def growth_coefficients(start_date, end_date, ref_date, alpha, samples):
    """
    Build a matrix of growth factors according to the CAGR formula  y'=y0 (1+a)^(t'-t0).

    a growth rate alpha
    t0 start date
    t' end date
    y' output
    y0 start value

    """

    start_offset = 0
    if ref_date < start_date:
        offset_delta = rdelta.relativedelta(start_date, ref_date)
        start_offset = offset_delta.months + 12 * offset_delta.years
        start_date = ref_date

    end_offset = 0
    if ref_date > end_date:
        offset_delta = rdelta.relativedelta(ref_date, end_date)
        end_offset = offset_delta.months + 12 * offset_delta.years
        end_date = ref_date

    delta_ar = rdelta.relativedelta(ref_date, start_date)
    ar = delta_ar.months + 12 * delta_ar.years
    delta_br = rdelta.relativedelta(end_date, ref_date)
    br = delta_br.months + 12 * delta_br.years

    # we place the ref point on the lower interval (delta_ar + 1) but let it start from 0
    # in turn we let the upper interval start from 1
    g = np.fromfunction(lambda i, j: np.power(1 - alpha, np.abs(i) / 12), (ar + 1, samples), dtype=float)
    h = np.fromfunction(lambda i, j: np.power(1 + alpha, np.abs(i + 1) / 12), (br, samples), dtype=float)
    g = np.flipud(g)

    # now join the two arrays
    a = np.vstack((g, h))

    if start_offset > 0:
        a = a[start_offset:]
    if end_offset > 0:
        a = a[:-end_offset]

    return a