diff --git a/conservancy_beancount/reports/core.py b/conservancy_beancount/reports/core.py
index cf5a39a7ca440f6073cfa68165533c37ceccc5ab..6b2845c70f7b3fcf29aef1af8c603fee4ab62ba0 100644
--- a/conservancy_beancount/reports/core.py
+++ b/conservancy_beancount/reports/core.py
@@ -132,10 +132,12 @@ class Balance(Mapping[str, data.Amount]):
         return type(self)(retval_map.values())
 
     def __eq__(self, other: Any) -> bool:
-        if (self.is_zero()
-            and isinstance(other, Balance)
-            and other.is_zero()):
-            return True
+        if isinstance(other, Balance):
+            clean_self = self.clean_copy()
+            clean_other = other.clean_copy()
+            return len(clean_self) == len(clean_other) and all(
+                clean_self[key] == clean_other.get(key) for key in clean_self
+            )
         else:
             return super().__eq__(other)
 
@@ -160,6 +162,17 @@ class Balance(Mapping[str, data.Amount]):
     ) -> bool:
         return all(op_func(amt.number, operand) for amt in self.values())
 
+    def copy(self: BalanceType) -> BalanceType:
+        return type(self)(self.values())
+
+    def clean_copy(self: BalanceType, tolerance: Optional[Decimal]=None) -> BalanceType:
+        if tolerance is None:
+            tolerance = self.tolerance
+        return type(self)(
+            amount for amount in self.values()
+            if abs(amount.number) >= tolerance
+        )
+
     @staticmethod
     def within_tolerance(dec: DecimalCompat, tolerance: DecimalCompat) -> bool:
         dec = cast(Decimal, dec)
diff --git a/tests/test_reports_balance.py b/tests/test_reports_balance.py
index 22271857aadd74662da6ad4c8d0b375e987356a1..706531e63689f3e5196803cb7f9aaa949e98f3c1 100644
--- a/tests/test_reports_balance.py
+++ b/tests/test_reports_balance.py
@@ -34,6 +34,8 @@ DEFAULT_STRINGS = [
     ({'JPY': '-5500.00', 'BRL': '-8500.00'}, "-8,500.00 BRL, -5,500 JPY"),
 ]
 
+TOLERANCES = [Decimal(n) for n in ['.1', '.01', '.001', 0]]
+
 def amounts_from_map(currency_map):
     for code, number in currency_map.items():
         yield testutil.Amount(number, code)
@@ -219,6 +221,15 @@ def test_eq(map1, map2, expected):
     actual = bal1 == bal2
     assert actual == expected
 
+@pytest.mark.parametrize('tolerance', TOLERANCES)
+def test_eq_considers_tolerance(tolerance):
+    tolerance = Decimal(tolerance)
+    mapping = {'EUR': 100, 'USD': '.002'}
+    bal1 = core.Balance(amounts_from_map(mapping))
+    mapping['USD'] = '.004'
+    bal2 = core.Balance(amounts_from_map(mapping), tolerance)
+    assert (bal1 == bal2) == (tolerance > Decimal('.002'))
+
 @pytest.mark.parametrize('number,currency', {
     (50, 'USD'),
     (-50, 'USD'),
@@ -294,6 +305,34 @@ def test_iadd_balance(mapping):
     expected = core.Balance(amounts_from_map(expect_numbers))
     assert balance == expected
 
+def test_copy():
+    amounts = frozenset(amounts_from_map({'USD': 10, 'EUR': '.001'}))
+    # Use a ridiculous tolerance to test it doesn't matter.
+    actual = core.Balance(amounts, 100).copy()
+    assert frozenset(actual.values()) == amounts
+
+@pytest.mark.parametrize('tolerance', TOLERANCES)
+def test_clean_copy(tolerance):
+    usd = testutil.Amount(10)
+    eur = testutil.Amount('.002', 'EUR')
+    actual = core.Balance([usd, eur], tolerance).clean_copy()
+    if tolerance < eur.number:
+        expected = {usd, eur}
+    else:
+        expected = {usd}
+    assert frozenset(actual.values()) == expected
+
+@pytest.mark.parametrize('tolerance', TOLERANCES)
+def test_clean_copy_arg(tolerance):
+    usd = testutil.Amount(10)
+    eur = testutil.Amount('.002', 'EUR')
+    actual = core.Balance([usd, eur], 0).clean_copy(tolerance)
+    if tolerance < eur.number:
+        expected = {usd, eur}
+    else:
+        expected = {usd}
+    assert frozenset(actual.values()) == expected
+
 @pytest.mark.parametrize('mapping,expected', DEFAULT_STRINGS)
 def test_str(mapping, expected):
     balance = core.Balance(amounts_from_map(mapping))