Ha, PE 37 - it seemed not that long ago when everybody was crying the sky will fall when the PE was in the 20-25 range (me included) - just crazy that the sentiment now is that it can't stop going up.

Intrinsic value of stocks is very tightly dependent on risk free rate.

https://www.investopedia.com/articles/basics/12/intrinsic-value.asp

V_{0} = BV_{0} + Σ RI_{t} / (1+r)^{t}

Where

BV_{0} = Current book value of equity

RI_{t} = Residual income at time period t

r = Cost of equity

This last thing, r = Cost of Funding => directly tied to the interest rate. I personally write code for a system where we have a cost of funding/equity curve that goes from 4% to 12%. With the current interest rate, it will generally be much closer to 4% than 12%.

I just did a quick spreadsheet with a company with book value = $100, yearly earning = $100, that folds after year 10. With 12% cost of funding, the intrinsic value came to be $665, and with 4%, it came to $747. i.e. a 12.33% increase in this simplistic model. In real world, the models get screwy close to 0% (far out cashflow is no longer discounted, and hence intrinsic value starts jumping much higher).

So the PE25 -> PE37 jump is not at all weird based just on the interest rates, assuming that you also assume that the interest rates are not going up (which seems to be a consensus the market has baked in).

There are other models of intrinsic value calculations. All of them are similarly affected by the interest rate, because all of them have to discount some future cashflow/earning.

I agree this is the math, but the thing not captured in this financial model is that when one buys a stock or a bond at a time when the risk free rate is <1%, there is a lopsided risk of loss due to either interest rate increases or earnings decreases. That is, as the risk-free rate goes lower and lower, the riskiness of risk assets increases. To illustrate:

When the risk-free rate is 6%, a risk model used to assess a bond might assign an equal probability to the chance that the rate drops to 2% as it does to the rate rising to 10%. Both are clearly possible. When the risk-free rate is 1%, however, the odds of it dropping by 4% (to negative 3%) is much less than the chance of it rising 4%. The risk distribution would need to be skewed to account for the near-impossibility of negative 3% rates. So as we approach zero the probability of a loss on the bond, or a stock, due to interest rate changes becomes relatively larger than the probability of a gain.

With regard to the equation, the (1+r)^t denominator used in both the residual income and DCF models produces the following results for 10 years at various risk free rates:

Risk Free Rate Denominator

15% 4.05

10% 2.59

5% 1.63

1% 1.10

0% 1

Thus, at around 0% the security is valued as the sum of residual incomes or cash flows. If loans were free and inflation was nothing, an asset would be worth exactly its future cash flows and a dollar today would be worth the same as a dollar in ten years.

Now what happens if we use a 15 year timeframe, where we have 50% more cash flows?

Risk Free Rate Denominator

15% 8.14 <100.9% higher than 10 year timeframe

10% 4.18 <61.4% higher

5% 2.08 <27.6% higher

1% 1.16 <5.5% higher

0% 1 <equal to 10y timeframe

So the higher risk-free rates go, the more sensitive these equations are to the timeframe we select for expected cashflows or RI. It's not linear like our cash flow or RI assumptions. This makes sense because carrying a loan at the risk-free rate or passing up the risk-free rate of interest has a cost that compounds and eventually overwhelms the value of those future cash flows. Another way to look at it: inflation depletes the value of those future cash flows in terms of today's dollars. At a 15% risk free rate, we get 50% more cash flows but have a 100% larger denominator! It would seem the equation was made for a world between 5 and 10%, where the increase in the denominator and numerator is more proportional.

If we chose 5 years or 2 years as the timeframe, we would get lower denominators for the high risk-free rates. But... at near zero percent the denominator would still be one in any timeframe. So now that rates are near zero, the price of an asset is the sum of its future cash flows (or BV plus that in the RI approach). Are assets worth 5 years' cash flows, 10 years', 11 years', or 20 years'?

Consider a company that earns $10 a year. At a 0% risk-free rate, its DCF value is literally the sum of its cash flows because the denominator is one. If you pick 10 years as your timeframe, the value is $100. If you pick 9 years, the value is $90. If you pick 40 years, the value is $400. Obviously, we're just picking timeframes for how long we expect the earnings to last and getting radically different answers. Our estimates of the company's earnings 10 years out are almost certainly wrong, and the risk-free rate may be higher at that time, but there is no discounting to address that uncertainty. There is also no probabilistic skew to what the expected risk free rate will be in each of the future years, unless you do some fancy guesswork. At zero, the $10 earned in every year X adds $10 to the value of the company, period.

DCF value of company earning $10/year

Risk Free Rate 15y 10y 5y

15% 58.43 50.19 33.52

10% 76.06 61.44 37.91

5% 103.79 77.22 43.29

1% 138.65 94.71 48.53

0% 150 100 50

Conclusions:

1) As interest rates get really low, stocks become more sensitive to projections of far-off earnings. I.e. At zero, if the earnings only last 10y instead of 15y, the stock needs to drop 33%. At 15% the stock would only drop 14%. This represents higher "bad news" risk for investors.

2) As interest rates get really low, the phenomenon of convexity becomes a bigger danger for stock investors. Look at the 10y DCF numbers. If rates go from 10% to 15%, the stock drops 18.3%, but the same size change going from 0% to 5% would cause the stock to drop 22.8%.

3) In theory, these higher risks to investors should be part of the discount rate. DCF and RI models assume these risks are reflected in the risk free rate and, by extension, inflation. To avoid spiraling deeper and deeper into risk as rates go down, we would need some sort of counterbalance. The risk-free rate plus some sliding-scale risk premium would add up to our "willingness to pay" rate, so that our r never approaches zero and so that very distant cash flows are adequately discounted when rates are low.

/dissertation

#and sorry for the time suck