MSCA-IF: A stable payroll setup strategy when the total costs are accounted in a currency different from the one used to pay the salary
Assumption: Funds are deposited in the same currency as the one used while auditing total costs.
Let,
- \(y_i\) be the gross salary to be paid in the \(i\)th month in Pound Sterling (£).
- \( \displaystyle b = \begin{cases} 0 & \text{opt-out of USS} \\ 1 & \text{else} \end{cases} \)
- \( \displaystyle \langle x \rangle := \begin{cases} 0 & \text{for } x \leq 0 \\ x & \text{for } x > 0 \end{cases} \)
- \(e_i\) be the Euro to Pound Sterling exchange rate when the salary £\(y_i\) is paid.
Then the total cost (accounting for the employer's NI cost) in Pound Sterling for the \(i\)th month is
\[ (1 + 0.26\,b) y_i + 0.138 \left\langle y_i - 676 \right\rangle \]If \(y_i > 676\), which is true for MSCA IF salaries, the total cost in Pound Sterling for the \(i\)th month is
\[ (1 + 0.26\,b) y_i + 0.138 \left( y_i - 676\right) = (1.138 + 0.26\,b) y_i - 0.138 \times 676 \]If £\(y_i\) is equivalent to \(x_i\)€, then we have \(x_i = (y_i/e_i)\). So, to pay a gross salary of £\(y_i\) in the \(i\)th month we have to arrage for the following amount in Euro.
\[ (1.138 + 0.26\,b) x_i - 0.138 \frac{676}{e_i} \]The total cost in Euro for an year should be equal to the living allowance \(C\) for that year.
\[ C = \sum_{i = 1}^{12} (1.138 + 0.26\,b) x_i - 0.138 \frac{676}{e_i} = 12 \left[ (1.138 + 0.26\,b) \bar{x}_{\text{A}} - 0.138 \frac{676}{\bar{e}_{\text{H}}} \right] \]where \(\bar{x}_{\text{A}}\) and \(\bar{e}_{\text{H}}\) are the arithmetic mean of all \(x_i\) and the harmonic mean of all \(e_i\), respectively.
\[ \bar{x}_{\text{A}} = \frac{1}{12}\sum_{i = 1}^{12} x_{i}, \quad \frac{1}{\bar{e}_{\text{H}}} = \frac{1}{12}\sum_{i = 1}^{12} \frac{1}{e_{i}} \]The average gross salary per month in Euro can be expressed as,
\[ \bar{x}_{\text{A}} = \frac{(C/12) + 0.138\,(676/\bar{e}_{\text{H}})}{1.138 + 0.26\,b} \]Naturally, we need to estimate \(\bar{e}_{\text{H}}\) which is not known a priori. Nevertheless, any error we make in the estimation of \(\bar{e}_{\text{H}}\) only affects a relatively insignificant term: \(0.138\,(676/\bar{e}_{\text{H}})\).
PROPOSAL: pay a gross salary of £\(y_i = e_i \bar{x}_{\text{A}}\) in the \(i\)th month.
Example: MSCA IF in the UK where the mobility and living allowance are included and paid as taxable salary and the fellow contributes to the USS
In this case the annual living allowance is \(C = 80327.40\)€. We can compute \(\bar{x}_{\text{A}}\) as follows.
\[ \bar{x}_{\text{A}} = \frac{(80327.4/12) + 0.138\,(676/\bar{e}_{\text{H}})}{1.138 + 0.26} = \frac{6693.95 + (93.288/\bar{e}_{\text{H}})}{1.398} = 4788.23 + \frac{66.73}{\bar{e}_{\text{H}}} \]The host institution deducts \(1.398 \bar{x}_{\text{A}}\)€ every month from the living allowance to pay a gross salary of £\(e_i \bar{x}_{\text{A}}\), the employer's NI contribution of £\(0.138 e_i \bar{x}_{\text{A}}\) and the total (employer + employee) USS contribution of £\(0.26 e_i \bar{x}_{\text{A}}\). Recall that \(e_i\) is the true exchange rate (a priori unknown and volatile) on the day the salary is paid and \(\bar{e}_{\text{H}}\) is an estimate.
Consider two extreme choices for the estimate: \(\bar{e}_{\text{H}} = 0.7\) and \(\bar{e}_{\text{H}} = 1.0\). This results in a difference \(\Delta \bar{x}_{\text{A}}\) of just
\[ \Delta \bar{x}_{\text{A}} = 66.73 \left(\frac{1}{0.7} - 1\right) = 25.6 \]A perturbation of just \(25.60\)€ per month! This is a win-win situation for the host and the fellow.