PORT 1 – Economics and Investment Markets

Every financial instrument is ultimately a claim on the real economy. A bond is a claim on the future tax receipts of a government or on the future operating cash flows of a company. A share is a claim on whatever is left after everyone else has been paid. A commercial property is a claim on rents that tenants can only pay if their own businesses are trading. Because the claims are real, the economy that generates them has to show up somewhere in their prices, and this reading is about exactly where.

The place it shows up is the present value model. Write the price of any financial asset i today as the sum of its expected future cash flows discounted at a rate built from three separate pieces:

Read the symbols carefully, because the whole reading depends on keeping them apart:

• CFⁱ_t+s is the uncertain nominal cash flow paid s periods from now, and E_t[·] is the expectation taken using only the information available today.
• l_t,s is the real default-free rate: the yield today on an investment that carries no default risk and no inflation risk, paying one unit of purchasing power s periods ahead. Think of the yield on a short-dated inflation-linked government bond.
• θ_t,s is the expected inflation rate between now and t + s. It is compensation for expected loss of purchasing power, not compensation for risk.
• ρⁱ_t,s is the risk premium demanded for the uncertainty in the cash flows of asset i. The superscript matters: this piece is asset specific, and the differences in it are largely what separate one asset class from another.

The additive form of the discount rate is an approximation of the exact multiplicative expression, which would carry interaction terms between the components. For the purposes of this reading the approximation is used throughout:

The three channels, and why the list is exhaustive

The first learning outcome of this reading is a statement about what economic analysis can and cannot do. An economic factor changes the market value of an asset only if it works through one or more of the following: the level of default-free interest rates across maturities, the timing or the magnitude of expected cash flows, or a risk premium. There is no fourth route. A piece of macroeconomic news that leaves all three untouched leaves prices untouched.

This is more useful than it sounds. It converts a vague question, such as what an unexpected acceleration in growth does to equity values, into three separate and tractable questions. Does it raise expected corporate cash flows? Almost certainly yes. Does it raise the real default-free rate? Also yes, for reasons developed in the sections that follow. Does it change risk premiums? Possibly. Because the first effect raises the numerator and the second raises the denominator, the net effect on equity values is ambiguous, and no amount of confidence about the direction of growth resolves that ambiguity on its own.

Two further observations follow immediately from the structure of the equation. Some assets have a single cash flow: a five-year zero-coupon bond has cash flows of zero in periods one through four and a single payment at N equal to five. Others, such as dividend-paying shares, have no terminal date at all, so N can be taken as infinity. And the amount of uncertainty attached to the cash flow varies enormously across the range. Investors may attach a very low probability to a developed-economy government failing to pay its debts on time and in full, even where a rating has been lost. At the other end, an equity investor is uncertain about both the size and the timing of dividends, and has to allow for the possibility of receiving nothing at all if the issuer fails.

What sits inside the risk premium

The generic term ρⁱ_t,s is a container, and the reading spends most of its length unpacking it, one asset class at a time. Compensation for default risk is only part of it. Liquidity risk, the possibility that an asset cannot be turned into cash quickly at anything close to fair value, is another part, and it is a defining feature of commercial real estate and of high-yield corporate debt. The 2008 to 2009 global financial crisis demonstrated the point sharply: instruments such as mortgage-backed securities became extremely illiquid at precisely the moment investors most wanted to be holding cash rather than risk.

Risk premiums also move. During recessions the premium demanded on assets that are not default-free may rise, because investors as a group become both less willing and less able to carry heightened default risk. That time variation, and its link to the business cycle, is the spine of the whole reading.

The table below previews where the argument is going. Each asset class inherits the discount rate of the class above it and adds one more premium.

The equity risk premium relative to default-free debt is written λ = γ + κ. Each symbol is developed in the section that introduces it.

Two observations fall straight out of the pricing equation, and both are easy to state and easy to forget under exam pressure.

The first is that asset values depend on expected future cash flows, never on cash flows already received. History matters only to the extent that it helps forecast. The second is that those expectations are conditional on information available today, which is what the time subscript t is doing in the notation. Any information that improves the accuracy or the precision of the forecast is relevant information.

Put the two together and a practical rule emerges. Information that was already anticipated is already embedded in the price. Only the part of a data release that differs from what was expected is news, and only news forces expectations to adjust. That adjustment generates a holding period return different from the return investors expected to earn.

Why markets sometimes fall on good numbers

This is the standard explanation for market reactions that look perverse from the outside. A company reports higher profits and the share price falls. An economy reports faster growth and the equity index drops. In each case the release was good in level terms but disappointing relative to the consensus that was already in the price. Investors judge data against their own prior expectations, not against zero. Prices can fall on good news when the expectation was for better news, and rise on bad news when the expectation was for worse.

A useful discipline is therefore to ask, before reacting to any release, what was already priced. If a central bank cuts rates by 25 basis points and the market had fully priced a 50 basis point cut, the cut is a hawkish surprise and short rates should rise.

Sentiment sits alongside the economic channels

This reading concentrates on economic factors, but it acknowledges one other force. Investor sentiment, meaning enthusiasm or despair that is not obviously grounded in fundamentals, also affects values. The route it takes is worth noting because it differs from the route economic factors take. Economic factors work mostly through direct effects on cash flows and on discount rates. Sentiment works primarily through discount rates, by pushing risk premiums up or down, with possible indirect effects on expected future cash flows as well.

Buying a financial asset means not consuming today. That is the opportunity cost of every investment, and when the individual decisions of millions of savers are aggregated, that opportunity cost is what sets the price of financial assets. This section works through the mechanism for the simplest possible asset: a bond that is default-free and index-linked, so that neither default risk nor inflation clouds the picture.

What return should such a bond pay? The tempting answer is zero, because there is no risk of losing money in either nominal or real terms. That answer is wrong, because it ignores the opportunity cost. An investor choosing to buy the bond can either pay a price P_t,s today for one monetary unit of income s periods ahead, or spend P_t,s on goods today. The choice between them is a relative price problem like any other.

Marginal utility and the rate of substitution

The trade-off is measured by the ratio of the marginal utility of consumption s periods ahead to the marginal utility of consumption today. Marginal utility of consumption is the extra satisfaction from one more unit of consumption. This ratio is called the inter-temporal rate of substitution and is written m_t,s.

Its behaviour follows from one property: marginal utility falls as consumption rises. In good economic times, incomes and current consumption are high, so the utility from one more unit of consumption today is low. In bad times, incomes and consumption are low, so one more unit today is worth a great deal. As wealth increases, fundamental needs are already satisfied and the marginal utility of further consumption diminishes. The direct consequence is that investors place a larger value on an asset that pays off more in bad times than on one that pays off in good times.

The pricing equation for the one-period real bond

The rate of substitution is a random variable, because an investor does not know today what other income, such as salary, will be available in the future. Even an investment with a certain payoff has an uncertain value in utility terms. The investor therefore prices the bond using the expectation of the rate of substitution conditional on information available today:

Formally m_t,s equals δ multiplied by MU(C_t+s) divided by MU(C_t), where δ is a discount factor capturing pure preference for consuming sooner. In the modern theory of asset pricing the same object is called the stochastic discount factor, or the pricing kernel. The three names refer to one quantity.

Once the price is known, the one-period real risk-free return follows as payoff minus price over price:

The inverse relationship is the point. A lower price means a higher return. And a lower price means investors are less willing to trade present consumption for future wealth. So the higher the return an investor can earn, the more valuable current consumption has become relative to future consumption.

The previous section priced an asset with a certain payoff. Now allow the payoff to be uncertain and a risk premium appears, even when there is no default risk at all.

The starting point is a property of utility. Make a payoff more uncertain and the expected marginal utility attached to it falls, because a given fall in wealth destroys more utility than an equal rise in wealth creates. An investor who requires compensation for that asymmetry is risk averse. The compensation is the risk premium.

Absolute risk aversion and wealth

A second property matters for valuation under uncertainty: absolute risk aversion is assumed to fall as wealth or income rises. Absolute risk aversion concerns the money amount held in risky assets at different levels of wealth, and is distinct from relative risk aversion, which concerns the fraction of wealth so held. Under decreasing absolute risk aversion, an investor puts larger amounts into risky assets as wealth rises.

The consequence is that a wealthier person demands less compensation to carry the same risk, because the typical loss of marginal utility involved in carrying it is smaller for that person. Wealthier investors are therefore more willing to hold a given risk and will buy more risky assets, since they value them more highly than poorer investors do. That process is self-limiting: as the wealthy buy more of the risky asset, their expected marginal utility declines, and in equilibrium wealthy and poorer investors end up with the same willingness to hold risky assets.

A bond with an uncertain interim price

Prices of instruments that are not default-free are established relative to the price of the default-free bond. To see how, take a default-free bond maturing in s periods, where s is at least two, and suppose the investor holds it for one period only. In one period the bond has a market value P_t+1,s−1, which is uncertain today because the bond will be sold before it matures. The pricing condition becomes:

There is no interim interest payment here because the bond promises a payment only at maturity. If a coupon were paid at t + 1, its value would be added on the right-hand side.

Now apply the standard statistical identity that the expectation of a product equals the product of expectations plus the covariance, and substitute the definition of the real risk-free rate. That gives an alternative and much more revealing form:

In the first term, the expected future price is simply discounted at the risk-free rate. A convenient name for that term is the risk-neutral present value, since it is the figure that would apply if nobody demanded any payment for carrying risk. The second term is the discount for risk, and it is a covariance, not a variance. This distinction is the intellectual core of the reading.

For a one-period default-free bond the covariance term is exactly zero, because the future payoff is a known constant and the covariance of a random variable with a constant is zero. For the two-period bond, the payoff two periods ahead is certain but the price one period ahead is not, so the covariance term is not zero.

Reading the sign of the covariance

For most risky assets held by risk-averse investors, the covariance term is expected to be negative. When the expected future price of an investment is high, times are good and the marginal utility of future consumption relative to current consumption is low. In bad times investors expect smaller labour income in the future, so the marginal utility of future consumption is high and so is the rate of substitution. Bad times are also when risky payouts fall: earnings and dividends decline and bonds default. The two therefore move in opposite directions, the covariance is negative, and the price of the asset is lower than the risk-neutral present value. A lower price today means a higher return over time, so a negative covariance produces a positive risk premium. The larger the magnitude of the negative covariance, the higher the required return and the lower the current price.

From the mechanics of the previous two sections it is a short step to the first empirical prediction of the reading: the level of real default-free interest rates in an economy is tied to how fast that economy grows and to how variable that growth is.

The growth channel

Suppose real GDP growth rises, and suppose for a moment that the change is known independently or can be forecast perfectly. The future stock of goods and services will be larger relative to what is available now. Future consumption is therefore less scarce, the willingness of investors to substitute future for present consumption falls, saving falls and borrowing rises, and the real default-free interest rate must rise to clear the market. That is precisely the inverse relationship between the rate of substitution and the real rate.

The volatility channel

GDP growth from one period to the next cannot in fact be anticipated perfectly. Under uncertainty, the real rate remains positively related to expected GDP growth, and it becomes positively related to the expected volatility of that growth as well. The reason runs through precautionary saving. Greater uncertainty about future income means a greater chance that consumption will fall short of expectations, so risk-averse investors save more. Higher saving raises expected future resources, which lowers expected future marginal utility, which lowers the equilibrium price of the bond and therefore raises its return.

Two practical implications

The analysis yields two statements that can be applied directly to cross-country comparisons.

• Higher trend growth means higher real rates. Other things being equal, real default-free rates should be higher in fast-growing developing economies such as India and China than in slower-growing developed economies such as those of Western Europe, Japan and the United States. A developing economy typically sits below its steady state growth path and grows faster to catch up. During that phase the marginal product of capital, the extra output from one more unit of capital holding everything else constant, is expected to be higher, so the real rate should be higher too. The advantage dissipates as the economy matures, as it did in Japan and Western Europe between 1950 and 2000.
• More volatile growth means higher real rates. Again other things being equal, an economy whose GDP growth is more variable should carry higher real interest rates than one whose growth is stable.

Applying the framework to conceptual questions

Three standard conclusions follow, and each is worth committing to memory in the form of the covariance argument rather than as an isolated fact.

• An asset risk premium is high when there is a negative relationship between its future payoff and investor marginal utility from future consumption. Zero covariance means the asset is risk free in this sense. Positive covariance means the asset is a hedge and will offer a return below the risk-free rate.
• Faster real GDP growth pushes the real risk-free rate up. It weakens the need to save for future consumption, so the rate has to rise before individuals will still choose to save.
• Greater volatility in real GDP growth pushes the real risk-free rate up too, because it raises the chance that the income available for consumption falls short of plan, and risk-averse investors want paying for that.

The framework also explains why a confident macroeconomic forecast does not translate into a confident valuation call. Suppose real GDP growth is expected to rise above trend, real corporate sector income follows real GDP, and there is no effect on inflation. The pricing equation applied to the aggregate equity market says the numerator rises, because expected corporate earnings rise by assumption. It also says the denominator rises, because higher real GDP growth raises the real risk-free rate. The two effects offset, and nothing in the information given determines which dominates. The correct answer is that the effect on aggregate equity market value is ambiguous.

To test the predictions of the previous section against data, an observable proxy for the real default-free rate is needed. Inflation-linked government bonds provide one. These instruments pay a real yield plus a return tied directly to a consumer price index, and they are issued by many developed-economy governments, among them Sweden, Italy, Germany, France, Canada, the United States and the United Kingdom, and by some developing-economy governments such as Brazil.

The indexation mechanics differ by market. In the United Kingdom index-linked gilt market, both the coupon and the principal are indexed to a measure of consumer prices. In the US Treasury Inflation-Protected Securities market, the principal is uplifted with the index and the coupon is then computed on that uplifted principal. Either arrangement means a rising consumer price index raises both the coupons and the principal eventually repaid, so for practical purposes both can be treated as inflation protected.

The cross-sectional evidence

Real yields on short-dated index-linked government bonds in July 2007, immediately before the global financial crisis, can be compared across seven developed markets against average GDP growth and the volatility of that growth over 1996 to 2007. Japanese real yields were the lowest of the group, consistent with Japanese growth having been historically low and not very volatile up to that point. Of the developed-economy yields shown, Australian government bonds offered the highest, which is consistent with relatively strong Australian growth.

The pattern is suggestive rather than clean. Across this limited sample the correlation between the bond yields and historical growth is 0.57, and the correlation with historical volatility is 0.74. Both signs support the theory, and the volatility correlation is the stronger of the two, but the sample is small.

One reason a sharper relationship does not appear is a timing mismatch. Real yield data are forward looking: they represent the real return investors require given expected future growth and expected future volatility. The GDP variables are backward looking. If investors use the past as a guide to the future, a reasonably high correlation should appear. If the past is a poor guide, as it is for rapidly developing countries or in the aftermath of a shock on the scale of the collapse of Lehman Brothers, the correlation weakens.

The time-series evidence

Index-linked markets are recent by the standards of government debt. The first index-linked bond issued by the US Treasury dates from 1997. The oldest such market is the United Kingdom one, where the first index-linked gilt was issued in 1981, so UK data offer the longest window on the relationship between the macroeconomy and the real risk-free rate.

The real yield on a short-dated constant maturity index-linked gilt trends clearly downward from 1985 to 2007, though with considerable volatility along the way. The obvious explanation, a matching decline in expected UK growth, does not survive contact with the data: real UK economic growth averaged 2.8% between 1985 and 1999 and 2.7% between 2000 and 2007. Average growth barely changed while real yields fell, so it is fair to assume growth expectations were relatively stable too.

The volatility channel does much better. UK real GDP growth volatility, measured as a three-year moving standard deviation, declined sharply between 1995 and 2007, and the same decline appeared across OECD economies. This episode of unusually low macroeconomic and financial volatility is known as the great moderation. A plausible reading of the UK evidence is that declining economic volatility drove real interest rates down between the early 1990s and 2007.

The absence of index-linked markets elsewhere over the same window does not mean the phenomenon was confined to the United Kingdom. The yield on a conventional government bond contains the real default-free rate as one of its components, so falling global economic volatility very likely lowered required real returns in other developed markets too, contributing to the general decline in conventional government bond yields.

The crisis period, and why the sign appears to flip

By the end of 2011, yields on short-dated index-linked bonds had fallen everywhere relative to their pre-crisis levels. On the face of it this contradicts the theory: the failure of Lehman Brothers, and the liquidity and sovereign debt crisis that came after it, drove economic and financial volatility sharply higher, so real yields ought to have risen.

The resolution is that other things were not equal. Investors also revised down their expectations of future real economic growth, and the growth channel dominated: if the equilibrium real yield is set by expected growth as well as expected volatility, a large enough downgrade to growth expectations can overwhelm the volatility effect. Real short-dated rates in the United Kingdom and the United States were not merely lower than before the crisis, they were negative. Most inflation-linked bonds issued by the US Treasury and all index-linked gilts issued by the UK government carried negative real yields at that point.

Two further explanations are worth carrying. First, easy monetary policy, including formal quantitative easing in the United States and the United Kingdom and informal easing in the eurozone, may have persuaded enough investors to fear high future inflation that they bought inflation protection despite low and even negative historical real yields. Second, index-linked bonds issued by developed-economy governments combine credit protection and inflation protection and are often in very limited supply. In times of crisis and acute uncertainty, investors may treat them as a safe haven for capital, and that demand drives yields down.

Pulling the section together

For a real default-free bond the general pricing equation collapses to a single-driver formula, because the cash flow is certain in real terms:

And l_t,s in turn is determined by real economic growth and the volatility of that growth over time, through the aggregation of the saving and consumption decisions of individual investors. Everything added in the sections that follow is an additional term in the denominator, layered on top of this one.

Now switch from a bond that pays a fixed real amount to one that pays a fixed nominal amount, still issued by a developed-economy government whose prospect of default is small enough to ignore.

In a world without inflation nothing changes: the investor still gives up current consumption, and the real formula applies. With positive inflation, the quantity of goods that the maturity payment will buy is no longer known. If inflation could be forecast perfectly, investors would demand l_t,s + θ_t,s, enough to preserve the intended real level of consumption. But unless the horizon is very short, investors have limited confidence in their inflation forecasts, and risk-averse investors demand compensation for that uncertainty as well as for the expected level. That compensation is a separate premium, written π_t,s, and it is distinct from the generic asset-specific premium ρⁱ_t,s:

Note carefully what is and is not uncertain. The payoff is still certain in nominal terms, because the issuer is assumed not to default. It is the real value of that payoff that is uncertain, and that is why the premium π_t,s is needed. The superscript i is suppressed on this premium because it is not asset specific: inflation uncertainty applies across asset classes.

The break-even inflation rate

The compensation for inflation is directly observable if two comparable bonds are available. Take a zero-coupon bond with no default risk whose payment is fixed in nominal terms, take a zero-coupon bond with no default risk whose payment is fixed in real terms, match the two maturities, and subtract the second yield from the first. What is left is the break-even inflation rate, usually shortened to BEI.

This decomposition is the whole content of the learning outcome on yield spreads between non-inflation-adjusted and inflation-indexed bonds, and the examinable trap is to forget the second term. Break-even inflation is not simply the best market guess of future inflation. It is that guess plus a risk premium compensating investors for uncertainty about the quantity of goods and services they will be able to consume in the future.

The consequences are practical. If expected inflation rises but uncertainty about inflation falls, the two components of the break-even rate move in opposite directions and the net change is indeterminate without knowing the magnitudes. If a central bank observes a rising break-even rate, it cannot conclude that inflation expectations alone have risen.

What the break-even series shows

Why the risk premium should be larger for long bonds follows from comparing horizons. The real payoff on a three-month Treasury bill is close to certain, so it is a good hedge against bad consumption outcomes: the real value of that payoff is very unlikely to fall if the investor loses a job within the three months. With a low, probably zero correlation between the payoff and bad consumption outcomes, the premium needed to attract the investor is close to zero. Over twenty years the position is completely different, and the premium demanded on a twenty-year conventional government bond should be materially higher than on a one-month or three-month bill.

Ten-year break-even rates derived from the Australian, UK and US government bond markets show a gradual decline since the mid-1980s in the two markets with the longer histories, tracking the changing inflation environment. Between 1985 and 1990, inflation averaged approximately 6.0% in the United Kingdom and 7.5% in Australia. Between 2000 and 2011, after falling steadily through the 1990s, it averaged 3.0% and 3.2% respectively. Ten-year break-even rates for the United States are available only from 1997, a period of relatively low and stable US inflation.

The 2008 to 2009 liquidity and credit crisis pushed ten-year break-even rates down across a range of developed economies, reflecting the weaker global economy and a weaker inflationary backdrop: lower demand for resources in a downturn means costs and prices rise more slowly. In Italy the 10-year measure dropped from 2.3% to 0.8%, which is what the eurozone crisis was doing to the Italian economy showing up in a market price.

Break-even rates are valuable to central banks precisely because they provide a market-based view of future inflation that can be set against the internal judgement of the central bank, though the two are not fully independent of one another.

Treasury bills are very short-dated nominal zero-coupon government securities, issued by most developed-economy governments or their agents to smooth government cash flow needs. Because they are short dated and because they are often the instrument through which monetary policy is implemented, their yields track the central bank policy rate closely.

Over a horizon of, say, three months, the uncertainty investors face about inflation is small enough that the premium π_t,s can be treated as negligible, and the pricing formula simplifies to a single payment discounted at the real rate plus expected inflation:

The consequence is a clean statement: nominal short rates rise and fall with short-dated real rates, and with the inflation investors expect over the same short window. Holding everything else constant, short rates ought to sit higher in economies that grow faster, grow more erratically, and average more inflation across time. Long runs of US and UK data show a close correlation between measured inflation and Treasury bill yields. Measured inflation is not the same as expected inflation, but current inflation plausibly plays a large part in forming expectations, particularly over the very short horizon of a bill.

The Taylor rule

Central banks set policy rates with reference to the position of the economy in the business cycle: cutting when activity or inflation is judged too low, raising when either is judged too high. John Taylor formulated a rule that puts numbers on that judgement and lets an observer gauge whether a policy rate is at an appropriate level.

Here pr_t is the policy rate, l_t is the real short-term rate at which long-term saving and long-term borrowing in the economy balance, ι_t is current inflation and ι*_t the target, and Y_t and Y*_t are the logarithmic levels of actual and potential real GDP. Their difference is the output gap, measured effectively in percentage terms.

The output gap is worth a moment. A positive output gap means the economy is producing beyond sustainable capacity, in the way that a marathon runner who sets off far too fast will eventually overheat and break down unless the pace comes down. A negative output gap means the economy is producing below capacity, like a runner who has set off too slowly and must at some point draw on conserved energy to speed up. Positive output gaps are usually associated with high or rising inflation; negative output gaps are usually accompanied by high unemployment.

The weight of 1.5 on inflation against 0.5 on output is not arbitrary. Inflation enters the rule twice, once as compensation for the current rate and once through the gap to target, and the larger combined weight is what stabilises inflation near the target over the longer term.

When inflation sits at target and the output gap is zero, the rule delivers a policy rate equal to l_t plus the target rate of inflation. That level is the neutral policy rate: the rate that neither spurs nor impedes real activity. Other things being equal, when inflation is above target the policy rate should be above neutral, and when the output gap is positive the policy rate should also be above neutral. The reverse holds in each case.

Policy error and the cycle

Taylor rule rates computed for the United States, Canada and the United Kingdom back to 1990, using conservative parameters, known inflation targets and an OECD estimate of the output gap, track actual policy rates fairly closely for the United States up to the point where the high-tech bubble burst in the early 2000s. On that measure the Federal Reserve kept policy rates too low for too long between 2002 and 2005, and a similar picture emerges for Canada. There is less evidence of the same pattern in the United Kingdom. Once the liquidity and credit crisis arrived, all three cut rates sharply, and measured against the rule, policy rates in all three economies were too low by the end of 2018.

No claim is being made that central banks follow the rule mechanically. The rule simply captures two of the main macroeconomic considerations that go into setting rates. But the possibility that a policy rate can be at too high or too low a level carries an important implication: the relationship between short-term interest rates and the business cycle is interdependent. Rather than smoothing the cycle with well-judged adjustments, central bankers can amplify it by responding badly. Rates held too low for too long risk a credit bubble; rates held too high for too long risk recessionary or even depression-like conditions.

Finally, the neutral rate is itself not a constant. It varies with the level of real economic growth and with expected growth volatility, for the reasons set out earlier, and it also shifts if the inflation rate targeted by the central bank changes. UK practice illustrates this: between 1992 and 1997 the UK inflation target was between 2% and 4%; in 1997 it became 2.5% with an allowance of 1 percentage point either side; in 2003 it was changed to 2.0%, with a different definition of inflation and the same 1 percentage point allowance.

Maturity changes how investors price a bond, so the natural next object of study is the whole curve rather than a single point on it.

Between July 2007, just before the wider financial crisis, and the end of 2011, the US zero-coupon Treasury curve shifted down by between 3 and 4 percentage points and became steeper. The short end was clearly driven by the Federal Reserve policy rate, which fell from 5.25% to virtually 0%. The gilt curve saw a comparable decline at the short end as the Bank of England took its policy rate from 5.75% down to 0.50% during the same crisis.

Two features stand out. The Treasury curve was upward sloping on each of the three dates. The gilt curve was downward sloping in July 2007: on that date the UK government could borrow one-year money at 6.25% but thirty-year money at 4.8%, and fifty-year money at just over 3.0%.

Splitting a curve shift into real and inflationary parts

When a curve moves significantly, it is informative to separate the movement into its real and inflation components by comparing the nominal curve with the break-even curve. Across 2007 to 2011 both break-even curves, the US one and the UK one, moved by a little under one percentage point at every maturity on the curve. That is consistent with bond investors steadily marking down their view of future inflation as the global recession developed. But the nominal curves shifted down by more than 1 percentage point, and the US curve fell by as much as 3 percentage points. The implication is that market participants saw the crisis as having a bigger impact on economic growth than on inflation. By December 2011 both break-even curves were upward sloping.

One caution attaches to this kind of dissection. It presupposes either that there is no risk premium inside investor return expectations, or that any such premium is constant over time. Neither is likely to be true, which is why the following section treats the premium explicitly.

Three factors describe almost all curve movement

Yield curves have three descriptive characteristics. Level indicates whether rates are high or low on average. Slope indicates the steepness, meaning how quickly rates change with maturity. Curvature indicates how far the curve departs from a straight line. All three can change over time, and principal components analysis can attribute observed changes to each. Applied to UK and US data with maturities from three months to ten years over January 1999 to January 2014, the results are as follows.

Figures are rounded to one decimal place. Results of this shape are typical for developed-economy government curves.

Level dominates in both markets, accounting for the large majority of movement. Slope accounts for a much smaller share, and curvature for a very small one. Taken together the three explain the vast majority of changes in both curves.

What drives each factor

The level of the curve should reflect the level of economic activity and, because these are nominal instruments, views about future inflation.

The slope is influenced by the risk premium that arises from the covariance between the future price of the bond and the inter-temporal rate of substitution over the investor horizon. A positive slope reflects that premium. But the premium is not the only driver. Because the policy rate is set on Taylor rule-like considerations, short rates tend to be lower in recessions, while the effect of monetary policy on longer-term rates is weaker, since the central bank is expected to return short rates to normal as the recession recedes and real rates rise with the recovery. The slope of the curve therefore increases during a recession.

Curvature moves for a related reason. If investors expect policy rates and short-term real rates to revert to normal as a recession recedes, the curve steepens at the short end while flattening at the long end, and curvature increases.

The crisis prompted a re-examination of what default-free actually means. Bonds issued by many European governments, including those of France and Italy as well as Greece, Portugal, Ireland, Belgium and Spain, were considered default-free before the euro financial crisis of 2010 to 2012. Even the default-free status of US Treasuries came into question with both investors and rating agencies. Any bond genuinely perceived as default-free does not have to compensate investors for default risk. Any bond carrying a non-zero probability that the issuer fails to meet its obligations must, and the compensation is the credit premium.

Adding it to the discount rate gives the pricing equation for credit risky debt, with the credit premium written γⁱ_t,s:

The numerator is now genuinely uncertain. The schedule of payments is known, but credit risk means there is uncertainty about whether they will be made as scheduled, and if a default occurs the amount recovered depends on a recovery rate that is itself unknown. Both uncertainties are compensated by γⁱ_t,s. Note that the inflation uncertainty premium π_t,s continues to apply: a credit risky bond carries every risk a default-free nominal bond carries, plus credit risk.

Credit spreads and the cycle

The difference between the yield on a corporate bond and the yield on a government bond of the same currency and maturity is the credit spread. It is demanded in compensation for the additional credit risk the investor bears.

Credit risky bonds share interest rate risk with default-free bonds. Shift the whole curve up in parallel and the proportionate price effect on a five-year corporate bond is close to identical to the effect on a five-year government bond of the same currency. Across time, the interest rate component of a corporate bond responds to exactly the forces that move government yields. What causes corporate and comparable government bond returns to diverge is the credit component and the evolution of the spread.

The premium should rise when economic weakness makes corporate default and bankruptcy more likely, and the historical record confirms it. A representative spread series for US corporate bonds rated AAA/Aaa, and a second for those rated BBB/Baa, each measured against US Treasuries, show two features. The Baa spread is always above the Aaa spread, reflecting lower credit quality. And spreads at both grades rise in the lead-up to and during a recession, then decline once the economy emerges. The AAA label is the rating category used by Standard and Poor’s and by Fitch Ratings; Aaa is the equivalent category used by Moody’s Investors Service.

The components of a credit spread

If investors were risk neutral, they would demand only enough extra yield to cover the loss they could expect to incur, which depends on the probability of default and on the recovery rate:

Under risk neutrality the expected return on a 10-year government bond would equal the loss-adjusted expected return on a comparable 10-year corporate bond. Investors are not risk neutral, so the expected return on the corporate bond is higher, even though a diversified portfolio removes a good deal of the credit risk. The main reason diversification does not remove the problem is that defaults cluster around downturns in the business cycle, so a well-diversified corporate bond portfolio remains substantially exposed to market risk.

The evidence on US corporate defaults since 1920 shows long stretches with very few defaults, punctuated by the depression of the 1930s and the recessions of the 1980s, 1990s and 2000s, each associated with relatively high default counts and rates. Historical default rates by rating class are sometimes used as a proxy for the probability of default in the expected loss calculation.

Recovery rates behave in the way the seniority structure predicts and in the way the cycle predicts. Senior secured debt carries a lien or a similar claim over some or all of the assets of the company. Senior unsecured debt has no explicit claim on assets in a bankruptcy, and that is why secured holders generally recover more than unsecured holders. Subordinated holders have an inferior claim to senior holders, and recovery rates on subordinated debt are often very low. Beyond the ranking, recoveries run higher in an expansion and lower in a contraction, for the simple reason that assets sold off to recover value for bondholders fetch better prices when trading conditions are buoyant than when they are stagnant.

Loss rates are the net result of default rates and recovery rates. They are counter-cyclical: they rise as economic activity declines. Note that this is a compounding effect rather than a single one. In a downturn the probability of default rises and the recovery rate falls, so both terms of the expected loss expression move adversely at the same time.

A credit risky bond has uncertain cash flows, but the investor at least knows the intended schedule of payments and how they are determined. Equity has neither guarantee. The dividend is not promised, it can rise and fall, and if the issuer becomes bankrupt it can cease altogether. Both the size and the timing of the cash flows are uncertain, and the cash flows may never materialise.

Writing the pricing equation for equity requires two changes. The summation runs to infinity, because there is no maturity, and a further term appears in the discount rate:

Expressing the equity premium relative to corporate debt is unusual, and it is done deliberately here for two reasons. If a company runs into financial difficulty, debt holders have the senior claim on cash flow and equity holders receive only the residue, which may be nothing. And if difficulties become severe enough to force bankruptcy, both bond and equity investors lose, but bond investors can usually expect to recover part of their investment depending on the quality of the company, whereas equity investors normally lose everything. Both are exposed, but the potential loss is greater for the equity holder. That is why a premium κⁱ_t,s is required over and above the credit premium γⁱ_t,s.

Both bond and equity holders bear what may be called corporate risk, and the combination of the two premiums is what is normally meant by the equity risk premium. Because equity risk is usually expressed relative to default-free debt of the same currency, the equation is more commonly written with a single premium:

Why equities make a poor consumption hedge

Whether investors should demand an equity risk premium at all reduces to the consumption hedging question established at the start of this reading. Investors demand a premium if equities tend not to pay off in bad times. The theory says the premium should be positive, which implicitly says equities are a poor hedge against bad consumption outcomes.

Pinning down the exact relationship between equity performance and consumption over time has proved very difficult, but a long history of real equity returns is suggestive. Across 1900 to 2010 in both the United States and the United Kingdom, the large drops in equity prices sit alongside recessions. The UK recession of 1972 cut real UK equity prices by more than half. The Great Depression took real US equity prices down by upwards of 40%. In 2009 the falls were 30% in the United Kingdom and 40% in the United States. Evidence of that kind makes the case for equities as protection against bad consumption outcomes very hard to sustain. So the premium ought to be positive, and given how large the drawdowns in bad times can be, it ought to be sizeable.

How large, in fact

The equity risk premium cannot be quantified ex ante. It can be observed ex post over long runs of data. Real annual returns on equities and on government bonds over 1900 to 2017 across a range of developed equity markets show equities outperforming government bonds in every country in the sample.

The US figure is an average over the 117 years described in the source. Premiums are measured against government bonds.

US equities outperformed US Treasuries by 4.5% a year on average, and the global equity market produced a premium of 3.2% a year against bonds. Nothing guarantees that a premium collected in the past repeats itself, but a record covering this many years does point toward an ex ante equity risk premium for developed markets of somewhere in the region of 3% to 5% a year.

One further relationship follows from the structure of the argument. Because λ contains γ, the equity risk premium and the credit premium of the same issuer are driven by overlapping forces, so they tend to be positively correlated over time and to respond to the business cycle in similar ways. And because equity cash flows are less certain than the scheduled payments on corporate debt, the equity premium should be the larger of the two, and it should rise in periods of economic weakness when uncertainty about future dividends is greatest.

Equity analysts spend most of their time on the numerator of the pricing equation, building a view of expected earnings and, from that view, of dividends and of free cash flow. Because aggregate profits are tied closely to the business cycle, an understanding of the cycle is essential to earnings projections, especially in the short term.

Real earnings growth can be tracked over a very long window for the United States and over a shorter one for the United Kingdom. Both series show that a sharp fall in real earnings almost always lands in a recession, which is what should be expected, since recessions bring falling employment, falling incomes, falling output and therefore falling profitability. Real US earnings collapsed in the Great Depression, and in 2009 they dropped by nearly 60%.

Profits as a leading indicator

The less obvious feature of the data is that sharp increases in profit growth occur at the end of a recession, and in some cases while recessionary conditions still persist. Corporate profitability can therefore lead an economy out of a recession as well as into one, and the mechanism is worth following step by step:

1. A negative demand shock causes demand and corporate profits to shrink.
2. Companies respond by laying off workers, which reduces their cost base but also adds to the recessionary backdrop by cutting household incomes.
3. When demand turns up, perhaps in response to monetary stimulus, that demand growth meets a lower cost base and produces a sharp increase in corporate profits.
4. Rising profits lead companies to invest and hire, and the process feeds on itself in the other direction.

Some analysts therefore treat corporate profitability as an important leading indicator of the cycle, on the view that it carries useful information about future growth.

Cyclical and non-cyclical businesses

The cycle does not affect every company equally. What matters is the type of product or service sold. Some products are relatively insensitive to general economic conditions. Toothpaste is the standard illustration: it takes a small share of the household budget, people continue to clean their teeth in a recession, and they do not clean them more often because the economy is booming. Demand is stable across the cycle, and companies and sectors of this kind are described as non-cyclical or defensive.

Airlines are the opposite case. In difficult conditions, consumers postpone or cancel holidays, or take them at home, far more readily than they reduce their consumption of toothpaste, and businesses cut back on air travel and substitute video conferencing. An annual family holiday is a large share of the household budget, and nobody needs one the way they need soap. In good times, when real incomes are rising, people take more holidays and more expensive ones, and rising business activity generates more travel to new and often distant markets. When cyclical earnings turn up after a spell of decline, economists and investment strategists often read that turn as a sign that broader growth is about to improve.

The same distinction shows up in national accounts data. Year-over-year growth rates of real GDP and of consumption of durable and non-durable goods for Canada and the United States over 1996 to 2012 show durable goods consumption to be markedly more sensitive to the cycle than non-durable consumption. It follows that the profits of companies producing durable goods should be correspondingly more volatile than those of companies producing non-durables.

Comparing real earnings growth in the discretionary consumer goods sector against the staple consumer goods sector over 1974 to 2012 gives the clearest evidence in the United States: real earnings growth in the cyclical sector rises and falls dramatically over the cycle, whereas the peaks and troughs for non-cyclical companies, though present, are less extreme. The UK comparison shows clear time variation in real earnings but a less clear difference between the cyclical and non-cyclical sectors, although the cyclical sector did tend to experience the more significant troughs over the period.

What else drives earnings

Product type is not the only determinant. How the business is financed matters. So does how capable and how experienced its managers are, and how easily a newcomer could arrive and compete abnormal profits away. But the interaction between the cycle and the nature of the good or service sold remains central, and there is a warning attached. In a booming economy, even poor managers of companies with weak financial structures can generate profits, or appear to generate them, as WorldCom and Enron did. Recessionary conditions expose weak companies, because demand turns down at the same time as financing becomes harder to obtain.

To compare equities within and across sectors, analysts monitor valuation multiples. Two dominate: the price-to-earnings ratio and the price-to-book ratio.

The price-to-earnings ratio

The P/E is the ratio of the current share price to earnings per share, and it states the price paid for each unit of company earnings. A stock trading on a low P/E relative to the market implies that investors are not willing to pay much for a dollar of earnings, perhaps because the market believes the prospect of strong future earnings growth is low. A high P/E relative to the market implies the opposite, and often reflects an expectation of rapid earnings growth.

Two versions are in use. When the earnings figure refers to the previous year, the P/E is historical or trailing. When it is based on an estimate of future earnings, it is leading or forward. The relationship between them follows arithmetically: if earnings per share are expected to grow, the denominator of the forward P/E is larger, so the historical P/E is greater than the forward P/E.

The price-to-book ratio

The P/B measures the share price against net assets per share, that is assets minus liabilities attributed to each share. It shows how far the net assets of the business back the value of the shares. Some of those assets are tangible, such as office buildings, and some are intangible, such as patents and copyrights. Some sit on the balance sheet and are therefore part of book value, and some do not. The ratio also indicates the strength of investor expectations about the ability of the company to generate a high risk-adjusted return on its net assets. A higher ratio means greater expectations for growth but a smaller safety margin if events disappoint. As with the P/E, what counts as high or low depends on the market, the sector and the individual stock.

Judging whether a multiple is high

The average trailing US P/E between 1900 and 1990 was 13.5, meaning investors were paying $13.5 for a dollar of the previous year of earnings. By the late 1990s and early 2000s they were paying $45. The expansion in the US P/E during the 1990s was a global phenomenon, and strategists justified it with a series of ad hoc explanations: the end of the cold war, better macroeconomic policy that would make major recessions a thing of the past, the internet revolution, and others.

The pricing equation offers a more disciplined list. A high P/E could result from any of the following, or from a combination of them:

• an increase in the expectation of future real earnings growth, raising the numerator;
• falling real interest rates l_t,s, possibly associated with falling volatility in real GDP growth;
• a decrease in inflation expectations θ_t,s;
• a decline in uncertainty about future inflation π_t,s; or
• a decrease in the equity risk premium λⁱ_t,s.

Other things being equal, any one of these, or all of them together, would justify a higher price relative to current earnings and therefore a higher equilibrium P/E. The general cyclical pattern is that the P/E tends to rise during expansions and to fall during recessions. Not everyone was convinced at the time that the multiples reached in the late 1990s could be justified, least of all on the argument that future earnings growth would be far higher. Alan Greenspan, then chairman of the US Federal Reserve Board, described equity market valuation in 1996 as essentially the result of irrational exuberance.

The cyclically adjusted P/E

Robert Shiller proposed an alternative multiple designed to strip out cyclical distortion in the denominator: the real cyclically adjusted P/E, or CAPE. It is constructed in the same way as the P/E, except that the price is the real, inflation-adjusted price of the equity market and the earnings figure is a ten-year moving average of real market earnings. Dividing the real price by a moving average of real earnings smooths the short-term volatility that makes a single year of earnings a poor denominator, particularly at the bottom of a recession when earnings collapse and the ordinary P/E can spike upward for reasons that have nothing to do with valuation.

Applied to the United States from 1900 to 2018, the CAPE still shows very high prices in 1929 and 1999, and to a lesser extent in 1965. What makes the series interesting is what followed those peaks.

The 1921 and 1980 entries are the two lowest values of the CAPE in the sample.

All three peaks were followed by negative average real returns over the subsequent decade, and both troughs by strongly positive ones. The gap is wide: the worst of the three peak outcomes, −5.4% a year after 1965, sits 17.7 percentage points below the best trough outcome of 12.3% a year after 1921. This is evidence about long-horizon expected returns, not a timing tool, and five observations are five observations.

The pricing framework is not restricted to bonds and equities. Commercial real estate demonstrates the extension, and it is a useful final case precisely because it combines features of everything covered so far.

Three components of a property investment

The rental stream is bond-like. The cash flow investors expect from commercial real estate comes from rents paid by tenants, normally collected net of ownership costs such as building upkeep, on a fixed schedule from the businesses that lease the property. Practices vary by country. Rental agreements are reviewed regularly and may be reset. In some countries rents are subject to upward-only restrictions, so existing tenants never see rents fall, only potentially rise. Rents may also be indexed to a pre-specified index of, usually, consumer prices. Rental income is therefore closely analogous to bond coupon income, and a well-diversified portfolio of commercial property resembles a well-diversified bond portfolio. Its credit quality is determined by the credit quality of the underlying tenants, in the same way that the credit quality of a bond portfolio is determined by the ratings of its issuers: the lower the tenant credit quality, the less likely the rent is to be paid on time or at all.

The terminal value is equity-like. When a bond matures the investor receives the face value plus the final coupon. When a lease expires the landlord takes back possession and must decide whether to re-let, to sell, or to redevelop for a future sale. The decision turns on the value of the property at that time, which may have risen dramatically or fallen. Two factors dominate that value: the location of the property and the state of the underlying economy. If the area has become more popular during the lease, a sale at a profit or a redevelopment may be attractive. If the lease expires when general activity is high and demand for property is strong, the same conclusion follows. If the location has become less desirable or the economy is weak, redevelopment may not be viable, future rents may have to be lower, and a sale at a loss may be the best available outcome. That potential for profit or loss, and the uncertainty attached to it, adds an equity-like increment to the cash flow, positive or negative. Hence the common description of commercial property cash flow as part bond and part equity.

The investment is illiquid. Anyone who has sold a home knows how much time and effort it takes to bring a property to market, find a buyer and complete a sale. Exiting a commercial property investment can take months and sometimes years, and high transaction costs discourage liquidation further. Set against that, in normal conditions government bonds of developed economies, investment-grade corporate paper and listed shares can all be turned into cash without much difficulty. Almost every asset class considered in this reading is liquid relative to commercial property.

The pricing formula

Adding an illiquidity premium φⁱ_t,s to the equity form of the equation captures all the salient features:

The expected cash flow is uncertain because tenants may default on the rental agreement, and the quality of the rental income depends on tenant quality in the same way that the reliability of corporate bond coupons depends on the credit standing of the issuer. The future value of the property is also unknown.

The composition of the discount rate depends on who the tenant is and how the lease is written. Consider three tenants and the bond analogue of each lease:

• A developed-economy government tenant paying rental income indexed to inflation: the lease resembles a real default-free government bond, discounted at 1 + l.
• A developed-economy government tenant paying a fixed nominal rent: the lease resembles a nominal default-free government bond, discounted at 1 + l + θ + π.
• A corporate tenant paying a fixed nominal rent: the lease resembles a credit risky nominal bond, discounted at 1 + l + θ + π + γ.

In every case two further premiums must be added. A premium analogous to the equity risk premium, κ, compensates for uncertainty about the value of the property at the end of the lease. And an illiquidity premium, φ, compensates for the fact that the investment cannot readily be turned into cash, which means it may not be possible to liquidate it in bad economic times. Illiquidity reduces the usefulness of an asset class as a hedge against bad consumption outcomes, and investors demand compensation for that. The three complete discount rates are therefore 1 + l + κ + φ, then 1 + l + θ + π + κ + φ, and then 1 + l + θ + π + γ + κ + φ. How large each piece is depends on how long the lease runs, how strong the tenant is, and where the building stands.

Property and the business cycle

The cash flows and the discount rate are both shaped by the economy, but they are shaped very differently, and separating the two is the main analytical point of this section.

UK commercial property income growth over a 30-year period is remarkably stable, averaging 6.5% a year in nominal terms, which is approximately 2.5% in real terms. The stability of that income stream across several business cycles suggests investors might discount it at a very low rate. Capital values behave completely differently. Between 1990 and 1992, as the UK economy went through a deep recession, UK commercial property prices fell by a cumulative 30%, and over the course of the UK recession of the 1990s the capital value of the market fell by 26%. The global recession of 2008 to 2009 hit capital values across the world, and in Ireland, arguably the worst affected developed economy, commercial property prices fell by 55.5%.

Strong economies produce the mirror image. The recovering and then strong global economy between 2003 and 2006 lifted world commercial property prices by nearly 20%, by 41% in the United Kingdom and by 51% in Ireland.

Because capital values are pro-cyclical, commercial property is not a good hedge against bad economic outcomes and investors demand a relatively high risk premium for holding it. The sharp declines in capital values during recessions resemble the declines equity investors experience, although such episodes occur more frequently for equity. The risk premium demanded on commercial property is therefore arguably closer to that demanded on equities than to that demanded on default-free government bonds. Putting a direct number on the property risk premium is hard. What can be said is that it almost certainly moves as economic conditions move, and that it should display a strong positive correlation with the premiums demanded on corporate debt and on shares.

One final distinction separates property valuation from equity valuation. Real estate does not trade in public markets, with the exception of real estate investment trusts. Compared with the valuation of publicly traded equities, the valuation of real estate should therefore reflect a discount for relative lack of liquidity.

Closing the loop

The reading began with a single equation and ends with the same equation carrying six terms in its denominator. Every extension has followed one rule: identify a source of uncertainty that makes the asset a worse hedge against bad consumption outcomes, and add a premium for it. Real rate, expected inflation, inflation uncertainty, credit, equity and illiquidity are not six unrelated ideas but six applications of one idea. Whenever a macroeconomic development is put in front of you, ask which of the three channels it works through, and then ask which of the premiums in the denominator it moves.