I have a credit card that gives me rewards. I get a nice 5% cash-back on purchases from Amazon and a lower cash-back rate on other purchases. Sometimes, there are promotions that provide a rate of 10% or even 15%. But what are these rewards worth?
To simplify, there are two reward options:
Option 1 adds to my Amazon gift-card balance. It’s attractive. When I’m checking out at Amazon, it shows me my reward balance and it also shows me what the total cost of my purchase could be if I applied the gift card. It’s like they’re trying to pressure me to redeem my rewards in this particular way.
Option 2 is simply to transfer my rewards as a payment on my credit card or as a credit to my bank account (for the current purposes, they’re identical). Either way, the rewards translate to the same number of dollars.
Say that I spend $1,000 at Amazon. Whether I choose option 1 or 2 has value implications.
The calculation is simple. If I spend $1,000 at amazon this month, then I can spend another $50 in gift card credits at Amazon next month. That’s the end. There are no more relevant cashflows. I used my credit card one month, and then was rewarded the next month. The only detail worth adding is the time value of money, which at 7% per year*, yields a present value of rewards at $49.72. Option 1 is nice in the moment. It’s so enticing to have a lower Amazon check-out balance due.
In grad school, I learned about the overlapping-generations model. The idea is that we simplify people down to the fundamental parts of their life-cycle. Each person lives for 2 periods. In the first period, they can produce only. In the second period, they can consume only. A popular conclusion of the model pertains to old-age benefit programs such as Social Security.
The first beneficiaries receive a gift that is free to them, then each subsequent generation accepts the debt, pays it off, and then passes on new debt to the proceeding generation. In this manner, the program benefit of the current generation is limited by the income of the following generation. Therefore, every single generation can consume as if they lived a generation later – and a generation richer – in time. That’s exciting.
But this model is not unique to governments. With a little bit of finance, we can model every person as their own self-encapsulated overlapping-generations model – with two similarly exciting conclusions. Let’s consider a person who has monthly consumption expenditures of $1k per month and let’s assume a discount rate of half a percent per month.
Life is pretty good for this person. They earn income each month and they spend $1k of it during the same period. Now let’s give the person a credit card. It doesn’t matter what the interest rate is – they’re going to pay it off each subsequent month. Now let’s see what’s possible.
What’s going on here? The difference in the consumption pattern is that the first month with a credit card can enjoy twice the consumption. How’s that? $1k of that January consumption is just the typical monthly spending. The other $1k is running up a month’s worth of spending on the credit card. So long one pays-off the card in the following month, there are no interest charges. But wait – if one pays-off the credit card in February, then how does one consume in February? By borrowing from March’s income, of course! And so the pattern repeats ad-infinitum. With a credit card one can borrow against next month’s spending. You too can borrow from your future self. And your future self won’t mind because they’ll do the same thing.
Conclusion #1: Having a credit card entitles you to one free month of double consumption.
The above example includes identical income over time. But, what if your income grows? Let’s assume that your income and commensurate consumption grow at a rate of one quarter percent per month. Our consumption without a credit card is tabulated below.
Obviously, having income and consumption that grow is more enjoyable than ones that are constant each period. Now let’s observe below what happens when we again introduce a credit card that one pays-off each month.
What’s going on here? Just as happened previously with a credit card, one can enjoy an extra boost to consumption in the first period. But what does growing income do for us besides greater complication? Just as previously, one can pay their debt each period and consume by borrowing against the next month’s income. But with growing income, having a credit card means that one can enjoy the next month’s level of consumption today. That is, next month’s higher consumption is shifted sooner in time by one month. Notice that, with growing income, consumption for July without a credit card ($1,018) is the same as the consumption in June with a credit card. Even without the first-month-gift, credit cards increase the present value of one’s consumption by making next month’s greater income available today – and the same is true for every single month.
Conclusion #2: Having a credit card today entitles you to next month’s greater income.
How big a deal is this? Obviously, it will differ with the discount rate and the rate of income growth. Using the numbers above, having a credit card permits one to consume with a present value that is 10.5% higher. Let that sink in. People who have access to credit consume as if they are 10.5% percent richer. Access to credit can make the difference between a pleasant Christmas, having quality internet, paying for car repairs, and so on. Being poorer is one thing. Being poorer and lacking access to credit is like taking an instant haircut to one’s quality of life. On the flip side, people can be made better-off without additional improvements to their productivity. Increasing access to credit may be a less costly improvement to the value lifetime consumption than many of the other less politically feasible improvements to labor productivity.
Hopefully by this time we all know about index funds. The idea is that by investing in a large, diversified portfolio, one can enjoy the average return across many assets and avoid their individual risk. Because assets are imperfectly correlated, they don’t always go up and down at the same time or in the same magnitude. The result is that one can avoid idiosyncratic risk – the risk that is specific to individual assets. It’s almost like a free lunch. A major caveat is that there is no way to diversify away the systemic risk – the risk that is common across all assets in the portfolio.
We can avoid the idiosyncratic risk among assets. But, we can also avoid idiosyncratic risk among times. Each moment has its own specific risks that are peculiar to it. Many people think of investing as a matter of timing the market. However, people who try to time the market are actively adopting the specific risks that are associated with the instant of their transaction. This idea seems obvious now that I’m writing it down. But I had a real-world investing experience that– though embarrassing in hindsight – taught me a heuristic for avoiding overconfidence and also drilled into my head the idea of diversifying across time.
I invested a lot into my preferred index fund this past year. I’d get a chunk of money, then I’d turn around and plow it into the fund. What with the Covid rebound, it was an exciting time. I started paying more attention to the fund’s performance, identifying patterns in variance and the magnitude of the irregularly timed and larger changes. In short, by paying attention and looking for patterns, I was fooling myself into believing that I understood the behavior of the fund price.
And it’s *so* embarrassing in hindsight. I’d see the value rise by $10 and then subsequently fall to a net increase of $5. I noticed it happening several times. I acted on it. I transferred funds to my broker, then waited for the seemingly regular decline. Cha-ching! Man, those premium returns felt good. Success!
Silly me. I thought that I understood something. I got another chunk of change that was destined for investing. I saw the $10 rise of my favorite fund and I placed a limit order, ensuring that I’d be ready when the $5 fall arrived. And I waited. A couple weeks passed. “NBD, cycles are irregular”, I told myself. A month passed. And like a guy waiting at the wrong bus stop, my bus never arrived. All the while, the fund price was ultimately going up. I was wrong about the behavior of the fund. Not only did I fail to enjoy the premium of the extra $5 per share. I also missed what turned out to be a $10 per share gain that I would have had if I had simply thrown in my money in the first place, inattentive to the fund’s performance.
I hate making bad decisions. I can live with myself when I make the right decision and it doesn’t pan out. But if I set myself up for failure through my own discretion, then it hurts me at a deep level. What was my error? Overconfidence is the answer. But why did it hurt me?