I have add a custom column in customer grid by copying core Grid.php file to local folder. I want to show last order date of customer in a column. So I have tried below code in collection.

protected function _prepareCollection()
    {
        $collection = Mage::getResourceModel('customer/customer_collection')
            ->addNameToSelect()
            ->addAttributeToSelect('email')
            ->addAttributeToSelect('created_at')
            ->addAttributeToSelect('group_id')
            ->addAttributeToSelect('track_device')
            ->addAttributeToSelect('inchoo_socialconnect_fid')
            ->addAttributeToSelect('inchoo_socialconnect_gid')
            ->joinAttribute('billing_postcode', 'customer_address/postcode', 'default_billing', null, 'left')
            ->joinAttribute('billing_city', 'customer_address/city', 'default_billing', null, 'left')
            ->joinAttribute('billing_telephone', 'customer_address/telephone', 'default_billing', null, 'left')
            ->joinAttribute('billing_region', 'customer_address/region', 'default_billing', null, 'left')
            ->joinAttribute('billing_country_id', 'customer_address/country_id', 'default_billing', null, 'left');
            

       // add last_order_date column     
            $sql ='SELECT MAX(o.created_at)'
                . ' FROM ' . Mage::getSingleton('core/resource')->getTableName('sales/order') . ' AS o'
                . ' WHERE o.customer_id = e.entity_id ';
            $expr = new Zend_Db_Expr('(' . $sql . ')'); 
        
            $collection->getSelect()->from(null, array('last_order_date'=>$expr));
     
     
        $this->setCollection($collection);

        return parent::_prepareCollection();
    }

here is the column code--

$this->addColumn('last_order_date', array(
            'header'    => Mage::helper('customer')->__('Last Order Date'),
            'type'      => 'datetime',
            'align'     => 'center',
            'index'     => 'last_order_date',
            'filter'    => true,
            'sortable'  => true ,               
            'gmtoffset' => true,        
        ));

column showing correct values but Sorting and Filtering not working for this column. What is wrong there.