Section I. Overview
Suppose an entity enters into an agreement to lease property and make rental payments each month, but that the fixed notional underlying the lease payments is denominated in some other currency. This introduces an exposure for the lessee (but not for the lessor) since now they must pay a domestic currency equivalent of some fixed amount in a foreign currency - in other words, they pay Lease Payment x Exchange Rate, whatever that might be at the time the payment becomes due. If the entity is a corporate entity, accounting regulations require the entity to "bifurcate" the embedded derivative from the contract and account for it as though it was a legitimate derivative, per the rules of derivative accounting. This introduces accounting complexities, but the problem must also of course be solved from a valuation point of view.
In this post we consider lease agreements as just described, as well as those with caps and floors on the exchange rate with strikes contractually written into the agreements.
Section II. Valuation Methodology
For leases determined to have embedded derivatives (from the point of view of the domestic entity), we value the embedded derivative as a strip of component derivatives corresponding to each future cash flow. That is, each cash flow represents the notional (denominated in the foreign currency CUR1) for each component derivative, and value of the lease embedded derivative is the aggregate value of these component derivatives.
Our FX convention is CUR1/CUR2 where this rate is the \# units of CUR2 per 1 unit of CUR1 - such a quantity has units [CUR2]/[CUR1]. We refer to CUR2 as the domestic, functional and settlement currency and CUR1 as the foreign, deal and notional currency.
Our valuation methodology is based on usual market-practice - in particular, no arbitrage and discounted cash flow principles. For options, we use the additional assumption of no-arbitrage for an asset price following a simple geometric Brownian motion (Black-Scholes-Merton model). Consider a present valuation date $t$, future maturity date $T>t$, future cash flow $N=N(T)$, corresponding strike rate $K=K(T)$, forward rate $F=F_{t}(T)$, discount rate $D=D_{t}(T)$ and volatility $\sigma=\sigma_{t}(K,T)$. Let $V=V_{t}(T,N,K,F,D,\sigma)$ denote the value of a derivative written on CUR1/CUR2 with the previous parameters. Then our previous assumptions lead us to the following valuation formulas:
$$(1)\;\;\;\;V^{\text{fwd}}_{t}(T)=N(T)\cdot(F_{t}(T)-K(T))\cdot D_{t}(T)$$
$$(2)\;\;\;\;V^{\text{call}}_{t}(\Phi(d_{+})F_{t}(T)-\Phi(d_{-})K)\cdot D_{t}(T),$$
and
$$(3)\;\;\;\;V^{\text{put}}_{t}(T)=(\Phi(-d_{-})K-N(-d_{+})F_{t}(T))\cdot D_{t}(T).$$
In (2) and (3) we define
$$d_{\pm}=\frac{1}{\sigma_{t}(K,T)\sqrt{T-t}}\left[\log\left(\frac{F_{t}(T)}{K(T)}\right)\pm\frac{1}{2}\sigma_{t}(K,T)^{2}(T-t)\right]$$
and
$$\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^{x}e^{-y^{2}}{2}\;dy,$$
the standard normal cumulative distribution function.
(Note the dependence of $\sigma_{t}$ on $(K,T)$ is due to the nature of FX option markets exhibiting term structure variation and ``smiles.'')
Section III. Extraction Methodology
Section III(a). Specifying the Strike
We extract the embedded derivative in accordance to the principle that the stated value of the cash flow at inception of the lease agreement should be such that the value of the embedded derivative at inception is $0$. We appy this principle to the forward component of the embedded derivative, and approximate it by assuming the cancellation between the cap and floor values (because one is a short position and the other is a long position - see below) would net $0$ if we assumed that they constitute a forward in combination. This is exactly true from put-call parity when the strikes are the same, but only approximately true if they are different (which they must be since otherwise the combination of the three instruments would net $0$ and there would be no embedded derivative). In particular, if the lease agreement has $i=1,2,3,\ldots,n$ future cash flows, a cap $\overline{S}=\overline{S}(T_{i})$ and a floor $\underline{S}=\underline{S}(T_{i})$, then for each corresponding component derivative we set (where again, $t=0$ is the inception date of the lease):
$$(4)\;\;\;\;K^{\text{fwd}}(T_{i})=F_{0}(T_{i}),$$
$$(4)\;\;\;\;K^{\text{cap}}(T_{i})=\overline{S}(T_{i}),$$
and
$$(4)\;\;\;\;K^{\text{flr}}(T_{i})=\underline{S}(T_{i}).$$
Accounting rules indicate that this is the proper approach from a valuation point of view.
Section III(b). Specifying the Derivative - A Decomposition
In keeping with our notation, we let $L_{t}(T_{i})$ denote the present fair value at time $t$ of the future cash flow made at time $T_{i}$. This is always a negative quantity from the entity's point of view. The idea in order to obtain the embedded derivative is to separate the risky portion of this value from the non-risky portion. In particular, we decompose $L_{t}(T_{i})$ as
$$L_{t}(T_{i})=B_{t}(T_{i})+Z_{t}(T_{i}),$$
where $B_{t}(T_{i})$ only depends on $t$ through the discounting at time $t$ $D_{t}(T_{i})$ (in particular, it is independent of market variables like $F_{t}(T_{i})$), and $Z_{t}(T_{i})$ is a function of all random market variables inherent in $L_{t}(T_{i})$. There are an infinite number of ways to structure such a decomposition, but accounting guidance discussed above is equivalent to certain initial and terminal conditions which allow us to uniquely solve for $B(T_{i})$ and $Z(T_{i})$.
Section III(c). Specifying the Derivative - Forward Only Case
If the lease payment $L_{t}(T_{i})$ lacks any optional features, then its payoff is
$$L_{T_{i}}(T_{i})=-N\cdot S_{T_{i}}$$
and therefore its fair present value for $0<t<T_{i}$ is given by
$$(7)\;\;\;\;L_{t}(T_{i})=-N(T_{i})\cdot F_{t}(T_{i})\cdot D_{t}(T_{i}).$$
Observe that this quantity has units of CUR2 and this is the present value of what the entity has to pay at time $T_{i}$. Since it depends on the forward rate $F_{t}(T_{i})$, it has an exposure to CUR1/CUR2 and is therefore risky. The ideas previous discussed involves decomposing $L_{t}(T_{i})$ into two parts
$$L_{t}(T_{i})=B_{t}(T_{i})+Z_{t}(T_{i}),$$
where $B_{t}(T_{i})$ only depends on $t$ through $D_{t}(T_{i})$ (in particular, it is independent of $F_{t}(T_{i})$), and $Z_{t}(T_{i})$ is a function of $F_{t}(T_{i}).$ The initial condition
$$Z_{0}(T_{i})=0$$
and
terminal payoff condition
$$L_{T_{i}}(T_{i})=B_{T_{i}}(T_{i})+Z_{T_{i}}(T_{i})=N(T_{i})\cdot F_{T_{i}}(T_{i})\cdot D_{T_{i}}(T_{i})=-N(T_{i})\cdot S_{T_{i}}$$
allow us to uniquely solve for the payoff of $B(T_{i})$ and $Z(T_{i})$, the principle of rational pricing and the fact that $B_{t}(T_{i})$ is a constant in $t$ (ignoring discounting) then gives us $L$, $B$, and $Z$ for all $0<t<T_{i})$. Indeed, from the terminal condition we have
$$Z_{T_{i}}(T_{i})=-N(T_{i})\cdot S_{T_{i}}-B_{T_{i}}(T_{i})$$
and from the initial condition
$$B_{0}(T_{i})=L_{0}(T_{i})=-N(T_{i})\cdot F_{0}(T_{i})\cdot D_{0}(T_{i})=-N(T_{i})\cdot K^{\text{fwd}}(T_{i})\cdot D_{0}(T_{i}).$$
Hence (dropping the ``fwd'' from $K$),
$$Z_{T_{i}}(T_{i})=-N(T_{i})\cdot(K(T_{i})-S_{T_{i}}).$$
This shows that the payoff of $Z(T_{i})$ is equal to a short position in CUR1/CUR2 with notion $N(T_{i})$. Applying our reasoning above, we discover $Z_{t}(T_{i})$ is given by (2). Explicitly,
$$(8)\;\;\;\;Z_{t}(T_{i})=-N(T_{i})\cdot(K(T_{i})-F_{t}(T_{i}))\cdot D_{t}(T_{i})$$br />
and
$$(9)\;\;\;\;B_{t}(T_{i})=-N(T_{i})\cdot K(T_{i}).$$
Section III(c). Specifying the Derivative - Ranged Forward Case (Caps & Floors)
If $L_{t}(T_{i})$ has optional features, then the initial condition $Z_{0}(T_{i})$ is replaced by the value of these optional features using the strike prices given by (5) and (6). For a ranged forward, we have a cap $\overline{S}(T_{i})$ and a floor $\underline{S}(T_{i})$. With our FX convention CUR1/CUR2, the terms ``cap'' and ``floor' are really as such from the counter-party's perspective, or from the entity's perspective when considering the value of the overall lease $L_{t}(T_{i})$ (a cap and floor on how much the entity has to pay). However, when considering the value of $Z_{t}(T_{i})$ from the entity's, the cap $\overline{S}(T_{i})$ is an upper-bound on how much CUR2 can weaken against CUR1, hence a floor ($=1/\overline{S}(T_{i})$) on their losses from their short position in the forward component of the embedded derivative. Conversely, the floor $\underline{S}(T_{i})$ is a lower-bound on how much CUR2 can strengthen against CUR1 hence a cap ($=1/\underline{S}(T_{i})$) on their gains.
The previous paragraph shows that $Z_{t}(T_{i})$ is a sum the sum of three distinct derivatives $\sum_{k=1}^{3}Z^{k}_{t}(T_{i})$ - a short position in a put option on CUR1/CUR2 struck at $\underline{S}(T_{i})$, a long position in a call option on CUR1/CUR2 struck at $\overline{S}(T_{i})$, and a short position in a forward on CUR1/CUR2 struck at $K(T_{i})=F_{0}(T_{i}).$ This can be proved as we did for the case of a forward, where the initial condition is taken to be
$$Z_{0}(T_{i})=\sum_{k=1}^{3}Z^{k}_{0}(T_{i})=V^{\text{call}}_{0}(T_{i})-V^{\text{put}}_{0}(T_{i})+\underbrace{V^{\text{fwd}}_{0}(T_{i})}_{=0},$$
as given by (1), (2) and (3), respectively.
The terminal condition is
$$L_{T_{i}}(T_{i})=\left\{\begin{array}{ll}-N(T_{i})\cdot\overline{S}(T_{i}),&S_{T_{i}}>\overline{S}(T_{i})\\-N(T_{i})\cdot S_{T_{i}},&\underline{S}(T_{i})\leq S_{T_{i}}\leq\overline{S}(T_{i})\\-N(T_{i})\cdot \underline{S}(T_{i}),&S_{T_{i}}<\underline{S}(T_{i}).\end{array}\right.$$
It follows that
$$B_{0}(T_{i})=-N(T_{i})\cdot K(T_{i})\cdot D_{0}(T_{i})$$
and hence
$$B_{t}(T_{i})=-N(T_{i})\cdot K(T_{i})\cdot D_{t}(T_{i})$$
for all $0<t<T_{i}.$ Now,
$$Z_{T_{i}}(T_{i})=L_{T_{i}}(T_{i})-B_{T_{i}}(T_{i})=\left\{\begin{array}{ll}K(T_{i})-\overline{S}(T_{i}),&S_{T_{i}}>\overline{S}(T_{i})\\K(T_{i})-S_{T_{i}},&\underline{S}(T_{i})\leq S_{T_{i}}\leq\overline{S}(T_{i})\\K(T_{i})-\underline{S}(T_{i}),&S_{T_{i}}<\underline{S}(T_{i}).\end{array}\right.$$
One verifies easily that this is equal to
$$Z_{T_{i}}(T_{i})=-(S_{T_{i}}K(T_{i}))+\max(S_{T_{i}}-\overline{S}(T_{i}),0)-\max(\underline{S}(T_{i})-S_{T_{i}},0)$$
which are the payoff functions of the indicated derivatives. Thus,
$$(10)\;\;\;\;Z_{t}(T_{i})=-A+B-C$$
for all $0<t<T_{i}$ where $A$ is given by (1), $B$ by (2) and $C$ by (3).
Section IV Lease Modifications - An Introduction
In a subsequent post I will elaborate on the bifurcation and valuation of modifications to lease agreements. For now, let us keep in mind the above consider a typical lease cash flow $L_{t}(T)$ with a notional of $N$. At the time the lease is entered into, FASB requires bifurcation of any implied derivative $Z$. Suppose $Z$ is just an FX forward (short CUR1/CUR2). At inception ($t=0$) the strike is $K_{0}=F_{0}(T)$, the forward rate corresponding to the future time $T$ as calculated at time $t=0$. The value of $Z$ at any time $0<t<T$ is
$$Z_{t}(N_{0},K_{0},T)=N_{0}\cdot(K_{0}-\cdot F{t}(T))\cdot D_{t}(T),$$
where $D_{t}(T)$ is the discount factor for term $T$ at time $t$. This valuation methodology makes the embedded derivative $0$ at inception of the lease.
Suppose at some time $0<\tau<T$ we have the modification $N_{0}\mapsto N_{\tau}<N_{0}$ (the lease payment decreases). Then this is economically equivalent to maintaining the unmodified lease and entering into another lease with notional $\Delta N_{0,\tau}:=N_{0}-N_{\tau}$ as a lessor at the time of modification $\tau$. FASB would then require the lessor to put the resulting embedded derivative on their balance sheet that time. The value of this derivative (since it is equivalent to a long position in CUR1/CUR2 or short CUR2/CUR1) is
$$\tilde{Z_{t}}(\Delta N_{0,\tau},K_{\tau},T)=\Delta N_{0,\tau}\cdot(F_{t}(T)-K_{\tau})\cdot D_{t}(T).$$
Now, from an operational lease accounting point of view, the P/L at the cash flow date is just $N_{0}-\Delta N_{0,\tau}=N_{\tau}.$ Therefore, the net embedded derivative of this overall lease contract is $Z+\tilde{Z}$ (the ``+'' is actually a ``-'' since we modeled $\tilde{Z}$ as a long position). Thus, the derivative's value at all times $\tau<t<T$ is
$$\begin{align*}
Z^{\tau}_{t}(N_{\tau},K_{\tau},T)
&=Z_{t}(N_{0},K_{0},T)+\tilde{Z_{t}}(\Delta N_{0,\tau},K_{\tau},T)\\
&=N_{0}\cdot(K_{0}-F_{t}(T))\cdot D_{t}(T)+\Delta N_{0,\tau}\cdot(F_{t}(T)-K_{\tau})\cdot D_{t}(T)\\
&=D_{t}(T)\Big[N_{0}K_{0}-N_{0}F_{t}(T)+N_{0}F_{t}(T)-N_{0}K_{\tau}-N_{\tau}F_{t}(T)+N_{\tau}K_{\tau}\Big]\\
&=N_{0}\cdot(K_{0}-K_{\tau})\cdot D_{t}(T)+N_{\tau}\cdot(K_{\tau}-F_{t}(T))\cdot D_{t}(T)\\
&=N_{0}\Delta K_{0,\tau}D_{t}(T)+N_{\tau}\cdot(K_{\tau}-F_{t}(T))\cdot D_{t}(T)\\
&=N_{\tau}(K_{\tau}-F_{t}(T))\cdot D_{t}(T)+C
\end{align*}$$
where the constant $C$ is the settlement price at time $T$ made at time $\tau$ and discounted to time $\tau<t<T$.
