Power Reverse Dual Currency (PRDC)

A PRDC is a product traded between a paying side (usually a bank) and a receiving investor, and the investor checks how the coupon changes with the exchange rate level and how the conditions that end the trade early take effect.

Exchanging an FX-linked coupon for a floating rate

A PRDC swap is a swap across FX and interest rates. One side pays a PRDC coupon whose amount is set by the exchange rate, and the other side pays a yen floating rate¹. It replaces the fixed side of an interest rate swap with a coupon linked to the exchange rate, and both payments apply to the same yen notional amount. Maturity can reach 30 years or more, longer than the typical interest rate swap, whose maturity is under 5 years¹. The basics of swaps across currencies are covered in Cross-Currency Swaps.

The source calls the side that pays the PRDC coupon (usually a bank) the issuer, and the side that receives it the investor¹. This page calls the receiving side the receiver and the paying side the payer. Consider a hypothetical example in which Company M, as the receiver, receives the PRDC coupon each year and pays the yen floating rate on a notional of 1 billion yen, for 30 years, with payments exchanged once a year. To keep the calculation easy to follow, assume the floating rate stays at 1% a year.

The payer is the dealer, who pays Company M a PRDC coupon set by the exchange rate on each payment date. Company M pays the dealer the yen floating rate applied to the same notional.

A bond whose principal is redeemed in yen and whose coupons are paid in a foreign currency, with several coupon swaps embedded to raise the coupon, is called a power reverse dual (PRD) note².

The formula that sets the coupon rate

The coupon rate $C$ for one payment is set from the exchange rate on the payment date (yen per dollar) by the following formula³.

$$C = \min\left(\max\left(c_f \times \frac{S}{f} - c_d,\ b_f\right),\ b_c\right)$$

$S$ is the exchange rate on the payment date, $c_f$ is the coupon rate of the foreign currency (the dollar), $c_d$ is the coupon rate of the yen, $f$ is the reference rate, $b_f$ is the lower limit (floor) of the coupon rate, and $b_c$ is the upper limit (cap). These values can differ from one payment date to the next. The reference rate is usually set at the forward exchange rate.

When the floor is 0 and there is no cap, the standard form, the formula can be rewritten as follows³.

$$C = h \times \max(S - K, 0), \quad h = \frac{c_f}{f}, \quad K = \frac{f \times c_d}{c_f}$$

$K$ is a level of the exchange rate, and the coupon is the amount by which $S$ exceeds $K$, multiplied by the rate $h$. The coupon for each payment has the same shape as a long position in a currency option call with strike price $K$, and the coupons of the whole swap can be seen as a series of long-dated currency options with different maturities³.

When $K$ is low relative to $S$, the coupon is more likely to be positive, but $h$ is usually also chosen low, so the level of the coupon is small. From the receiver's side this is a low-leverage situation, and when $K$ and $h$ are both high, leverage is high³.

As an example of calculating $K$, in the bond example in the Bank of Japan material ($c_f$ of 15%, $f$ of 123 yen, and $c_d$ of 9.6%), $K$ is about 78.7 yen².

One payment and the gain or loss

Company M's transaction has $c_f$ of 12%, $c_d$ of 9%, and $f$ of 150 yen on every payment date (the reference rate is held constant for simplicity). Assume the exchange rate when the trade is made is 155 yen per dollar, and the reference rate of 150 yen is set as a value corresponding to the forward exchange rate. The floor is 0 and there is no cap. $K$ is 150 × 9% ÷ 12% = 112.5 yen, and $h$ is 0.08% per yen. The payments for one period at each payment-date exchange rate are as follows.

When the exchange rate is below 112.5 yen, the coupon is zero, and Company M only pays the floating rate of 10 million yen. The net is exactly 0 at 125 yen, where the coupon rate equals the floating rate of 1%. The weaker the yen, the larger the coupon, reaching 54 million yen at 180 yen. In this example, Company M's largest payment for one net settlement is the floating rate of 10 million yen. That is only because the floating rate is held constant; in practice it changes with yen interest rates.

Figure description

The coupon rate has the same shape as a long position in a currency call option. It stays flat at 0% up to the level that corresponds to the strike price, and rises to the right, on the weaker-yen side, above that level. There is no cap, so the rise does not level off on the weaker-yen side. The vertical axis is the coupon rate on the notional and does not include the floating rate payment.

Comparing high-leverage and low-leverage forms

Company M's form (A) has $c_f$ of 12% and $c_d$ of 9%, and $K$ is 112.5 yen. Consider a hypothetical example in which a comparison form (B) has $c_f$ of 8% and $c_d$ of 5%, so $K$ is 93.75 yen. $f$ is 150 yen in both. A, with both $K$ and $h$ high, is the high-leverage form, and B is the low-leverage form. The table compares how they respond to the exchange rate.

At the reference rate of 150 yen, the two forms have the same coupon rate of 3.0%. A pays a higher coupon than B when the yen is weak and reaches zero before B when the yen is strong. For the receiver, both the gains and the losses when the exchange rate moves are larger with A.

Value that moves with the exchange rate and two interest rates

The value of a PRDC moves with the exchange rate and with the interest rates of the two currencies, yen and foreign⁴. The reference rate is usually set at the forward exchange rate at the start of the trade, and the forward exchange rate reflects the difference between the interest rates of the two currencies. The forward exchange rate $F(t, T)$ for maturity $T$ can be written as follows⁵.

$$F(t, T) = \frac{P_f(t, T)}{P_d(t, T)} \times S(t)$$

$P_d(t, T)$ and $P_f(t, T)$ are the prices at time $t$ of the yen and foreign zero-coupon bonds maturing at $T$, each in its own currency⁶. When yen interest rates are lower than foreign interest rates, the forward rate falls steeply as maturity gets longer, pointing to a large strengthening of the yen⁷.

The receiver takes the side that the yen will not strengthen as much as the forward rate implies, because the coupon grows as the exchange rate gets higher (a weaker yen)⁷. Conversely, if the yen strengthens to the reference rate, the coupon rate falls to $c_f - c_d$. For Company M's form ($c_f$ of 12% and $c_d$ of 9%), that is 3.0%.

Three conditions that end the swap early

A PRDC often has a condition that ends it partway through its term, and the three typical forms are below. In each, the swap can end only after the period before the first coupon (the no-call period, during which the swap is not ended) is over⁸.

A typical target amount is 10% to 50% of the notional, depending on the leverage⁸. The swap ends when the cumulative gain reaches the target, which is similar to TARFs. How the target is counted and how it is valued are covered in Target Redemption Forward (TARF).

Suppose a target amount of 20% is attached to Company M's example (A). If the exchange rate stays at 150 yen, the coupon is 3.0% every year, and the cumulative total is 18% after six payments. On the seventh, Company M receives only the 2.0% of the 3.0% coupon that brings the total to the target, so the cumulative total is 20% and the swap ends after 7 years. The coupon on the last date is cut to the amount that does not exceed the target⁹. Company M receives 200 million yen in coupons (20 million yen of it on the last date) and pays 70 million yen in floating payments, a net receipt of 130 million yen. In an actual contract, the treatment of the last date can differ.

If instead the exchange rate stays at 105 yen, the coupon is zero, the cumulative total does not grow, and the swap runs for 30 years. Company M pays the floating rate of 10 million yen every year for 30 years, 300 million yen in total (a simple sum that ignores discounting). The weaker the yen, the sooner the cumulative coupon reaches the target and the shorter the swap. The period in which Company M can receive high coupons is shortened by the same amount.

The same applies to knockout. If the level is 170 yen, the swap ends on the first payment date on which the exchange rate is above 170 yen (the coupon rate that day is 4.6%), and Company M cannot receive the high coupons after that.

In a form where the payer holds the right to cancel, the receiver receives a higher first fixed coupon in return for selling that right to the payer⁷. The knockout and target-redemption conditions also work, from the receiver's side, as a way to raise the yield⁷.

Risks the receiver bears and differences from similar products

The receiver bears three main risks. The first is a continuing strong yen. The coupon stays at zero and the receiver keeps paying the floating rate until maturity (30 years in Company M's example), as in the 105 yen example in the previous section. The second is that the swap ends while high coupons are still being paid. By accepting a knockout or target-redemption condition, the receiver loses the gains from a weak yen partway through. The third is that the valuation depends on the model and on how the skew is treated. Even for the same trade, the price changes with the assumptions (see the next section).

The differences from similar products are as follows. The coupons have the same shape as a series of long currency calls, but a PRDC is a swap that exchanges them for payments of a yen floating rate, and both payments apply to the same notional. A TARF is an FX contract made up of a series of agreements to buy or sell a currency at a set rate, and the remaining agreements disappear when the cumulative gain reaches a target. A PRDC differs in that its payments are an exchange of an FX-linked coupon for a yen floating rate.

Valuation models and practical difficulties

A PRDC depends on the exchange rate and on the interest rates of both currencies, so valuing it requires a model that moves at least three factors together⁴. The standard combination puts a short-rate model (Hull-White type) on each of the two interest rates, and models the exchange rate with either a lognormal distribution or local volatility, which expresses the skew⁴. The correlation among the three is usually estimated from historical data¹⁰.

For FX, long-dated currency options show a large skew (differences in volatility by strike, a form of the volatility smile), which a lognormal distribution cannot capture⁴. A PRDC with conditions such as knockout is especially affected by the skew, because of how the strike of the coupon and the level of the condition are set⁴. The price therefore changes with how the FX skew is treated (model risk). The doctoral thesis analyzes how much the skew affects the price of a PRDC.

The model's parameters are set to match market prices (calibration). The interest rate side can be set from swaption prices. For the FX side, a PRDC has a long maturity and a wide range of strikes, so currency options with many maturities and strikes are needed, which takes effort¹⁰.

The price is found by solving the model's equations numerically. One method solves a partial differential equation with three variables, updating state such as the cumulative total on each payment date. A target-redemption condition makes payments depend on the past path, so a forward method such as Monte Carlo is considered the natural choice¹¹.

Deep diveEarly cancellation as a swaption

Cancelling a PRDC swap on a payment date $T_\alpha$ is the same as continuing the original swap and, on the same day, entering into an identical swap with the payments in the opposite direction¹². The payments on later dates offset each other to zero, which gives the same result as cancelling. So the payer's right to cancel can be valued in the same form as a Bermudan swaption (the right can be exercised on several dates), which is the right to enter into the opposite swap on each payment date¹².

What this page does not cover

Hedging procedures, counterparty credit risk, and how easily it can be traded in the market are not covered on this page.

Notes

1. Duy Minh Dang (University of Toronto, 2011), Section 2.2.1 "Description of PRDC Swaps" (pages 22–23). The notional of the payments, the maturity, and the terms issuer and investor are on page 22. ↩ ↩² ↩³

2. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "仕組商品投資のリスク把握と管理" (Risk assessment and management of investment in structured products; our translation) (2019), page 4, "(1) PRD (power reverse dual) notes" (in Japanese). The reverse dual note, the embedding of coupon swaps, and the figures in the example (dollar 15%, 123 yen, yen 9.6%; term of 15 years, zero floor). The $K$ of about 78.7 yen is a calculation from the figures in the example (123 × 9.6 ÷ 15). ↩ ↩²

3. Duy Minh Dang's doctoral thesis, Section 2.2.1, equation (2.8) (page 23) and the text around equation (2.9) (page 24). The coupon formula, the fact that the reference rate $f$ is usually the forward exchange rate, the standard form (floor of 0, no cap), the symbols $h$ and $K$ (written $e_\alpha$ in the source), and the explanation of high and low leverage. The symbols in the formula are the same as the source's $c_f$, $c_d$, $b_f$, and $b_c$. ↩ ↩² ↩³ ↩⁴ ↩⁵

4. Duy Minh Dang's doctoral thesis, Section 2.2.2 "The Model" (pages 27–28). The standard models, the skew in long-dated currency options, and its effect on conditional PRDCs. The Hull-White type is on page 28, equations (2.10b) and (2.10c). ↩ ↩² ↩³ ↩⁴ ↩⁵

5. Duy Minh Dang's doctoral thesis, Section 2.1.1 "Definitions and Notation", "Forward FX Rate", equation (2.5) (page 17). ↩

6. Duy Minh Dang's doctoral thesis, Section 2.1.1, the continuation of "Tenor Structure" (page 17). That $P_i(t, T)$ are the prices of the zero-coupon bonds of the yen (domestic) and the foreign currency. ↩

7. Duy Minh Dang's doctoral thesis, Section 2.2.1, "Economics of PRDC Swaps" (pages 26–27). That the forward exchange rate falls steeply with maturity when yen interest rates are lower than foreign interest rates, that the receiver takes the side that the yen will not strengthen as much as the forward rate implies, and that the cancellation right, knockout, and similar features raise the receiver's first coupon. ↩ ↩² ↩³ ↩⁴

8. Duy Minh Dang's doctoral thesis, Section 2.2.1, "Exotic Variations" (pages 25–26). The three conditions, the no-call period, that knockout usually ends after the payments, that there are forms in which the level moves over time, and that a typical target amount is 10% to 50% of the notional. The "Who decides the end" column in the table is organized from the source's description (the issuer for cancellation, automatic by condition for the others). ↩ ↩² ↩³

9. Duy Minh Dang's doctoral thesis, Section 3.6.1 (equations 3.12 and 3.13). The cumulative total is updated as "cumulative + min(target − cumulative, coupon for that date)", so the coupon on the last date is the amount that does not exceed the target, and the swap ends when the cumulative total reaches the target. ↩

10. Duy Minh Dang's doctoral thesis, Section 2.2.3 "Calibration Overview" (pages 29–30). That the interest rate models are matched to swaptions, that adjusting the FX local volatility needs a wide range of currency options because of the long maturity and the choice of strikes of a PRDC and takes effort, and that correlations are estimated from historical data. ↩ ↩²

11. Duy Minh Dang's doctoral thesis, Chapter 1 "Introduction" (page 2) and Chapter 8 "Summary and Future Work" (page 164). That partial differential equation methods are widely used for problems with fewer than 5 dimensions, and that for FX-TARN PRDCs forward methods (such as Monte Carlo) are natural. ↩

12. Duy Minh Dang's doctoral thesis, Section 3.4.1 "Key Observation" (pages 42–43), "Bermudan Swaption" in Section 2.1 (pages 20–21), and "Economics of PRDC Swaps" in Section 2.2.1 (page 27). ↩ ↩²