The Shiller P/E (CAPE) Ratio: Measuring Long-Run Market Valuation

Alexandre LANGEVIN

In this article, Alexandre LANGEVIN (ESSEC Business School, Global Bachelor in Business Administration (BBA), 2022-2026) explains the Shiller P/E ratio, also known as the CAPE ratio: a valuation tool that adjusts for the business cycle to give a more reliable picture of whether equity markets are cheap or expensive.

Introduction

Every investor knows the price-to-earnings (P/E) ratio: divide the current market price by earnings per share and you get a simple measure of how much the market is paying for each dollar of profit. It is one of the most widely quoted metrics in equity analysis. But it has a structural flaw: earnings are cyclical. In a recession, they collapse, making the P/E look artificially inflated even when prices have barely moved. In a boom, they surge, making markets appear cheap when they may not be. A single year of earnings is a poor basis for a long-term valuation judgment.

Robert Shiller, a Yale professor and 2013 Nobel laureate in economics, proposed a simple fix. His ratio replaces one year of earnings with the average of the past ten years, adjusted for inflation. The result is a smoother, more stable measure of valuation that filters out the noise of the business cycle and allows for meaningful comparisons across time.

The Problem with Standard P/E

Consider the S&P 500 in 2009, shortly after the financial crisis. Prices had fallen sharply, but earnings had fallen even further, with many companies reporting losses. Standard P/E spiked above 100 at certain points, not because markets were expensive, but because the denominator had collapsed. An investor reading that number at face value might have concluded the market was dangerously overvalued, when it was near a generational buying opportunity.

The opposite problem occurs at cycle peaks. Strong earnings in boom years compress P/E ratios, making markets look reasonable just before a downturn. Standard P/E captures both price and the cyclical position of earnings simultaneously, making it hard to separate valuation from timing.

The CAPE Ratio: Construction and Formula

Shiller’s solution is to replace single-year earnings with the average of real earnings over the previous ten years. A ten-year window spans a full business cycle, smoothing out both recessions and booms. The formula is:

CAPE ratio formula

where P is the current market price, Et are reported earnings in year t, CPI0 is the current price index, and CPIt is the price index in year t. The inflation adjustment ensures that past earnings are expressed in today’s dollars, making them directly comparable to recent figures.

In the Excel model, each annual earnings figure is the average of the 12 monthly observations in Shiller’s dataset. Shiller himself constructs monthly earnings by interpolating S&P four-quarter totals, so the monthly series is a smooth continuous estimate rather than actual reported monthly results. The current S&P 500 price used is the April 9, 2026 closing price of $6,824.66, sourced from Yahoo Finance. The CPI reference is the February 2026 release from the U.S. Bureau of Labor Statistics.

Historical Record and Market Signals

Shiller’s dataset goes back to 1871, giving the ratio an exceptionally long historical record. The average CAPE over that full period is approximately 17.7 and the median around 16.6. These serve as rough benchmarks: readings significantly above the average suggest the market is expensive relative to long-run earnings capacity, while readings well below suggest the opposite.

The ratio’s most cited applications came before two of the largest crashes of the modern era. In December 1999, at the peak of the dot-com bubble, the S&P 500 CAPE reached 44.2, more than double its historical average. Shiller published Irrational Exuberance that same year, arguing on the basis of CAPE that US equities were severely overvalued. The S&P 500 subsequently fell by nearly 50% over the following two years. In August 2007, CAPE rose above 26 before the financial crisis and another major decline.

At the other extreme, CAPE dropped to around 8.5 in August 1982, one of its lowest post-war readings, preceding one of the strongest bull markets in US history. As of April 9, 2026, our model gives a CAPE of approximately 38.8, well above the historical average.

Figure 1. CAPE ratio at key historical market turning points (S&P 500, selected monthly readings). Source: Robert J. Shiller, econ.yale.edu; computation by the author.
CAPE historical chart
Source: computation by the author.

Excel Model

The Excel model below computes the CAPE ratio from Shiller’s raw data. It contains four sheets: a source data sheet copied directly from Shiller’s dataset, a CAPE Calculator that pulls ten-year annual averages and applies the inflation adjustment, a Historical Context sheet with key turning points, and a Read Me. The starting year of the ten-year window is adjustable, and the model updates automatically when price or CPI inputs are changed.

Figure 2. CAPE Calculator: ten-year window of inflation-adjusted earnings and resulting CAPE ratio.
CAPE calculator Excel screenshot
Source: computation by the author.

Download the Excel file

Interpretation and Limitations

What CAPE tells you. Shiller’s own research found a strong negative relationship between starting CAPE and subsequent 10-year real returns for the S&P 500: high CAPE tends to precede lower decade-long returns, and low CAPE tends to precede higher ones. The relationship is not mechanical and does not predict timing, but it is one of the more robust long-run return predictors in the academic literature.

The interest rate objection. The most common criticism is that CAPE ignores the level of interest rates. When rates are structurally low, investors rationally accept higher valuations because the alternatives offer little return. Some analysts argue that elevated CAPE readings since 2010 partly reflect lower rates rather than pure overvaluation. This debate is unresolved.

Accounting changes. Reporting standards for earnings have evolved significantly since the 1870s, particularly around goodwill and write-offs. Some researchers argue that modern reported earnings are not strictly comparable to historical figures, making century-long CAPE comparisons imperfect.

Not a timing tool. Investors who sold equities in 1996 because CAPE was already above its long-run average missed four more years of exceptional gains before the dot-com peak. CAPE is a signal about long-run expected returns, not a predictor of short-term price moves.

Why should I be interested in this post?

Valuation metrics appear in equity research, asset allocation decisions at investment managers, and macro discussions at private banks. The CAPE ratio is referenced in strategy notes, central bank research, and academic papers on return predictability. Understanding what it measures, how it is built, and what its limits are is practical knowledge for anyone working in equities or asset management — and one of the cleaner examples of how academic research translates directly into a practitioner tool.

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Useful resources

Academic research

Campbell, J.Y. and Shiller, R.J. (1988) Stock Prices, Earnings, and Expected Dividends, Journal of Finance, 43(3), 661-676. Available at scholar.harvard.edu.

Bunn, O. and Shiller, R.J. (2014) Changing Times, Changing Values: A Historical Analysis of Sectors within the US Stock Market 1872-2013, NBER Working Paper No. 20370. Available at nber.org.

Data sources

Shiller, R.J. Online Data, Yale University. S&P 500 price, earnings, CPI, and CAPE data from 1871 to present.

S&P 500 current price: Yahoo Finance.

CPI reference: U.S. Bureau of Labor Statistics, Consumer Price Index release.

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About the author

The article was written in April 2026 by Alexandre LANGEVIN (ESSEC Business School, Global Bachelor in Business Administration (BBA), 2022-2026).

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Duration and Convexity: Measuring Bond Price Sensitivity to Interest Rates

Alexandre LANGEVIN

In this article, Alexandre LANGEVIN (ESSEC Business School, Global Bachelor in Business Administration (BBA), 2022-2026) explains how duration and convexity allow investors and risk managers to measure and anticipate how bond prices react to changes in interest rates, and why the distinction between the two matters in practice.

Introduction

Bond markets sit at the heart of the global financial system, with outstanding fixed income markets exceeding $145 trillion worldwide (SIFMA, 2025). Yet one of the most fundamental challenges in fixed-income investing is deceptively simple to state: when interest rates move, bond prices move in the opposite direction. The harder question is by how much, and how accurately can we predict it?

Two risk measures answer that question: duration and convexity. Duration provides a first-order, linear approximation of price sensitivity to yield changes. Convexity accounts for the curvature in the price-yield relationship, improving accuracy when rate moves are large. Together, they form the analytical backbone of fixed-income risk management, from portfolio construction to regulatory capital requirements at banks.

Bond Pricing: The Starting Point

The price of a fixed-rate bond is the present value of all its future cash flows: periodic coupon payments and repayment of the face value at maturity, discounted at the bond’s yield-to-maturity. The yield-to-maturity (YTM) is the single discount rate that equates the present value of all cash flows to the current market price. With nominal value N, annual coupon rate c, maturity T, and YTM r, the bond price P is:

Bond price formula

As r rises, each discount factor grows, reducing the present value of every future cash flow and pushing the total price down. A useful benchmark: when the coupon rate equals the YTM, the bond prices at par. When the coupon rate exceeds the YTM, the bond trades above par — this is a premium bond, identifiable directly from the parameters before computing anything.

Duration

Macaulay Duration

Duration was formalized by Frederick Macaulay in 1938. Macaulay duration is the weighted average of the times at which a bond pays its cash flows, where each weight is the share of total present value arriving at that date. It answers: on average, how long does an investor wait to receive their money back?

A zero-coupon bond has a duration equal to its maturity, since all cash flow arrives at the end. A coupon bond always has a shorter duration than its maturity, because intermediate coupon payments pull the weighted average forward. For a given maturity, a higher coupon rate or a higher yield both reduce duration.

Modified Duration

Modified duration is Macaulay duration adjusted by dividing by (1 + r). It has a direct use as a price sensitivity measure: a bond’s percentage price change is approximately equal to minus its modified duration multiplied by the change in yield.

Modified duration definition

Duration price approximation

If a bond has a modified duration of 6, a 1% rise in yield reduces its price by roughly 6%. This is practical and widely used, but it is only a linear approximation and loses accuracy as yield changes grow larger.

In practice, traders and risk managers also use DV01 (Dollar Value of a Basis Point): the monetary price change for a 1 basis point (0.01%) shift in yield, equal to D* × P × 0.0001. DV01 is the standard unit for setting position limits on bond desks and for computing interest rate risk under Basel III.

Convexity

Why Duration Is Not Enough

The price-yield relationship of a bond is not a straight line — it is a convex curve. Duration approximates this curve with a tangent line at the current yield. For small yield moves this works reasonably well, but for larger moves the error accumulates in a predictable direction: duration always underestimates the true price. When rates fall, the actual price gain is larger than duration predicts. When rates rise, the actual price loss is smaller. This asymmetry, always working in the bondholder’s favor, is the essence of convexity.

The Convexity Correction

Convexity is the second derivative of the bond price with respect to the yield, divided by the price. Adding it as a second-order correction gives a substantially more accurate estimate:

Duration and convexity price approximation

The convexity term is always positive regardless of yield direction, which creates the favorable asymmetry: it always adds to the price estimate, making gains larger and losses smaller than the duration-only figure.

A Numerical Illustration

Consider a 7-year bond with a face value of $1,000, an annual coupon rate of 4%, and a current YTM of 3.5%. Since the coupon exceeds the yield, this is a premium bond. The Excel model gives a bond price of $1,030.57, a Macaulay duration of 6.26 years, a modified duration of 6.04, and a convexity of 44.91.

Figure 1. Cash Flow Analysis table and key results (N = $1,000, c = 4%, T = 7 years, r₀ = 3.5%).
Excel bond calculator screenshot
Source: computation by the author.

Now suppose the yield rises 2 percentage points, from 3.5% to 5.5%. The exact bond price falls to $914.76, a decline of 11.24%. The duration approximation predicts $906.00, overestimating the loss by nearly $9. The duration-convexity approximation gives $915.26, bringing the error down to under $0.50. Figure 2 shows this comparison across the full yield range.

Figure 2. Bond price as a function of YTM (N = $1,000, c = 4%, T = 7 years, r₀ = 3.5%): exact price (blue), duration approximation (red), duration + convexity approximation (green).
Bond price vs yield chart T=7
Source: computation by the author.

Excel Model

The Excel file below replicates these calculations for any bond. It contains a Cash Flow Analysis sheet computing present value, duration contribution, and convexity contribution for each year; a Price-Yield Chart comparing all three methods; and a Read Me tab. All inputs are editable in yellow cells, and the model supports maturities from 1 to 20 years.

Download the Excel file

A Note on Long-Duration Bonds

The limitations of the duration approximation become more pronounced for longer-maturity bonds. A 20-year bond with the same 4% coupon carries a modified duration of roughly 13-14 years. Applied to a large yield shift, the linear formula can produce a negative estimated price, because the correction term eventually exceeds the bond’s starting price. This does not happen in reality. It is simply a demonstration of how far the linear approximation strays when pushed outside its valid range. The duration-convexity approximation remains far better behaved across the same range. For long-duration bonds in volatile rate environments, accounting for convexity is not optional.

Figure 3. Price-Yield chart for a 20-year bond: the duration approximation turns negative at high yields while the convexity approximation tracks the exact price.
Bond price vs yield T=20
Source: computation by the author.

Applications in Fixed-Income Risk Management

Portfolio immunization. A portfolio manager protecting a bond portfolio against parallel rate shifts will match portfolio duration to the investment horizon. Price losses from rising rates are offset by higher reinvestment income on coupons, leaving total return roughly unchanged.

Risk limits and regulatory capital. Banks use DV01 to set position limits for fixed-income traders and to estimate interest rate risk under Basel III. A trader might be authorized to hold a maximum DV01 of $50,000, meaning no more than $50,000 of profit or loss per basis point move.

Convexity as a source of value. In volatile rate environments, investors seek bonds with high convexity. The asymmetric payoff profile — larger gains than losses for equal rate moves in either direction — is a property the market prices accordingly. Long-dated government bonds are a typical example.

Limitations. Both measures assume a parallel shift in the yield curve. In practice, the curve can steepen, flatten, or twist. For more granular risk measurement, practitioners use key rate durations, which isolate sensitivity at individual maturities. Duration and convexity remain the essential starting point.

Why should I be interested in this post?

Duration and convexity appear in fixed-income interviews, in the CFA curriculum, and in the daily work of bond traders and risk officers. Whether you are targeting investment banking, asset management, or financial risk management, these are concepts you will encounter early. The distinction between linear and non-linear sensitivity also recurs throughout quantitative finance, from option Greeks to credit portfolio models. Being able to work through it from first principles and build a functioning model is a meaningful differentiator at the MSc Finance level.

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Useful resources

Academic research

SIFMA (2025) Capital Markets Fact Book 2025. Available at sifma.org.

Cerovic, S., Pepic, M., Cerovic, S. and Cerovic, N. (2014) Duration and Convexity of Bonds, Singidunum Journal of Applied Sciences, 11(1), 52-66. Available at journal.singidunum.ac.rs.

Winkel, M. (2011) Duration, Convexity and Immunisation, Lecture Notes, Department of Statistics, University of Oxford. Available at stats.ox.ac.uk.

Crack, T.F. and Nawalkha, S.K. (2000) Common Misunderstandings Concerning Duration and Convexity, Working Paper. Available at ssrn.com.

Jeffrey, A. (2000) Duration, Convexity and Higher Order Hedging (Revisited), Yale International Center for Finance, Working Paper No. 00-22. Available at ssrn.com.

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About the author

The article was written in April 2026 by Alexandre LANGEVIN (ESSEC Business School, Global Bachelor in Business Administration (BBA), 2022-2026).

   ▶ Discover all articles by Alexandre LANGEVIN.