With historically low interest rates it’s hard to miss a common refrain from the financial pundits, that bonds and/or stocks are over-valued and/or that they are ‘in a bubble’. Yes, it is true that low interest rates have driven up the price of both stocks and bonds relative to average historical levels. It’s easy to find those who argue stocks are more over-valued than bonds or vice versa. Today, I won’t argue which asset class is relatively more expensive. What I will do introduce a new way of thinking about the risk associated with stocks and bonds given price levels (and interest rates).

Let’s start with a review of bonds and the interest rate risk measure known as ‘duration’.

Bond Duration

If I hold a bond paying an interest rate of x %, than ignoring default risk and inflation risk for the moment, we expect that we will receive our money back with interest at the term of the bond. Interest rate risk is the risk an investor faces in reduction in the value of the bond if interest rates should rise. A convenient measure called ‘duration’ is used to express the percentage (%) change in the value of the bond for each 1% change in interest rates. For example, if we hold a collection of bonds with an average duration of 7, we know that the value of the bond portfolio will drop 7% for each 1% in interest rate increases. Sounds pretty bad, right? You may even here an industry pundit claim something such as: ‘bonds are highly over-valued and if interest rates rise they will go down, therefore stocks are a better bet right now’.

Extending Bond Duration to Stocks

For stocks, a key measure of value is the price-to-earnings ratio (or PE). We know that we are near historical highs for PE. When we look at trailing earnings, the PE ratio for S&P 500 companies is about 25. Now the math involved in determining what is a ‘fair’ price for stocks needs to account for a few more variables than bonds. Stock shareholders have a right to future earnings. Return comes via dividends and/or capital appreciation. Using a cash flow based model, the price we are willing to pay should be a function of some required return and our expectation of growth in the economy. One simple model suggests that the required return should be the return to bonds plus a set of premiums for taking on extra risk. The return of the bonds themselves ought to also reflect the underlying inflation rates and expectations. Using a simple earning model, we should be willing to pay a PE equal to 1 / (r-g), where ‘r’ is the required return and ‘g’ is our expectation for earnings growth. Are stocks overpriced?, well if we assume r and g remain steady at 6% and 2% respectively, the market is ‘fairly’ valued with a PE of 25. (1/(.06 – .02)) = 25 . But, this doesn’t give us the complete picture. We should also ask how sensitive are stock prices in the face of interest rate and growth rate assumptions.

Using the equation above, we can see the stock investor has multiple risks. For starters, if interest rates rise then ‘r’ goes up and the justified PE should go down. To get a roughly equivalent measure of bond duration, we need to know how much the PE will change with 1% rise in interest rates. PE justified with r = .07 and g = .02 is 20. This represents a 25% drop in stock prices if nothing changes except for interest rates! So this would be equivalent to a bond duration of 25. Now of course one problem with this view is that in all likelihood the growth rate assumption could change with interest rates. Generally we expect economic growth and higher rates of return to hand in hand. Let’s define this measure as ‘stock market interest duration’.

What about changes in PE based on changes in growth expectations? Let’s call that this ‘stock market growth duration’, the change in price based on changes in expectations for growth. OK going back to our example if our growth assumption goes from 2% to 1%, our new PE would be: 1/(.06 – .01) = 20. Once again, this represents a 25% reduction in the justified price, given a reduced expectation of growth.

From the discussion above, we can see from the simple earnings growth model that the stock market price is not immune to changes in interest rate and growth. In fact, when PE ratios are elevated given low required return expectations, the change in price is more sensitive to changes in interest and growth than would otherwise be the case. For example, if the starting justified PE ratio was 20 based on an assumed required return of 7% and assumed growth of 2%, if the required return rose 1% due to interest rate increases, the new justified PE would be 16.67. Prices would fall 16.67% from a PE of 20 to a PE of 16.67.

Summary

In this article, we interested the concept of stock market interest duration and growth duration as an analogy to bonds to get an appreciation for how the stock market prices could change with changes in required return (driven by interest rate increases) or growth changes (for example a decline caused by economic slowdown). From the discussion, we showed the transmission mechanism from bond interest rates to increases of return to stocks and hypothetical declines in stocks. As importantly, we showed that that the sensitivity in stock prices to changes in interest rates increases or growth declines on a percentage basis are higher when stock price is high as a starting point. . For this reason, it is possible to conclude that stocks are fairly valued but, at the same time are relatively more risky as measured by sensitivity to changes in interest rates and growth rates.