There has been a lot of discussion recently of Fed policy, tapering, and asset price “bubbles.” One point to bear in mind is that when interest rates are low even rationally determined asset prices may fluctuate wildly. Consider, the simplest Gordon model of asset prices in which future dividends are expected to be $100 forever, then the asset price is $100/r where r is the interest rate. If r is .1, for example, then the stock will be worth $1000. At an interest rate of 10% the price of an asset that pays $100 forever is just $1000 because the future is heavily discounted. If the interest rate were to fall to 9%, the asset price would rise to 1111.11 ($100/.09). The asset is worth more at a lower interest rate because the future counts for more but not that much more since the far future is still discounted to near zero. On the other hand, if the interest rate were .02 the asset would be worth $5000, much more since the future is discounted less heavily. Most importantly, notice that if the interest rate were to fall the same amount as before to .01 then the asset price doubles to $10,000. Thus, when interest rates are low we can expect very wide swings in asset prices. The figure illustrates:

The Gordon model is simple, of course, but adding other factors such as possible variations in dividend growth rates tends to reinforce the conclusion. At low interest rates, for example, even small variations in dividend growth rates will also generate large swings in asset prices.

The lesson is that QE doesn’t have to generate bubbles to generate wide swings in asset prices it just has to lower interest rates.

Here’s a bond convexity primer for anyone interested in a bit more;

http://blogs.cfainstitute.org/investor/2013/06/25/a-bond-convexity-primer/

That is a horrible description of convexity. What is says is accurate enough, but it misses the point.

Optionality drives convexity. Options are convex. And giving someone an option makes and an investor negatively convex. e.g. the borrow might have the option to refinance their debt if rates fall, then the bond has less positive upside if rates fall than its downside if rates rise.

Credit risk can also be modeled as an option (and gives negative convexity). The company has the option to stop paying its debts. There may be serious consequences for doing so, but it is an option nonetheless.

A bit harsh. The explanation is a beginner intro to the term as applied to bond markets and not a more advanced discussion of optionality. As it stands, noncallable bonds will exhibit positive convexity in certain rate environments, and that is the topic the explanation is discussing. Yes even for non-callable bonds there is a natural “option” of refinancing but that’s not the gist of convexity.

I’m confused about how the notion “interest rates are” became some fringe Austrian theory. Oh well, so be it.

Isn’t there generally an assumption that unusually high or low interest rates will revert to the man in the long run? That is, the prices of assets that pay out over a long period of time should be much more stable than predicted by this model, as we see for 30-year treasury bonds, for example.

How long is long? Long-term Treasury yields were 3% in 1950, 15% in 1981, and 3% again in 2012. One cycle- 31 years up, 31 years down.

There’s a great Friedman quote on this…

“A fourth effect, when and if it becomes operative, will go even farther, and definitely mean that a higher rate of monetary expansion will correspond to a higher, not lower, level of interest rates than would otherwise have prevailed. Let the higher rate of monetary growth produce rising prices, and let the public come to expect that prices will continue to rise. Borrowers will then be willing to pay and lenders will then demand higher interest rates-as Irving Fisher pointed out decades ago. This price expectation effect is slow to develop and also slow to disappear. Fisher estimated that it took several decades for a full ad- justment and more recent work is consistent with his estimates.”

That’s the inverse of what they’ve been doing since 1980, or…

“Paradoxically, the monetary authority could assure low nominal rates of interest-but to do so it would have to start out in what seems like the opposite direction, by engaging in a deflationary monetary policy.”

Oh, Alex, you are comparing a 10% change in rates to a 50% change in rates and calling them both the same thing!

The right comparison for the 10 to 9 move is a 2 to 1.8 move, not a 2 to 1 move.

But in the real world interest rates aren’t changed on a total percentage basis, they’re adjusted via BPS. I haven’t seen any real world examples of the Fed weighting interest rate hikes or cuts at the lower end where each BPS move represents a greater proportional change. Various 50 BPS moves seem to be treated the same no matter where they are on the scale. So, IMO, Alex’s point is still relevant for practical reasons.

I figured someone would say this but I don’t think it is valid. We expect fewer 50 point shifts in stocks when stock prices are 100 than when they are 1000 to keep percentage changes, i.e. returns, similar but the same is not true for bonds. Bond variances do not decrease anywhere near what equal percentage changes would imply. Interest rates are somewhat more volatile at higher rates but not markedly so for U.S. levels.

Angus seems to be defining an equivalent interest rate move by what it would be to result in an equivalent asset-price move. By all means, tell The Fed about this.

Are we sure that, for example a 0.1% change in interest rates is as likely in a low rate environment as it is in a high rate environment? If instead it is the volatility of bond prices that is the same in these two environments, that would imply that the volatility of rates decreases as rates go down. It’s an empirical question though.

Regardless, asset prices ARE more uncertain in a low-rate environment for the simple reason that far in the future earnings comprise a greater share of the valuation, and far in the future earnings themselves are more uncertain. If interest rates were at 20%, next year’s earnings would be a huge part of the valuation, and we have a decent estimate of next year’s earnings.

Interesting. How does this fact help explain the stock market bubble, which occurred when interest rates were much higher, or the housing bubble, when ditto?

After The Fed jacked up interest rates the assets crashed. I am not implying causation…but…

But how were those interest rates

changingat those times?the model is slightly more complicated than what is shown here.

price = D / (k-g)

D = dividends — although Free Cash flow and earnings can also be used

k = risk free rate + risk premium

g = expected rate of dividend growth.

The model fails if dividends are zero or earnings / free cash flow are negative. But there are tweaks for that.

The model also becomes unstable when g approaches k. And fails when g exceeds k.

but we can have “rational” valuations with a 30x P/E ratio for certain values of k and g, and extreme price sensitivity to changes in the assumptions.

You’re missing the original point. Alex is showing that even with definitive cash flows, asset price variation will be high when rates are low. They are more volatile when the cash flows are more complicated or involve options.

The stock market bubble and the housing market bubble did not involve a lot of asset price volatility. Both just involved a multi-year run up in the price of stocks and houses.

Perhaps I am missing something.

What factors generate the “interest rate(s)?”

What conditions generate those factors?

The supply of money and the demand for money…

Isn’t it the other way around? Asset prices determine the yield, a very high asset price means that the yield is smaller.

The only way the causation could work the other way is a price fixing scheme by someone who would buy up a very large quantity, large enough to set the asset price. If the price dropped, the yield would increase, so the buyer would increase the pace and quantity of purchases to maintain the high price and low yield. The ‘dividend’, or investment return would be from either the increasing asset value, or the use of such a valuable asset as collateral for something else. The yield is a pittance.

The artificial high price of the asset would collapse if there was any hint that the price fixer would stop or slow down it’s purchases. A cascading collapse because of secondary unwinds. There is no return holding the bonds, only returns from owning them as collateral. And if you bought high you would dump them quickly before you lost money.

“The only way the causation could work the other way is a price fixing scheme by someone who would buy up a very large quantity, large enough to set the asset price.”

We won’t name names since we don’t want to embarrass them…

The interest rate environment does not determine the yield of any particular asset but it does restrict the bound of that yield.

No. Price and yield are simultaneously determined. If you hold a fixed income asset and interest rates fall, the value of that asset rises because it is paying a relatively higher yield than it was before rates fell compared to the treasuries that now pay less. If that’s too hard to grok, think about it in terms of opportunity cost. The sensitivity of asset prices to rate changes is known as the duration.

There are actually many different “yields” depending on optionality and your investment objective.

A simple, but important point.

Kocherlakota notes it as well in an April speech on low rates. The quote is as follows:

“The second consequence of low real interest rates is that asset returns should be expected to be highly volatile. When the real interest rate is very high, only the near term matters to investors……But when the real interest rate is unusually low, then an asset’s price will become correspondingly sensitive to information about dividends or risk premiums in what might seem like the distant future.”

http://www.minneapolisfed.org/news_events/pres/speech_display.cfm?id=5090

At low interest rates, for example, even small variations in dividend growth rates will also generate large swings in asset prices.

I think this is the much better argument because we don’t have to argue about whether a 10%-9% change is equivalent to a 2%-1% change or a 2%-1.8% change. That being said, is asset price volatility an acceptable price to pay for really cheap capital? Remember, economists have always been in love with lower real interest rates. This been has the ostensible goal for conservative policy for years. It makes capital investment by business cheaper, thus encouraging capital investment and, theoretically, growth.

Fixed income math is all based around (1+y). There is nothing magical about y = 0. Heck, it rates can even go negative.

So a rate fall from 10% to 9% is roughly equivalent to a rate fall from 2% to 1% as it is from 1% to 0%.

It is only when we extend expected cash flows out to infinity do rates have to stay positive.

Everything makes sense to me except for the connection from the stuff above the last sentence to the last sentence. Mathematically, the stuff above the last sentence says the (partial) derivative of price with respect to interest rate is larger in magnitude when the interest rate is lower. That is perfectly fine. The last sentence, on the other hand, says, we can generate a large standard deviation (I assume that is what mean by “swings”) of price by lowering the interest rate (and keep the rate fixed). I don’t see the connection here. Building on everything above the last sentence, we can only say, to generate a large standard deviation of price, we can change the interest rate (either increase or decrease) – but we’d better do it in low interest rate levels. Am I missing something?

QE has arguably pushed down rates and generated large asset price moves to the upside. That is not necessarily a “bubble”, but it’s a large move. Going forward, to the extent that low rates are/were a result of the Fed, then the Fed can be further blamed for future large asset swings if rates move back up off a low base. This is of course part of what we saw in the EM selloff on Fed tapering talk. If 1) asset price moves are larger when rates are lowered / low and 2) QE lowers rates then 3) QE results in larger asset price moves and 4) larger asset price moves (to the upside) aren’t necessarily “bubbles”.

Not sure if that is precisely what you were asking but this makes a good deal of sense to me.

For the stock market changes in rates impact the market through PE changes.

If the relationship between rates and the PE is one-to-one the market impact depends on where the PE is. If the PE is 20 a 100 basis point rise in rates would drop it to 19, a 5% drop. But it the PE was 10 a fail to 9 would be a 10% drop.

The one-to -one relationship between bond yields and the PE is a good rough and ready rule of thumb.

A P/E of 20 is an earnings yield (E/P) of 5%. A 1% rise in earnings yeild is 16.7 P/E.

A P/E of 10 is an earnings yield of 10%. A 1% rise in earnings yeld is 9.1 P/E.

dP/dY / P — the price elaticity with respect to yield increases as Y gets smaller.

The analysis is flawed. First, one should look at the % change in interest rate. A 1% increase when rates are 1% is huge. So redraw the graph and use log scale. Second, that assumption is that the “g” in Gordon model is just staying constant while rates are changing makes no sense. Finally, the numerator is changing as well as rates change. For example, if rates are increasing because of higher inflation or better economy, then both “g” and dividends will increase. There is no empirical evidence that markets are more volatile at low levels of interest rates. In fact, the evidence is the opposite. Interest rates are more volatile when they are relatively high leading to more volatility in asset prices. However, if one were to apply this to bond prices, then it is true that bonds become more volatile at lower levels of interest rates because bonds durations increase.

The duration of a stock is the inverse of its dividend yield, so yeah, the log function will show ‘nearly’ a straight line. 1/D is a rectangular hyperbola.

Alex presented an example of one of the simplest equities possible for illustration purposes. He isn’t claiming that the DDM is the best or only model for calculating the value of an equity asset or its duration.

Equities have durations of 20 to 100 years which tells you a lot about interest rate risk. Interest rate risk in an equity instrument is beat thought of in terms of opportunity cost. When interest rates rise, the value of this expected cash flow will fall because other assets with similar risk characteristics are now paying higher yields.

Not sure how significant it is to your point(s) here (if at all), but: don’t *proportionally* similar changes in interest rates (i.e. 1% down .1% vs. 4% down .4%) yield proportionally similar changes in asset values?

If we assume that volatility the rate of change in interest rate is relatively constant (which is roughly supported by data), then the graph should be in log scale. Just redraw the chart and use a log scale for the horizontal axis. You will get an almost straight line.

Interest rate volatility is pretty much independent of the level of the rate.

Any thoughts as to what has occurred in Japan? Low and falling interest rates for many years…..associated with declining asset prices during the same period of time?

Good point to illustrate convexity. But can you apply this analysis to defined benefit pension liabilities? How about to a rational saver who is imputing his own retirement future consumption “liability”? Might this explain some of the depressed investment and consumption today?

I can tell from the linked paper that you know this, but I think the post is misleading. The Gordon Growth model uses a discount rate comparable to required stock return. The post reads like it is some long-term interest rate (something like 2%).

To use another simple model, the CAPM:

R = rf + B*MRP [where B is the market beta]

Most of the discount rate is made up of the market risk premium. Changes in the risk free rate are still important, but 10% to 2% change in the discount rate seems quite unrealistic.

The numbers chosen and the words used just seemed misleading on this point. Though perhaps there was nothing actually wrong written.

I initially had these same thoughts – some fast and loose playing with the numbers. But the example is apt. Alex presented a very simple example of how price volatility can be higher when rates are lower for the most basic financial instrument. He isn’t saying that this is how all or most or some assets behave. I think it is more correct to view this post as a counterexample to conventional wisdom than an argument on its own merits that he must defend. It also demonstrates the effect of margin or spread compression when rates are low.

Another reason why ZLB is problematic, the Fed should be keeping us out of ZLB. But this says ZLB maximizes Fed influence. I don’t know that the Fed is immune from incentives.

Of course if you expect 10% deflation that 1% interest rate isn’t very attractive…

This is cute, but what is the point in conjecturing with a model when you can simply look at history?

Were markets more volatile when interest rates were lower? The data is out there. Do the work!

Dividend models of stock valuation (for example) have been some of the worst at predicting prices, so it might not be shocking if other factors impacted markets more. The Great Depression, for example, was a time of low interest rates. Weren’t markets fairly placid (post crash)?

Pre Crash, rates were much higher. That didn’t seem to stem the volatility.

This is where Scott Sumner smacks you in the head and says, “Never reason from a price change.”

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