Backtesting of macro trading strategies requires good approximate profit-and-loss data for standard derivatives positions, particularly in equity, foreign exchange, and rates markets. Practical calculation methods of generic proxy returns not only deliver valid strategy targets but are also the basis of volatility adjustments of trading factors and for calculating nominal and real “carry” of macro derivatives. A methodological summary for equity index futures, FX forwards, and interest rate swaps shows that generic return and carry formulas need not be complicated. However, decisions on how to simplify and set conventions require good judgment and adjustment to institutional needs.

The below post is a simplified summary of a technical paper on proxy return and carry calculations by Dr. Lasse de la Porte Simonsen, Director of Systems & Advanced Analytics at Macrosynergy.

## The importance of proxy returns and carry

Backtesting of macro trading strategies typically requires good approximate profit-and-loss (PnL) data for standard derivatives positions, particularly for three major types of macro asset classes:

**Equity**: The performance of company shares is approximated by returns of equity index futures, based on the most liquid or broadest liquid local currency-denominated index of stocks listed in a currency area. The future’s return represents the PnL of exposure to the equity index funded in local currency.**Foreign exchange**: The most common derivatives are foreign exchange forward contracts with maturities of 1-3 months. They approximate the performance of base currency positions in terms of a quote currency. For example, the return on an AUDUSD forward depends positively on the future price of the Australian dollar in terms of the U.S. dollar. Under full convertibility, the return on a long forward in the base currency is close to the PnL of a base currency deposit funded in the quote currency.**Duration**: The performance of long-duration positions can be approximated by returns of fixed receiver positions in interest rate swaps (IRS), typically with maturities of 2, 5, or 10 years. The return on the fixed receiver represents the PnL of exposure to longer-term interest rate risk that has been funded in local currency, where the interest paid on funding is determined by a shorter-maturity floating rate.

Returns of the above derivatives are better called “excess returns” because they are based on PnLs that conceptually strip out funding costs or opportunity costs of cash deployment. These derivative returns can be scaled in line with the leverage the investment manager is willing and able to take. On a day-by-day basis, all these broad macro excess returns mostly depend on changes in equilibrium prices of the economy, i.e., the price of capital, the price of long-term rates exposure (approximate default-free credit), and the price of the local currency in terms of a base currency.

Approximate macro derivatives returns are not only necessary target data for evaluating the predictive power and economic value of macro trading factors. They are also the basis of trading factors’ volatility adjustments, estimating the volatility scaling for position sizing and calculating the “carry” of macro derivatives.

__Carry can be defined consistently across asset classes and contracts as the return that would accrue for unchanged market prices__. It is a popular building block of many macro trading strategies. That is because carry is easy to calculate in real-time, based on standard return formulas, and is often related to risk premia and implicit subsidies (view post here). We can distinguish between nominal and real carry. __Real carry does not assume that all prices in the economy are unchanged but allows an expected inflation rate to drive relative price drifts between real assets, such as capital, and nominal assets, __such as deposits and loans, and between nominal assets across currency areas.

In the following sections, we summarize the conceptual highlights of a technical paper that explains generic return and carry calculation in the J.P. Morgan Macrosynergy Quantamental System (JPMaQS) based on standard quoted market prices. Some returns and carry data panels are available without a subscription (view dataset on Kaggle) and are highly useful for macro factor analysis. The full set of generic return data is explained on the quantamental academy (view downloadable Jupyter notebooks here), as is a broad set of generic carry data (view downloadable Jupyter notebooks here).

## Excess returns and carry of equity index futures

An equity index future is a financial derivative contract whose payout is directly linked to the price of a basket of stocks at a specific future date, typically representing a country or sector. In the short run, the price of the future changes proportionately to the price of the underlying index. The buyer of a future does not have to put up the full amount of the underlying notional in cash but is typically required to post initial and maintenance margins to diminish the risks of counterparty default. The proxy return calculations below disregard marginal costs and credit value adjustments.

Principally, an index future’s excess return is merely its price change over the return period relative to its price at the end of the base period.

Here F_{t} is the index’s future price at the end of period t.

The futures price at the end of each period equals the index spot price plus funding costs of the index cash position until the future’s expiry date and minus the expected dividend yield on the index cash position until the future’s expiry date.

Here S_{t} is the “spot” price of the cash index at the end of period t, i_{t,h }the relevant interest rate proxy for funding costs at the end of the period for the time horizon h until expiry, E_{t} is the expectation operator based on information at the end of period t, and DPS_{t,t+h }the dividend payments per share between the end of t and the end of the future’s expiry date. For a conceptual and empirical overview of equity index futures returns in developed and emerging markets, view the documentation notebook here.

Applying the general definition of carry requires more thought for equity than for other asset classes. That is because cash payout is not the only economic benefit that accrues to the share owner, even if buybacks are considered a form of payout. Instead, non-distributed earnings theoretically increase the value of a company and drive a wedge between the quoted price of a company share and the abstract price per unit of capital in the corporate sector. For the calculation of carry, a decision must be made on what price is held stable. If the stock price is held stable, carry is largely determined by the dividend yield. If the price of a unit of capital is held stable, the earnings yield takes this role. Here, we assume that carry is based on a weighted average of the two concepts. Following this approach, the approximate annualized carry is the annualized weighted average of dividend and earnings yield over a specific horizon (typically one year) minus the funding costs over that same horizon.

Here EPS_{t,t+1y }are the earnings per share accrued from the end of period t over the following year, DPS_{t,t+1y} are the dividends per share paid from the end of period t over the following year, it,1y the one-year interest rate that is a relevant proxy for funding costs of an equity index position, and θ is the share of the dividend yield in the calculation. In the J.P. Morgan Macrosynergy Quantamental System, this has been set at 0.5.

Real carry of an equity index future is nominal carry plus expected inflation. This is because inflation increases the nominal value of real assets, such as equity, but does not affect the nominal amount of local-currency debt, which is the conceptual basis of funding costs.

Here π_{t,t+1y }is the rate of broad inflation from the end of period t one year out, typically approximated by a reasonable prediction of consumer price inflation over the next year.

For a more detailed and technical description of equity index future return and carry, see section 2 of the underlying paper. For a conceptual and empirical overview of equity index futures carry in developed and emerging markets, view the documentation notebook here.

## Excess returns and carry of FX forwards

An FX forward is a financial derivative contract that stipulates the exchange of two currencies at a predetermined exchange rate on a future settlement date. Similarly, a non-deliverable FX forward (NDF) contract, often used if one currency is not fully convertible, stipulates the cash settlement of the difference between a predetermined rate and the spot rate on the settlement date. The settlement date is typically in the future, ranging from a few days to several months.

The excess return of an FX forward that is long the local currency and short the quote currency is simply the change in the forward exchange rate for a specific settlement date, whereby the exchange rate gives the price of the base currency in units of the quote currency:

Here f_{t,h }denotes the forward exchange rate at the end of period t for a contract that expires in h periods, i.e., at the end of period t+h. The above change implies that the forward horizon shortens as the forward contract is held without a roll. This means that the maturity declines until a rules-based restoration of the original maturity, for example, an exchange of short to longer maturity contract when a minimum maturity has been reached. For a conceptual and empirical overview of 1-month FX forward and NDF returns with monthly rolls in developed and emerging markets, view the documentation notebook here.

The forward exchange rate is linked to the spot exchange rate via the forward implied carry, i.e., which itself is based on forward points quoted by the market. Due to arbitrage, the forward points typically, but not always, have a close link to the differentials of on-shore low-risk deposit rates across the two currencies. The latter relation is called the covered interest rate parity.

In the above equation, s_{t} is the spot exchange rate, the price of the base currency in units of the quote currency, i_{base,h }is the interest rate in the base currency for the time-to-expiry of the forward contract and i_{quote,h }is the equivalent interest rate in the quote currency area. Thus, forward implied appreciation of the base currency versus the quote currency, which means a higher forward price than spot price of the base currency, corresponds to a lower interest rate in the base currency than in the quote currency.

Nominal forward-applied FX carry of the base currency relative to the quote currency is simply the annualized implied rate of forward-implied depreciation, i.e., the return that would accrue if the spot exchange rate just remained unchanged:

As for the case of equity index futures, we can calculate real carry, i.e., the return that would be earned if relative prices drifted in accordance with expected inflation. The carry for being long the base currency versus the quote currency would be diminished if expected inflation in the base currency area was higher than in the quote currency area:

Here π_{i,t,t+1y }is the broad inflation rate in currency area i from the end of period t one year out, typically approximated by a reasonable prediction of consumer price inflation over the next year.

For a more detailed and technical description of FX forward returns and carry see section 3 of the underlying paper. For a conceptual and empirical overview of FX forward and NDF carry in developed and emerging markets, view the documentation notebook here.

## Returns and carry of interest rate swaps (fixed receivers)

Interest rate swaps are derivatives contracts that allow parties to exchange interest rate cash flows over a specified period, with one party typically paying a fixed interest rate while the other party pays a floating rate linked to a variable reference rate, such as an OIS rate or LIBOR. During the life of the swap, periodic net differences of interest payments are exchanged. A contract that entitles a party to receive the fixed rate and obligates them to pay the floating rate constitutes a fixed receiver position.

The excess return on an interest rate swap fixed receiver position between the end of two trading days can analytically be presented as the sum of three effects: the **market price change** for a contract with rest maturity on the return day, the effect of the decline in maturity (“**rolldown**”) since the day before, and the **interest rate differentia**l between the fixed and the floating interest rate of the swap. For the assumption of a daily contract roll, these three additive terms give the excess return as per below:

Here y_{t,h }is the yield to maturity of the fixed leg, h is the number of periods to maturity (e.g., days), i_{t} is the floating rate, and D_{mod,t} is the modified duration of the swap contract at time t. Modified duration is a measure of the sensitivity of the price of a fixed-income contract to changes in interest rates. It quantifies the percentage change in the price of the security for a one percentage point change in its yield to maturity if all other factors remain constant.

For approximate returns, the excess return formula is often simplified by using constant maturity prices, i.e., by effectively excluding the rolldown and approximating the price change based on the constant maturity. Many market databases only contain yields for selected tenors, and this biased estimation many data databases only contain yields for selected tenors can be justified for short-term returns, where rolldown effects are small.

For a conceptual and empirical overview of IRS receiver returns in developed and emerging markets, view the documentation notebook here.

Since carry is the annualized return for unchanged prices, it can be derived from the above excess return formula by adding the assumption that the yield curve is unchanged. Then, it is the return that ensues from the passage of time alone, which simplifies it to rolldown (decline in maturity by one year) and interest rate differential.

Real carry and nominal are identical for interest rate swaps since both legs of the trade would be equally affected by inflation.

For a conceptual and empirical overview of interest rate swap receiver carry in developed and emerging markets, view the documentation notebook here.