**Carry Trade Strategy: Profiting From Interest Rate Differentials** – In two previous articles, I showed how to calculate the price of one interest rate swap and DV01 with a book containing thousands of swaps.

I now focus on the calculation of Carry and Roll-Down for a single swap, defined as the total amount (drawn + unexecuted) earned by holding the swap until a given time T, called the horizon. The assumption is that the yield curve at time T has exactly the same shape as today.

## Carry Trade Strategy: Profiting From Interest Rate Differentials

The precise definition of CFinterm’s cash flows received between T0 and T depends on whether amounts accrued during this period but paid after T are considered part of those cash flows. Similar attention is paid to the definition of PV(T).

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If the amount accrued — but not paid by T — is treated as “earned” by T, the treatment is the same as calculating the bond’s net value.

Both approaches have their advantages and disadvantages, but the “pure” definition is preferred, perhaps because the cumulative amount of T is considered “earned” from an economic point of view.

Carry and Roll-Down CR consists of two parts. The first is PV(CFinterm) corresponding to the realized profit from holding the position until T, and the second is PV(T) – PV(T0) corresponding to the unrealized profit (investment profit) due to price appreciation. Swap position between time T0 and time T.

It is common to call the first part of the swap the Carry C and the second part the Roll-Down R.

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The following chart will help you visualize these two metrics for a 7-year receiver replacement with an annual frequency of one year.

Since the swap went live about 4 months ago, the first coupon is paid before the time expires.

The above definition applies to absolute amounts with direct financial interpretation. For swaps, different concepts or tenor swaps cannot be used without scaling.

Therefore, swap traders prefer the relative Carry and Roll-Down concept, which is derived by dividing the above absolute amount by the underlying swap spot or forward (as T) flat DV01.

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Therefore, the definition can apply to any financial product imaginable, and one fact is that Curry’s original concept was used in the world of forex with so-called carry trades, whereby a forex trader could profit by simply borrowing currency. lending in high-interest currencies (eg, USD) versus low-interest (eg, Yen). Therefore, between T0 and T, the trader will pay a lower interest rate on the borrowed currency, and earn a higher interest rate on the borrowed currency, making a net income. If the spot rate of the currency between T0 and T does not change too much, the trader closes the position without losing the exchange rate at time T. Note that fx car trading is not an arbitrage free trading strategy. Expected gains may not materialize because they depend on the assumption that the spot exchange rate will remain the same.

The next major sector that uses Carry and Roll-Down is the fixed income market, particularly the bond market.

Most fixed income portfolio managers must follow certain guidelines regarding total portfolio duration. For example, the guidelines for a particular portfolio may have an average maturity of 5 years. This means that over time, the portfolio manager must periodically, say monthly, replace certain bonds with other bonds with longer maturities. The higher the “Cry and Roll-Down” of a particular bond, the higher the profit from switching it.

What is the Carl bond trading spot rate is the general shape of the transportation bond trading yield curve. Just as a foreign exchange trader expects the spot exchange rate not to change much between T0 and T, a bond trader is concerned with the slope and curvature of the yield curve.

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For example, if the yield curve is upward-sloping and assumed to hold its shape until T, the bonds’ coupons at time T will, on average, have a lower discount than at time T0.

The consequence is that bond prices in period T will tend to be higher than those in period T0.

This is exactly the price increase that a trader with an upward sloping curve would expect. This represents a positive Roll-Down size, and trading strategies that use this effect are called “rolling down the yield curve”.

Deriscope calculates the shipment at any date T by summing the discounted values of all cash flows before and after T .

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The calculation algorithm depends on whether the accumulated but unpaid amount T is considered “earned” as part of the cash flow in the interval (T0, T). Deriscope allows users to choose between these two treatments. Additional input key facility Ignore Accumulation that the user can set to TRUE or FALSE.

For now, I’ll explain the Ignore Accrual=TRUE case, because it’s conceptually simpler and more elegantly handled, whereby the swap is not changed, only the curve is transformed in a certain way. The last detail is a good exercise in understanding and managing yield curves in general.

Therefore, in this situation, the NPV of the swap is calculated in the usual way, but using a discount yield curve with an additional twist equal to the original discount curve, making the discount factor equal to 0 for all periods above T.

This technique has the advantage of treating all swaps in the developed portfolio on the same basis without having to check what cash flows occur before or after T .

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First, a new discount and hypothetical yield curve is formed by “shifting” to the right along the time axis at intervals equal to T – T0.

The word “transpose” is in quotation marks because the operation is not simply a crude parallel transposition of the original curve.

What happens is that the produced curve contains forward discount factors equal to the time-shifted spot discount factors of the corresponding original curve.

More specifically, by setting Δ = Τ – T0, the new “shifted” forecast curve is constructed so that the discount factor DFRf(T1) for any period T1 > T is given by:

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These formulas show that the “shifted” forecast curve represents the forward rate beyond T and matches what would have been observed at T if the curve had been the same as today’s curve.

This transformation is sufficient for the prediction curve, but the discount curve needs to be transformed again. This should be further modified by dividing all discount factors by the original discount factor of T .

This is necessary because we want the NPV calculated by today’s shifted curve to represent T’s observed NPV.

In fact, the final discount curve represents the discount factor DFRd(T1) for any period T1 > T, which is equal to

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Then, when I use these two curves to value any swap, the resulting NPV should be equal to the NPV that would be observed at time T if the curves had the same shape as they do at T0.

Subtract the NPV of T0 from the above NPV to obtain the final absolute Roll-Down value.

Bloomberg shows the date T represented by the given yield curve, but only if T falls at the beginning of the swap period. Broken joints are not supported.

The carry is calculated in bps per annum minus the nominal exchange rate for the period from swap inception to expiration.

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Roll-Down is calculated by subtracting the nominal rate from the start (minus the expiration date) of the swap, in bps per annum.

Assume that the swap has N periods, and the intervention time is set at the end of the kth period.

Bloomberg’s formula should be considered an approximation of the true Carry and Roll-Down in the absence of a more accurate estimate, such as Deriscope, which implements the definition by calculating the NPV of the relevant cash flows.

The Carry formula is only valid for nominal swaps. But the swap is only equal at the outset. Then, when the market rate changes, the value of the swap is no longer zero and the derivative loses its value.

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It also breaks down for nominal swaps with non-standard characteristics, such as amortization/fixation and increase/decrease.

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