Future value of loan: interest calculated daily and compounded at end of each calendar month

You can use this formula

where s is the principal, i is the rate, and dk is the number of days in month k

demonstrated here using Mathematica.

d[1]=  9; d[2]= 31; d[3]= 30; d[4]=31; d[5]= 31;
d[6]= 28; d[7]= 31; d[8]= 30; d[9]=31; d[10]=30;
d[11]=31; d[12]=31; d[13]=30;

t[n_] := s Product[1/365 (365 + i*d[k]), {k, n}]

s=50000;
i=0.055;

t[13]

52892.

Or in Excel. Ironically this only seems to work in Excel 365.

=C1*PRODUCT(1/365*(365+C2*A1:INDEX(A:A,B1)))

Longhand version

s = 50000
i = 0.055

s/365^13*(365 + 9 i)*(365 + 31 i)*(365 + 30 i)*
  (365 + 31 i)*(365 + 31 i)*(365 + 28 i)*
  (365 + 31 i)*(365 + 30 i)*(365 + 31 i)*
  (365 + 30 i)*(365 + 31 i)*(365 + 31 i)*(365 + 30 i) = 52892.02