Capital Market Expectations, Part 2: Forecasting Asset Class Returns: formula sheet

Key formulas

CME process (order)

Specify expectations and horizon → Research historical record → Specify method, assumptions and limits → Determine best information sources → Interpret current environment → Provide and document expectations → Monitor and refine

Learn the order and what each step asks. Step 1 is to specify the expectations needed, including the horizon; the client's objectives and IPS inform it.

Error families

Data limitations | Biases and judgment errors | Model and parameter uncertainty | Ex post risk and structural change

This grouping is a study aid, not a CFA classification. Place every pitfall in one family before naming it, but use the curriculum term in your answer.

Smoothed data effect

Appraisal-based data → reported σ lower and correlation with other assets lower than true values

Real estate and private equity are typical cases. Diversification looks better than it is.

Ex post vs ex ante

Ex post = realised history; ex ante = forward-looking estimate

A calm sample period can produce a low ex post risk that is not a good ex ante guide.

Expected return of a bond (decomposition)

E(R) ≈ Yield income + Rolldown return + E(Change in price from yield/spread changes) − E(Credit losses) + E(Currency gain/loss)

Use only the terms that apply. A hedged domestic government bond has no credit or currency term.

Rolling yield

Rolling yield = Yield income + Rolldown return

Rolldown return is the price gain from the bond moving to a shorter maturity on an unchanged upward-sloping curve. It is zero on a flat curve.

Price change from a yield change

%ΔP ≈ −ModDur × ΔY + ½ × Convexity × (ΔY)²

ΔY in decimals. Convexity term is small for small moves but matters for large moves.

Rolldown return (approximate)

Rolldown return ≈ −ModDur at horizon × (Yield at shorter maturity − Current yield)

Use the modified duration of the bond as it will be at the horizon (the shorter-maturity bond), because it is repriced at the new yield. Using today's longer duration is only a rough simplification and overstates the gain. A lower yield gives a positive return.

Nominal yield decomposition

Nominal yield = Real yield + Expected inflation + Risk premium

The risk premium includes the term premium for long bonds and the inflation risk premium.

Credit bond yield

Yield = Government yield + Credit spread

The spread covers expected loss, risk premium for credit risk and liquidity.

Expected credit loss

Expected loss ≈ Probability of default × Loss severity

Loss severity = 1 − recovery rate. Subtract the expected loss (PD × severity) from yield when estimating expected return.

Spread change effect

%ΔP from spread ≈ −SpreadDur × ΔSpread

Use spread duration for the credit component of price change.

Gordon growth expected return

E(Re) = D1 ÷ P0 + g

Dividend yield uses next year's dividend. g is long-run sustainable dividend growth.

Grinold-Kroner expected return

E(Re) ≈ D/P − ΔS + i + g + ΔP/E

This is the simple additive approximation. D/P is the dividend yield. ΔS is the percentage change in shares outstanding, so −ΔS is the net buyback yield. i is expected inflation. g is real total earnings growth. ΔP/E is the expected annual percentage repricing. The approximation ignores small cross-product terms, such as the interaction between repricing and earnings growth. Use the form supplied in the question.

Income and capital gains view

Expected return = Expected income return + Expected nominal earnings growth return + Expected repricing return

Income return = D/P − ΔS. Nominal earnings growth = i + g. Repricing = ΔP/E.

Building block equity premium

Required equity return = real risk-free rate + expected inflation + equity risk premium

Expected inflation plus real risk-free rate gives the nominal risk-free rate. The ERP is added on top.

Equity risk premium from expected return

ERP = E(Re) − nominal risk-free rate

Use a rate of matching maturity to the horizon.

Singer-Terhaar integrated vs segmented

Integrated premium = market Sharpe ratio × σ_asset × correlation with market. Segmented premium = asset's own Sharpe ratio × σ_asset. Blended premium = φ × segmented premium + (1 − φ) × integrated premium.

The integrated premium uses the global market Sharpe ratio and the asset's correlation with the market. The segmented premium uses the asset's own Sharpe ratio, which is an input given in the question, not something you derive from the premium. φ is the degree of market segmentation: φ = 0 means fully integrated and φ = 1 means fully segmented.

Earnings yield and the Fed model

Fed model: earnings yield (E/P) of the equity market ≈ yield on long-term government bond

It is a relative-value comparison, not a valid equilibrium. It compares real and nominal items, so use it with care. The Yardeni model adjusts for earnings growth: E/P = Moody's A-rated corporate bond yield − d × long-term earnings growth forecast, where d is the weight placed on the growth forecast.

Equity risk premium

ERP = E(R equity) − Risk-free rate

Use the risk-free rate matching the horizon in the question. Long-horizon questions usually use a long-term government bond yield.

Historical ERP

ERP (historical) = Mean(equity return) − Mean(risk-free return)

Check whether the question wants arithmetic or geometric averages. State the sample period and risk-free proxy used.

Gordon growth ERP

ERP = D1 ÷ P0 + g − Risk-free rate

D1 ÷ P0 is the forward dividend yield. g is the long-run expected dividend growth rate, which is supplied or justified by the analyst and may be linked to nominal economic growth.

Supply-side ERP (Ibbotson-Chen style)

E(R equity) ≈ (1 + i)(1 + g real EPS)(1 + g P/E) − 1 + Income return; ERP = E(R equity) − Risk-free rate

Here i is expected inflation, g real EPS is real earnings growth per share, g P/E is the percentage change in the P/E, and income return is the dividend yield. Long-run estimates often set the P/E term to zero when no re-rating is expected, but the question may give a non-zero value, and then you must use it. Use the exact form in the question and check which terms are given.

Approximate supply-side form

E(R equity) ≈ i + g real EPS + g P/E + Income return

Use the approximation only when the question allows it. g P/E is the percentage change in the P/E. It is often set to zero for long-run estimates when no re-rating is expected, but use any non-zero value the question gives.

Cap rate

Cap rate = NOI (year 1) ÷ Property value

Use forward-looking NOI. It is a yield, not a total return.

Expected real estate return (cap rate approach)

E(R) ≈ Cap rate + NOI growth rate − Capital expenditure/depreciation rate

Growth is often set near expected inflation for a stable property. Drop the last term if the question ignores it.

Value from cap rate

Value = NOI ÷ Cap rate

Value falls when the cap rate rises, other things equal.

Unsmoothing (first-order)

R*(t) = [R(t) − (1 − λ) × R(t−1)] ÷ λ

R(t) is the observed smoothed return. λ is the weight on the current true return. Here R(t) = λ × R*(t) + (1 − λ) × R(t−1), so a lower λ means heavier smoothing. Check that the question uses this form.

Unsmoothed standard deviation (approximation)

σ* = σ(observed) × √[(2 − λ) ÷ λ]

Valid for the smoothing form above when true returns are uncorrelated over time. It follows from observed variance = λ × σ*² ÷ (2 − λ). A lower λ gives a larger scale-up.

Build-up for alternatives

E(R alt) = Public-market return + Premiums (illiquidity, leverage, skill) − Fees

State each premium and its justification. Use net-of-fee return.

Absolute PPP

S(P/B) = CPI(P) ÷ CPI(B)

Law of one price applied to a basket. Holds only loosely in practice.

Relative PPP (approximate)

%ΔS(P/B) ≈ π(P) − π(B)

Currency with higher inflation depreciates. Base currency rises if price-country inflation is higher.

Relative PPP (exact)

S(t+1) = S(0) × (1 + π(P)) ÷ (1 + π(B))

Use when the question gives precise inflation rates or a long horizon.

Covered interest rate parity

F(P/B) = S(P/B) × (1 + i(P)) ÷ (1 + i(B))

Same horizon for rates and forward. Higher-rate currency trades at forward discount.

Forward premium or discount

F − S ≈ S × (i(P) − i(B))

Approximation. If i(P) > i(B), then F(P/B) > S(P/B), so the price currency is at a forward discount and the base currency is at a forward premium.

Uncovered interest rate parity

E[S(t+1)] = S(0) × (1 + i(P)) ÷ (1 + i(B))

Expected spot equals the covered forward in theory. Often fails empirically.

International Fisher effect

i(P) − i(B) ≈ E[π(P)] − E[π(B)]

Assumes equal real rates across countries.

GARCH(1,1) variance

σ²(t) = ω + α × ε²(t−1) + β × σ²(t−1)

ε(t−1) is last period's return shock. ω, α, β are estimated parameters. Stationarity needs α + β < 1.

Long-run variance (GARCH 1,1)

σ²(long run) = ω ÷ (1 − α − β)

Valid only when α + β < 1. Forecasts revert toward this level.

ARCH(1) variance

σ²(t) = ω + α × ε²(t−1)

Simpler model with no lagged variance term.

Shrinkage estimator

Σ(shrunk) = δ × F + (1 − δ) × S

S is the sample covariance matrix, F is the structured target, δ is the shrinkage intensity between 0 and 1.

Annualizing volatility

σ(annual) = σ(monthly) × √12

Assumes returns are independent over time. Convert variance by multiplying by 12, then take the root.

Unsmoothing returns

R(true,t) = (R(obs,t) − λ × R(obs,t−1)) ÷ (1 − λ)

λ is the smoothing weight on the prior return. Unsmoothing raises the standard deviation relative to the observed series.

Quick revision

• Forecasts have limits: data problems, model and input uncertainty, regime change and behavioral biases.
• A bond return forecast breaks into yield income, roll-down, and the effect of expected yield and credit changes.
• Use a forecast horizon that matches the client's horizon.
• The Grinold-Kroner model splits equity return into income, growth and repricing parts.
• Know the three method groups: discounted cash flow, risk premium and equilibrium.
• Equity risk premium methods: historical, forward-looking such as DDM-based, and survey.
• Historical estimates depend on the sample period and may suffer survivorship bias.
• Appraisal-based real estate data is smoothed, so volatility looks too low and correlations look too low.
• Exchange rate forecasts use parity relationships and capital flow views, and parity holds poorly in the short run.
• Emerging markets add data limits, higher and more variable risk, and political and liquidity risk.
• Show each calculation step and give the number in the form the question asks.

Common mistakes

• Confusing data-mining bias with time-period bias Fix: Data mining means many tests until something fits. Time-period bias means the result depends on the window chosen, and a different window would change it.
• Treating smoothed appraisal data as accurate risk Fix: Say the volatility and correlation are understated, and that the data should be unsmoothed before it is used in optimisation.
• Treating yield to maturity as the expected return. Fix: YTM equals expected return only if the curve is unchanged, the bond is held to maturity, coupons are reinvested at YTM and there is no default. Build the return from its parts.
• Getting the sign of rolldown wrong. Fix: Use −Duration × (future-maturity yield − current yield). On an upward-sloping curve the rolldown return is positive because the yield falls.
• Using the dividend yield alone as the income return. Fix: Always check whether the question gives a share count change. Income return = dividend yield minus the percentage change in shares.
• Getting the sign of the share count change wrong. Fix: A fall in shares (buyback) gives a positive contribution. An increase in shares (issuance) reduces return.
• Forgetting to subtract the risk-free rate and giving the expected equity return as the ERP. Fix: Always write ERP = E(R equity) − Rf as the final line.
• Mixing real and nominal inputs, such as nominal growth with a real risk-free rate. Fix: Label every input nominal or real before combining. Keep the ERP on a consistent basis.
• Treating the cap rate as the total expected return. Fix: Remember it is only the income yield. Add expected growth, then subtract any depreciation or capex drag.
• Saying unsmoothing lowers volatility. Fix: Smoothing hides volatility. Unsmoothing reveals it, so volatility and correlation with other assets rise.

Exam tips

• Learn the pitfall names exactly. Graders look for the standard term.
• For every pitfall, be ready to say the direction of the error: risk understated, return overstated, or correlation too low.
• Use the vignette's facts. A generic definition without the case detail often earns less.
• Give only as many responses as asked. Extra ones are not evaluated.
• Link the fix to the client's IPS horizon where the question allows.
• Read the command word. "Calculate" needs the number with steps shown. "Justify" needs the view that supports each input, in one short sentence.
• Always write the decomposition first. If your arithmetic slips, you still earn credit for the correct structure.
• Check the curve shape before calculating rolldown. Flat means zero, inverted means negative.