Economics and Investment Markets: formula sheet

Key formulas

Framework sequence

Specify expectations and horizon → Research history → Choose method and model → Determine data sources → Interpret current environment → Provide and document expectations → Monitor and refine

Use it to place a vignette's issue at the right step.

Risk premium build-up

E(R) = risk-free rate + risk premium(s)

Used in risk premium approaches; fit the premium to the asset class.

Gordon growth equity return

E(R) = D1 ÷ P0 + g

A DCF-style estimate of the expected equity return from current price and long-run growth.

Grinold-Kroner equity return

E(R) ≈ D/P − %ΔS + %ΔE + %ΔP/E

D/P is the dividend yield, %ΔS is the percent change in shares outstanding (so −%ΔS is the net repurchase yield), %ΔE is nominal earnings growth, and %ΔP/E is the repricing component.

Shrinkage estimate

Shrinkage estimate = (weight × sample estimate) + ((1 − weight) × target)

Blends noisy sample estimates with a more stable target; weights sum to one.

Change in inventory contribution to growth

Change in GDP = Change in final sales + Change in inventory investment

Growth in GDP reflects both final sales growth and the change in inventory investment. Production can swing more than sales because inventory adjustment adds to or subtracts from output.

Inventory-sales ratio

Inventory-sales ratio = Inventories ÷ Sales

A rising ratio with weak sales suggests unplanned stock build and later production cuts. A falling ratio to low levels suggests restocking ahead. Its timing class (coincident or lagging) depends on the source, so focus on the change in inventories relative to sales.

Indicator timing rule

Leading → turn before; Coincident → turn with; Lagging → turn after

Classify the indicator first. Do not assume an indicator that looks 'important' is leading.

Yield curve signal

Slope = Long-term yield − Short-term yield

A flattening or inverted curve is a leading signal of slowdown. A steep curve often accompanies early recovery. It is a tendency, not a rule.

Neutral policy rate

Real neutral rate = real trend growth rate; Nominal neutral rate = real trend growth rate + inflation target

Policy is expansionary if the policy rate is below the nominal neutral rate and contractionary if above.

Taylor rule

i* = r_neutral + π + 0.5 × (π − π*) + 0.5 × (y − y*)

π is current inflation, π* target inflation, y − y* the output gap in percent. The 0.5 weights are the standard Taylor values; use any weights the question gives. Here r_neutral is the real neutral rate, so adding π once gives a nominal rate. If the neutral rate is given in nominal terms, do not add π again: i* = r_nominal_neutral + 0.5 × (π − π*) + 0.5 × (y − y*).

Fiscal multiplier (simple)

Multiplier = 1 ÷ [1 − MPC × (1 − t)]

MPC is the marginal propensity to consume and t the tax rate. The balanced budget multiplier is a separate concept and equals 1 in the simple case.

Growth accounting (Cobb-Douglas)

ΔY/Y = ΔA/A + α(ΔK/K) + (1 − α)(ΔL/L)

α is capital's share of income. ΔA/A is TFP growth, the residual.

Labor productivity growth form

Δ(Y/L)/(Y/L) = ΔA/A + α × Δ(K/L)/(K/L)

Use when the vignette gives growth in capital per worker.

Sustainable growth rate

g = growth rate of labor force + growth rate of labor productivity

Potential GDP growth trend.

Covered interest rate parity

F(P/B) = S(P/B) × (1 + i_P × t) ÷ (1 + i_B × t)

P is price currency, B is base currency. Interest rates must match the horizon.

Forward premium rule

Base currency trades at a forward premium (F > S) when i_P > i_B.

Higher price-currency rate means F > S, so the base currency trades at a forward premium.

Relative PPP (expected change)

ΔS(P/B) ≈ π_P − π_B

Currency with higher inflation is expected to depreciate. Exact: S1 = S0 × (1 + π_P) ÷ (1 + π_B).

Absolute PPP

S(P/B) = CPI_P ÷ CPI_B

Law of one price applied to a basket.

Uncovered interest rate parity

E[ΔS(P/B)] ≈ i_P − i_B

Holds only if investors are risk neutral. Often fails empirically.

International Fisher relation

i_P − i_B ≈ π_P − π_B (equal real rates)

Combines real rate equality with Fisher effect.

Grinold-Kroner expected equity return

E(Re) ≈ D/P − %ΔS + i + g + %ΔP/E

D/P is the dividend yield, %ΔS is the percentage change in shares outstanding (so −%ΔS is the net repurchase yield), i is expected inflation, g is real earnings growth, %ΔP/E is the annualised repricing return. It is an approximation.

Components of the Grinold-Kroner model

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

The three parts you add. Check each one against the vignette before summing.

Simple DCF (Gordon) equity return

E(Re) = D1/P0 + g

Assumes dividends grow at a constant rate g indefinitely and that the required return exceeds g.

Bond expected return building blocks

E(R) ≈ YTM + rolldown return + E(price change from yield view) − E(credit losses) + E(currency gain or loss)

Rolldown return is the price gain from moving down the curve with the curve unchanged. Use only the terms the vignette gives.

Price change from a yield view

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

Use when you expect yields to change. A yield rise gives a negative first term.

Risk premium (build-up) approach

E(R) = Rf + risk premium(s)

For bonds the premiums can be term, credit and liquidity. For equities it is the equity risk premium.

Equilibrium (CAPM) expected return

E(Ri) = Rf + βi × [E(RM) − Rf], with βi = σi × ρ(i,M) ÷ σM

Beta rises with the asset's volatility and its correlation with the market. Use the global market in an integrated market and the local market in a segmented market.

Real estate DCF view

E(R) ≈ cap rate + expected NOI growth, where cap rate = NOI1 ÷ P0

An approximation that assumes a stable cap rate and no change in leverage.

Number of pairwise covariances

N(N − 1) ÷ 2

Distinct covariances (or correlations) among N assets. It grows fast as N rises. A full covariance matrix also needs N variances, so N(N + 1) ÷ 2 parameters in total.

Shrinkage estimator

Σ_shrunk = δ × Σ_target + (1 − δ) × Σ_sample

δ is the weight on the target matrix, between 0 and 1. Higher δ means more shrinkage.

EWMA variance

σ²_t = λ × σ²_(t−1) + (1 − λ) × r²_(t−1)

λ is between 0 and 1. A higher λ gives more weight to old variance and a smoother estimate. The form shown assumes mean return is about zero.

ARCH(1) variance

σ²_t = γ₀ + γ₁ × ε²_(t−1)

The variance depends on the previous squared error. Requires γ₀ > 0 and γ₁ ≥ 0.

GARCH(1,1) variance

σ²_t = γ₀ + γ₁ × ε²_(t−1) + β × σ²_(t−1)

Stationary if γ₁ + β < 1.

GARCH long-run variance

σ² = γ₀ ÷ (1 − γ₁ − β)

The level to which forecasts revert. Valid only when γ₁ + β < 1.

Covariance from correlation

Cov(A,B) = ρ × σ_A × σ_B

Use to convert a correlation forecast into a covariance.

Quick revision

• Capital market expectations are forecasts of asset class return, risk and correlation, built through a stated process.
• Always state the investment implication, not just the economic conclusion.
• Identify the cycle phase from indicators before judging policy or asset returns.
• Leading indicators turn before the economy, coincident ones move with it, lagging ones follow.
• Policy mix matters: combine monetary and fiscal stance to judge the effect on rates and yield curve.
• Expansionary policy tends to push in one direction on growth and the opposite on debt concerns, so read the vignette context.
• Long-run growth depends on labour, capital and productivity; check which driver the vignette highlights.
• Exchange rate views need the relevant model's inputs; check whether the question gives real or nominal data.
• Forecasts built from historical data inherit its errors, such as limited samples and regime change.
• Correlations and volatility can shift in stress, so do not assume past estimates hold.
• Check units, time horizon and whether returns are nominal or real before computing.
• Never leave an answer blank; there is no penalty for a wrong answer.

Common mistakes

• Confusing survivorship bias with backfill bias. Fix: Survivorship bias: failed funds are dropped from the database. Backfill bias: funds enter the database after good performance and their earlier history is added.
• Saying smoothed appraisal data overstates risk. Fix: Smoothing understates volatility and correlations with other assets, so diversification looks better than it is.
• Treating the inventory-sales ratio as a clearly leading indicator. Fix: The ratio is typically treated as coincident or lagging, and the class depends on the source. Do not assume it leads. Use changes in inventories relative to sales to judge the inventory cycle.
• Assuming a rising ratio means firms will keep producing more. Fix: If sales are weak and stock builds unintentionally, firms cut production to correct it. That is negative for near-term growth.
• Adding inflation twice or not at all in the Taylor rule Fix: Read the neutral rate label. With a real neutral rate, add current inflation once, then add the gap terms.
• Using the inflation gap and output gap with the wrong sign Fix: Always use actual minus target and actual minus potential. Positive gaps raise the rate.
• Inverting the quote so the forward premium goes the wrong way. Fix: Always write P/B first. In F = S × (1 + i_P) ÷ (1 + i_B), the price currency rate goes on top.
• Treating uncovered interest rate parity as reliable. Fix: Remember CIP is arbitrage-based and holds; UIRP needs risk neutrality and often fails. That is why carry trades have historically earned positive returns on average, but they carry crash risk, so profits are not guaranteed.
• Using the wrong sign on the share count term in Grinold-Kroner. Fix: If shares outstanding fall by 0.5% a year, %ΔS = −0.5%, so −%ΔS adds +0.5% to the return. Call it the net repurchase yield.
• Using the total P/E change over several years as the annual repricing return. Fix: Convert to an annual rate: (P/E end ÷ P/E start)^(1/n) − 1. Then add it.

Exam tips

• Learn each bias and limitation as a label plus a direction of error; questions ask for both.
• Always check the sample period and data source in the vignette first; these hide most of the flaws.
• Remember the seven-step framework sequence so you can say what step the analyst skipped.
• When two options both name valid biases, pick the one that matches the analyst's exact behavior, not the broadest term.
• There is no penalty for wrong answers, so never leave a question blank.
• Memorise one or two standard examples per indicator type, but classify unfamiliar ones by when the underlying activity occurs.
• Read the direction of inventories relative to sales against sales growth before choosing an answer.
• When two options both look plausible, pick the one that uses all the vignette facts: inflation, policy and curve together.