a finite lifetime RoR, you should set up a payback date.
If the friend does pay you interest over the loan, you get a finite lifetime RoR, and you don't have to worry about a specific payback date.

Yes, if you have 100 clients of your loan, you can use the normal distribution with the EV and variance. Then you can use the same RoR formulas you use for blackjack. Your double-before-bust RoR should lower considerably (as you spread the risk), and if you give your friend a payback deadline you can also calculate a lifetime RoR.

 

[assume_R]

Gracias

Thanks for your help MangoJ; you're quite numerically savvy!

 

[Automatic Monkey]

One important point is that after he makes the first interest payment, your risk of ruin (in the sense that we use "ruin" in blackjack) is zero unless you reinvest the cash. After 8 annual payments your risk of losing money is zero.

This is important when dealing with "junk bonds" and similar paper that can pay a very high yield, but who knows for how long? Your 14% investment would probably be categorized as "junk" in the current market; that is not necessarily a disparaging term unless used by a financial manager who is forbidden by policy from trading such investments.

 

[assume_R]

This is important when dealing with "junk bonds" and similar paper that can pay a very high yield, but who knows for how long? Your 14% investment would probably be categorized as "junk" in the current market; that is not necessarily a disparaging term unless used by a financial manager who is forbidden by policy from trading such investments.

So if something is yielding a very high yield, and I can spread my investment across multiple high yield accounts (with an average default rate of 2% - 3%), it seems that this is an excellent way to invest my money, being as my calculated RoR is much lower than what I expect in blackjack for a given investment.

However, if I can put in 300 hours a year in blackjack, which I'm on track to (between 300 and 400 hours), a $10k investment could yield about 100% return in blackjack ($33 / hour is reasonable for me for a $10k bankroll). But that's with a substantially higher RoR (approximately 10%-20%), and my results will vary much more wildly than traditional investments, or these "junk bond" investments. In addition, it requires active time taken for the year. So the 14% investment will have a lower EV than blackjack, but then again much higher than traditional stocks.

I wanted to run these numbers so I could essentially compare apples to apples in terms of EV and RoR for traditional investment vehicles vs. blackjack and other money-making opportunities.

 

[MangoJ]

Yes, if you compare different investments you need to include EV, RoR and sometimes your own work efforts (i.e. for blackjack). At least for comparison of different investments, there is a unifying quantity which is CEV (certainty equivalent value), which you can use for comparision.
More formal
log(bankroll + CEV) = <log(bankroll + investment*Payout)>
where Payout is a random variable (in your loan case it is the interest (with 98%) and the loss of your loan (2%).
<...> is the statistical expectation value. If bankroll is infinity, then
CEV = <investment*Payout> = investment * EV.
However, if your bankroll is not finite, then CEV is smaller.
A Kelly bet is the investment which maximizes CEV for a 2-outcome bet (only two options in Payout).

Regarding your 100 clients: You must be careful. If all your clients invest your loan on the very same object (think of all play the same numbers on lotto each week), your RoR is dramatically increased. Chances are, if one of them busts, other will bust too. They don't have to invest into the very same property, it is sufficient that their investment performance is correlated (i.e. same industry). You need to account for that in your RoR calculations.

Famous example for this correlated RoR is the economic crisis in 2007/2008. Loans from banks where bundled (your 100 clients), which all were used for mortage with VARIABLE interest rates. What happened, the interest rates increased, and the majority of your clients could not pay the interest rates and busted. Furthermore their securities (their houses) where on the same market, and hence prices dropped simultaneous (as those busted clients had to sell simultaneous).
The next step was, since those loans were lucrative for banks, they all invested in those loans, and had problems - you guess it - simultaneous.

Be aware of correlation, it dramatically adds to your risk. However if you overbet your risk, you will bust despite of any advantage.

 

[assume_R]

Yes, if you compare different investments you need to include EV, RoR and sometimes your own work efforts (i.e. for blackjack). At least for comparison of different investments, there is a unifying quantity which is CEV (certainty equivalent value), which you can use for comparision.
More formal
log(bankroll + CEV) = <log(bankroll + investment*Payout)>
where Payout is a random variable (in your loan case it is the interest (with 98%) and the loss of your loan (2%).
<...> is the statistical expectation value. If bankroll is infinity, then
CEV = <investment*Payout> = investment * EV.
However, if your bankroll is not finite, then CEV is smaller.
A Kelly bet is the investment which maximizes CEV for a 2-outcome bet (only two options in Payout).

Regarding your 100 clients: You must be careful. If all your clients invest your loan on the very same object (think of all play the same numbers on lotto each week), your RoR is dramatically increased. Chances are, if one of them busts, other will bust too. They don't have to invest into the very same property, it is sufficient that their investment performance is correlated (i.e. same industry). You need to account for that in your RoR calculations.

Famous example for this correlated RoR is the economic crisis in 2007/2008. Loans from banks where bundled (your 100 clients), which all were used for mortage with VARIABLE interest rates. What happened, the interest rates increased, and the majority of your clients could not pay the interest rates and busted. Furthermore their securities (their houses) where on the same market, and hence prices dropped simultaneous (as those busted clients had to sell simultaneous).
The next step was, since those loans were lucrative for banks, they all invested in those loans, and had problems - you guess it - simultaneous.

Be aware of correlation, it dramatically adds to your risk. However if you overbet your risk, you will bust despite of any advantage.

Indeed, my investments are, as far as I can tell, uncorrelated, which I feel is a safe assumption. They don't all have the same bank, same investment objective, etc. I suppose they are all correlated very loosely with the global economy and if it completely crashes, but for the purpose of this discussion I reasonably assume uncorrelatedness.

Now, regarding CEV, I've always used the equation:

CEV = EV - Var / (2 * KellyFraction * Bankroll)

which I can't remember where I got or derived that from. KellyFraction is how much you bet, in that you put 1 / (KellyFraction) x f* and you resized your bets accordingly. Is that a direct derivation from your equation? I'll try to go through the math if I can.

 

[Automatic Monkey]

So if something is yielding a very high yield, and I can spread my investment across multiple high yield accounts (with an average default rate of 2% - 3%), it seems that this is an excellent way to invest my money, being as my calculated RoR is much lower than what I expect in blackjack for a given investment.

However, if I can put in 300 hours a year in blackjack, which I'm on track to (between 300 and 400 hours), a $10k investment could yield about 100% return in blackjack ($33 / hour is reasonable for me for a $10k bankroll). But that's with a substantially higher RoR (approximately 10%-20%), and my results will vary much more wildly than traditional investments, or these "junk bond" investments. In addition, it requires active time taken for the year. So the 14% investment will have a lower EV than blackjack, but then again much higher than traditional stocks.

I wanted to run these numbers so I could essentially compare apples to apples in terms of EV and RoR for traditional investment vehicles vs. blackjack and other money-making opportunities.

There is something to consider in that market investments have covariance just like hands at a BJ table. If one piece of paper crashes chances are good many others will too, particularly in the same industry. I myself took a hit on a subprime mortgage preferred issue just like that. Only got 20% of my investment back before it croaked. But if I spread my money out over a whole bunch of subprimers, chances are good they would have all suffered the same fate.

On the other hand, we risk a small portion of our bankroll on a game, and there is no covariance from table to table or even shoe to shoe, unlike a market where the covariance can extend over many years and all market segments. This is what keeps casino AP reasonably safe. There are some market segments that have negative covariance relative to one another, but those relationships are mostly empirical and subject to unexpected change.

 

[MangoJ]

This is the reason I never took much interest in stock markets and stuff.

If you win, someone else loses and vice versa. Further, both pay fees to their broker. Hence it is a -EV game if all you can do is randomly buy something. (at least it is -EV compared to overall economic growth)

Therefore, the key to success is to find someone else who is even dumber (than yourself) within the market. If you know nothing about the market, you are one of the dumbest - again it is -EV.

I've never traded anything on stocks, but once I did spot an advantage - being a customer of a company I noticed some severe change in their operations, which I figured would cost the company millions of dollars over the weekend (which it did!). I really wanted to short-sell those stocks, but honestly I didn't know how as I never had traded something before (and didn't know someone who did).
In the following midweek, the stocks of that company took a quite sharp 15% drop. What a nice opportunity left uncatched.

 

[assume_R]

If you win, someone else loses and vice versa. Further, both pay fees to their broker. Hence it is a -EV game if all you can do is randomly buy something. (at least it is -EV compared to overall economic growth)

That's the key, though. If you assume the stock market will on average grow (which it has in many 10-year spans), then it is like paying commission in a negative house edge game (i.e. a player edge off the top, minus commission). You will on average make money, unless the commission gets too high (i.e. day trading).

Putting $1k in an index fund that you think will on average grow 5% might charge $5 commission. That's 0.5% commission for a +5% EV, and only for the first year. After that it's 0 commission if you keep it. Yet that $1k is usually bet on something with an APR, or per year, while in blackjack we can make many bets which makes our total amount bet >> $1k in 1 year with the same $1k.

 

[Automatic Monkey]

This is the reason I never took much interest in stock markets and stuff.

If you win, someone else loses and vice versa. Further, both pay fees to their broker. Hence it is a -EV game if all you can do is randomly buy something. (at least it is -EV compared to overall economic growth)

Therefore, the key to success is to find someone else who is even dumber (than yourself) within the market. If you know nothing about the market, you are one of the dumbest - again it is -EV.

I don't agree- stocks ownership makes you a part owner of a theoretically wealth-producing industry and unlike a poker game or parimutuel bet you get more out of the pot than you put in. It is possible for everyone to make money, but most people treat the markets like a casino.

I've never traded anything on stocks, but once I did spot an advantage - being a customer of a company I noticed some severe change in their operations, which I figured would cost the company millions of dollars over the weekend (which it did!). I really wanted to short-sell those stocks, but honestly I didn't know how as I never had traded something before (and didn't know someone who did).
In the following midweek, the stocks of that company took a quite sharp 15% drop. What a nice opportunity left uncatched.

:1st: That's the way you do it! You do the work of researching companies to get information everyone else doesn't have, and reward the most productive with your investment. In return you are rewarded for your research and willingness to risk.

If I ever get back into the markets on a large scale I'm going to spend all my time sneaking up on companies, hanging out in their parking lots and making casual but prying conversation with their employees in the luncheonettes. It's not that different from backcounting or tracking down flashing dealers.

 

[assume_R]

Now, regarding CEV, I've always used the equation:

CEV = EV - Var / (2 * KellyFraction * Bankroll)

which I can't remember where I got or derived that from. KellyFraction is how much you bet, in that you put 1 / (KellyFraction) x f* and you resized your bets accordingly. Is that a direct derivation from your equation? I'll try to go through the math if I can.

Bump on this. Any thoughts, MangoJ?

Edit: Nevermind, I got it. Assuming maximizing bankroll grown, or maximizing log of the bankroll,

log(BR + CEV) = 98% * log(BR + $14) + 2% * log(BR - $100)
CEV = exp(98% * log(BR + $14) + 2% * log(BR - $100)) - BR

If BR = $200, CEV = $10.77
If BR = $10,000, CEV = $11.71

 

[Shoofly]

Therefore, the key to success is to find someone else who is even dumber (than yourself) within the market.

It's called the "Greater Fool Theory." Usually in raging bull markets. The idea is to buy a stock (even though you know it is overpriced) with the idea that some greater fool will buy at an even higher price. It is somewhat like a Martingale system. It works beautifully until.....it doesn't.

A good example is what happened in the housing market a couple of years ago.

 

[MangoJ]

Bump on this. Any thoughts, MangoJ?

Edit: Nevermind, I got it. Assuming maximizing bankroll grown, or maximizing log of the bankroll,

log(BR + CEV) = 98% * log(BR + $14) + 2% * log(BR - $100)
CEV = exp(98% * log(BR + $14) + 2% * log(BR - $100)) - BR

I'm sure you get to your approximated formula
CEV = EV - Var / (2 * KellyFraction * Bankroll)
if you expand the logarithm log(1+x) ~= x - x^2/2, and then plug in into the remainings the expressions for EV and variance.
The factor 2 in the denominator gives you that hint, as it is the Taylor coefficient of that expansion.