Part X · Edges, one by one · Chapter 10E

Carry

And how the foundation premia fit together

Where we are

Chapters 10B to 10D took three foundation premia, and this chapter takes the fourth: cross-asset carry, which the Edge Book files as a foundation premium but sizes as a supplement. It then puts the four together. How much of the portfolio's risk should each carry, how do they move together in a crash, and what does the whole foundation lose in the stress scenarios?

What you need first

Futures, forwards and the cost of carry (Chapter 4B).
The equity risk premium, the variance risk premium and trend following (Chapters 10B to 10D).
Tail correlation (Chapter 1F), the stress scenarios S1 to S3 (Chapter 1D) and drawdown (Chapter 1E).
The five categories of edge (Chapter 10A), and for the last check question, the conviction score (Chapter 13B).

1 · Carry: the return if nothing moves

In plain terms

Carry is the return an asset is expected to earn if prices do not change. Currencies with high interest rates, bonds on steep yield curves, commodities in backwardation and equities with high dividend yields all pay positive carry. Across asset classes, high-carry assets earn higher returns on average than low-carry ones. Their underperformance is concentrated in liquidity events, when everyone who owns them tries to sell at once.

The classic carry trade is in currencies. It borrows where interest rates are low and lends where they are high, and it earns the gap for as long as the exchange rate holds still.

Worked example: Buying Australian dollars with yen

Step 1: the trade. Buy the Australian dollar (AUD), a high-yield currency, and short the Japanese yen (JPY), a low-yield one. The interest rate differential is about 3% to 4% a year. Suppose, hypothetically, that you borrow yen at 0.5% and deposit Australian dollars at 4.0%, a carry of 3.5%, or $3,500 a year on a $100,000 position.

Step 2: the hope. If exchange rates are random walks (Chapter 3B), the exchange rate is as likely to move for you as against you, and you earn the differential. Empirically, AUDJPY, the price of an Australian dollar in yen, tends to drift up in calm periods and crash in stress, as it did in 2008 and 2020.

Step 3: the crash. In 2008 AUDJPY dropped 40% in 4 months as carry trades reversed. At 3.5% a year, a 40% loss is $0.40/0.035 \approx 11$ years of carry, gone in a season. The compensation for bearing that crash risk is the carry premium.

2 · The formal object

Take a futures or forward contract on asset $i$ , with spot price $S_{i,t}$ and futures price $F_{i,t}$ at time $t$ . Its carry is what a long position earns if spot stays where it is while the futures price converges to it by expiry:

C_{i,t} = \frac{S_{i,t} - F_{i,t}}{F_{i,t}}

(10E.1)

A futures price below spot, which is backwardation, gives a long position positive carry, and a price above spot, contango, gives it negative carry. The Edge Book writes the simplified futures-spot ratio as $(F_{i,t} - S_{i,t})/S_{i,t}$ . That ratio has nearly the same size and the opposite sign: it is the premium of the futures price over spot, which is the carry of a short position. The form above matches the book's own definition, the return if prices do not change, and its example that commodities in backwardation pay positive carry.

Chapter 4B's cost of carry sets the futures price at $F_0 = S_0 e^{(r-q)T}$ , where $r$ is the financing rate and $q$ the yield from holding the asset. Put into (10E.1), that makes carry about $(q - r)T$ : what holding the asset yields, less what financing it costs. Each asset class has its own $q$ :

An equity index: $q$ is the dividend yield, so high-dividend markets pay positive carry.
A currency: $q$ is the foreign interest rate and $r$ the home one, so carry is the interest rate differential.
A commodity: backwardation arises when holders value physical supply, the convenience yield, and it gives the long a positive roll yield.
A bond: carry is its yield above the financing rate, which is larger when the yield curve is steep.

Worked example: The carry of an AUD forward

Step 1: the forward price. Suppose AUDJPY trades at 100 yen, the yen rate is 0.5% and the Australian rate 4.0%. Holding Australian dollars earns 4.0% and financing them in yen costs 0.5%, so for one year $F = 100\,e^{0.005 - 0.040} = 100\,e^{-0.035} = 96.56$ yen.

Step 2: the carry. By (10E.1), $C = (100 - 96.56)/96.56 = 3.56\%$ , almost exactly the 3.5% differential of the earlier example. If the exchange rate stays at 100, the forward bought at 96.56 is worth 100 at expiry.

A carry strategy goes long the high-carry assets and short the low-carry ones. As in trend following (Chapter 10D), each position is scaled by its volatility, and the positions are diversified across currencies, bonds, commodities and equities.

3 · The evidence

Koijen, Moskowitz, Pedersen and Vrugt (2018), "Carry", in the Journal of Financial Economics (JFE), find Sharpe ratios of 0.37 to 1.80 across asset classes. The average for a single asset class alone is about 0.74, and a diversified combination reaches 0.6 to 0.9.
The strategy's common bad periods cluster in global recessions.

4 · Why it exists

Carry is a risk premium (Source 1 of Chapter 1G), with an institutional component. High-carry assets bear "bad-times" risk: currencies crash during liquidity events, commodity backwardation reverses when demand collapses, and high-yielding bonds underperform when defaults rise. The carry premium compensates for that risk. Its losses come in the same bad times that hurt equities, which is why its tail correlation with equities is high, about 0.7 in liquidity events (Chapter 1F).

5 · In practice

In practice

Vehicles: retail access is limited. Mutual funds in the style of QSPRX and QSPIX have closed or restricted new money. Some return-stacked products carry it (Chapter 10D). You can run FX carry yourself through currency futures, though the effect has been small since 2008.
The haircut: from an academic combined Sharpe ratio of 0.6 to 0.9 to a realistic retail 0.15 to 0.30.
Operations: complex, with many asset classes and futures execution in each of them.

The haircut is far wider than the equity premium's. An index ETF delivers the ERP at a Sharpe ratio of 0.35 to 0.50 in your account, close to its academic 0.4 to 0.5 (Chapter 10B). Carry needs positions in many futures markets, rolled and rebalanced, and the mutual funds that packaged it have closed or restricted new money.

6 · When it fails

When it fails

Carry fails in liquidity events. Carry trades unwind violently: in 2008 AUDJPY dropped 40% in 4 months as carry trades reversed. Stop-losses do not help, because the gap moves are larger than typical stops. Suppose a stop is set 5% below the price and the rate gaps 12% overnight. The stop does not sell at −5%: it sells at the next price, −12%.

Familiar ground

Judging carry by its calm years is judging a treatment on short follow-up. A Kaplan-Meier curve that is flat for three years says little about events that cluster in year five. Carry's losses arrive together, in the liquidity events that end calm periods, so a record without one understates the risk however long it looks.

7 · A decision rule

Decision rule

Make a small allocation, 5% to 10%, and only if you can access the strategy. The honest position is that cross-asset carry is institutional-quality, but the retail implementation gap is wide. Do not treat it as a foundation; treat it as a supplementary diversifier. The kill criteria are two:

Halt new positions when the TED spread exceeds 200 bps or the MOVE index exceeds 150.
Reduce the size when the ratio of the carry spread to its volatility falls below 0.3.

The TED spread is the gap between a short-term interbank lending rate and the yield on Treasury bills: what banks charge one another over what the government pays (Chapter 10H). A spread of 200 bps means banks are paying 2 points over Treasury bills to borrow, a sign of stress in funding. The MOVE index is the implied volatility of Treasury options, the bond market's counterpart of the VIX.

The carry-to-volatility ratio is a Sharpe ratio for the carry alone: the carry you expect to earn, divided by the volatility of the position that earns it. A trade carrying a hypothetical 3.5% a year with 12% volatility has a ratio of $3.5/12 = 0.29$ , below 0.3, so the rule reduces it. The premium no longer pays enough for the swings you bear to collect it.

The rule also explains a tension in the Edge Book. Its taxonomy lists carry among the four foundation premia (Chapter 10A), because by source it is a risk premium. Its dossier sizes carry as a supplement, because the retail gap is wide. The category says why carry pays; the gap says how much of it you can hold.

Common mistake: "FX carry is easy retail money"

Since 2008, the dispersion of interest rates across the major currencies has been compressed. Most central banks of the G10 currencies, the ten most traded, moved toward zero rates and then back up together. Carry premiums in FX have shrunk substantially, because carry is the differential: when two central banks both set rates near zero, there is nothing to earn. Diversified cross-asset carry, across FX, bonds, commodities and equities, still has a documented Sharpe ratio. But FX-only carry is much weaker than its reputation from the 1990s and 2000s suggests.

8 · Putting the foundation premia together

The allocation

The Edge Book gives each foundation sleeve a share of portfolio risk:

Sleeve	Share of portfolio risk
Long equity (ERP)	50% to 60%
VRP harvest (XSP put-writing)	10% to 15%
Trend following	10% to 15%
Cross-asset carry, if accessible	5% to 10%

The total foundation comes to about 70% to 80% of portfolio risk. The remaining 20% to 30% is for factor tilts, structural overlays and tactical positions (Chapters 10F and 10G). Two numbers in the Edge Book do not agree with this total. The four ranges in the table add up to 75% to 100%, and its taxonomy gives the foundation 60% to 70% (Chapter 10A). Treat all of these figures as rough guides rather than a formula.

How the premia move together in stress

The four premia correlate to varying degrees in stress:

ERP and VRP are highly correlated in the left tail, at about 1.0: both lose in crashes.
ERP and carry are moderately to highly correlated in liquidity events.
ERP and trend are negatively correlated in slow drawdowns. This is the key reason trend is in the foundation.
VRP and trend are mostly uncorrelated, and trend can be positive while the VRP is negative.

The trend allocation is therefore the single most important diversifier. Without it, the foundation is "different ways to be long equity beta and short skew". With it, the foundation has a genuine crash hedge.

Stress profit and loss on the whole foundation

Apply the stress scenarios to a hypothetical $1 million portfolio weighted 60/15/15/10 across the four sleeves. S1 to S3 are the scenarios of Chapter 1D, a fall in SPX with a rise in the VIX. S4 is a one-day gap like 1987's.

Scenario	ERP P&L	VRP P&L	Trend P&L	Carry P&L	Total
S1: SPX −10%, VIX +10	−$60k	−$25k	Flat	−$3k	−$88k
S2: SPX −20%, VIX +25	−$120k	−$50k	+$10k	−$8k	−$168k
S3: SPX −35%, VIX +50	−$210k	−$110k	+$30k	−$15k	−$305k
S4: a 1987 gap	−$132k overnight	−$80k	+$5k	−$10k	−$217k

Trend partially offsets the equity-correlated sleeves in S2 and S3. Without trend, the same scenarios would lose another 5% to 10%. This is why the allocation matters.

Worked example: Reading the stress table

Step 1: the equity column. The ERP sleeve holds $600,000, so an SPX fall of 10%, 20% and 35% costs $60,000, $120,000 and $210,000. The S4 loss of $132,000 is a gap of 22% on the same sleeve.

Step 2: the totals. In S2, $-120 - 50 + 10 - 8 = -168$ , in thousands of dollars, and in S3, $-210 - 110 + 30 - 15 = -305$ . The S3 loss is 30.5% of the portfolio, which by Chapter 1E needs a gain of $0.305/0.695 = 44\%$ to repair.

Step 3: without trend. Suppose the trend sleeve had been held in cash. S2 would lose $178,000 instead of $168,000, about 6% more, and S3 $335,000 instead of $305,000, about 10% more. That is the "another 5% to 10%" of the text, measured against the loss.

The figure applies the same table to weights you choose. Move the trend sleeve into cash, then into equities, and watch the S2 and S3 losses grow.

ERP 60%

VRP 15%

trend 15%

carry 10%

trend sleeve

in cash = 0%

Scenario	Loss	Gain to recover
S1	8.8%	10%
S2	16.8%	20%
S3	30.5%	44%
S4	21.7%	28%

Trend gains $10k in S2 and $30k in S3, the only sleeve that gains. Held in cash instead, those losses would be 6% and 10% larger.

What the whole foundation loses, and what trend saves. A hypothetical $1 million portfolio split across the four foundation sleeves, with anything left over in cash. Each sleeve's profit and loss per dollar in each scenario is taken from the chapter's stress table and scaled to the weight you set, so the default weights, 60/15/15/10, reproduce the table. The line across each bar is the total, and the table beneath gives it as a loss of the portfolio with the gain needed to recover from it. Move the trend sleeve into cash or into equities to see how much of the offset in S2 and S3 is trend's.

9 · Where this fits

The foundation premia are the heart of the framework: the part that anchors the portfolio. Chapter 10F covers the factor tilts that modulate the equity sleeve, Chapter 10G the structural overlays, Chapter 10H the regime filters and Chapter 10I the rejected category, what not to trade and why. Chapter 13E then walks through the conviction analysis for each of these strategies in different market conditions, which is where concept becomes judgment.

What to carry forward

Carry is the return if prices do not change, $(S - F)/F$ for a long futures position: about the asset's yield less the cost of financing it.
It pays for bad-times risk, with Sharpe ratios of 0.37 to 1.80 by asset class and 0.6 to 0.9 combined, but a retail account keeps 0.15 to 0.30. Hold 5% to 10% if you can access it, as a supplement rather than a foundation.
The foundation takes about 70% to 80% of portfolio risk: 50% to 60% equity, 10% to 15% each for the VRP and trend, and 5% to 10% for carry.
The ERP, the VRP and carry all lose in crashes. Trend is the one foundation premium that pays in slow drawdowns, and in S2 and S3 it offsets part of the others' losses.

10 · Check yourself

Question 1

Name the four foundation premia. Why is each a foundation rather than an overlay?

Answer

The equity risk premium, the variance risk premium, trend following and cross-asset carry. Each is a compensated risk premium, at least in part, and exists as long as the risk it pays for is real. That makes it durable enough to hold for decades as its own sleeve. An overlay only adjusts exposure to other premia and can vanish when its mechanism changes.

Question 2

Which foundation premium has only a middling standalone Sharpe ratio but the best portfolio contribution? Why?

Answer

Trend following. Its Sharpe ratio is middling, 0.20 to 0.40 in a retail account, but it is negatively correlated with equities in slow drawdowns and mostly uncorrelated with the VRP. By Chapter 10D's hurdle, a sleeve with near-zero correlation to the portfolio helps whenever its Sharpe ratio is positive. In the stress table it is the only sleeve that gains in S2 and S3.

Question 3

Apply the conviction score of Chapter 13B to a hypothetical 12% allocation to XSP put-writing in a calm regime, with the VIX at 14 and the term structure in contango. What is C?

Answer

The conviction score multiplies five scores between 0 and 1: evidence $E$ , mechanism $M$ , regime $R$ , execution $X$ and sizing survivability $S$ . Scores for this case might be $E = 0.85$ for T1 evidence and $M = 0.90$ for a clear risk premium. The regime might score $R = 0.85$ for a low VIX in contango, execution $X = 0.90$ for liquid XSP options, and sizing $S = 0.90$ for a cash-secured sleeve well within survivable size. Then $C = 0.85 \times 0.90 \times 0.85 \times 0.90 \times 0.90 = 0.53$ , above the 0.5 threshold for high conviction, so allocate. At 12% rather than a larger size, sizing survivability is at least as good, so C is at least about 0.53.

Question 4

Why does cross-asset carry have a wider retail haircut than the ERP?

Answer

The ERP comes in one cheap, liquid, tax-efficient ETF, which keeps a Sharpe ratio of 0.35 to 0.50 of an academic 0.4 to 0.5. Carry needs futures across many asset classes, rolled and rebalanced, and the mutual funds that packaged it, such as QSPRX and QSPIX, have closed or restricted new money. The FX part alone has weakened since 2008. So an academic 0.6 to 0.9 becomes a realistic 0.15 to 0.30.

Question 5

What is the operational difference between buying SPY and writing at-the-money (ATM) puts on XSP? When would you prefer each?

Answer

Buying SPY holds the index: linear exposure to its rises and falls, with dividends, no expiry and no roll, taxed like a stock by holding period (Chapter 2E). Writing at-the-money XSP puts sells insurance on the index: you collect a premium each month, keep none of the upside above the strike, and take nearly the whole downside, a negatively skewed payoff. You must roll each month and hold the strike's full value in cash, and gains are taxed under Section 1256. Prefer SPY for the equity premium, the portfolio's anchor. Prefer XSP puts for a 10% to 15% sleeve that harvests the VRP, sized for stress and halted when the VIX/VIX3M ratio exceeds 1.05.