Try the following;

    global $wp_query;
    global $paged;
    $temp = $wp_query;
    $wp_query = null;

    $args  = array( 
        'post_type' => 'portfolio', 
        'paged' => $paged, 
        'posts_per_page' => -1, 
    );

    $wp_query = new WP_Query( $args );
    while ($wp_query->have_posts()) : $wp_query->the_post();

    //do your output here...

    endwhile;

    //do your pagination here...
    wp_paginate();
    $wp_query = null; 
    $wp_query = $temp;

...also this, 

    <?php
    /*
    Template Name: Home
    */
    ?>

...indicates that you are trying to use this template for the homepage right? Would this file happen to be named also `home.php` ? 

In that case can you tell us what you have set under the following;

![enter image description here][1]

Update: (lets debug the query)

Beneath this line;

    $wp_query = new WP_Query( $args );

Please paste..

    var_dump('<pre>', $paged, '</pre>');
    var_dump('<pre>', $wp_query, '</pre>');

So it looks like;

    ...etc
    $wp_query = new WP_Query( $args );
    var_dump('<pre>', $paged, '</pre>');
    var_dump('<pre>', $wp_query, '</pre>');
    ...etc

Then copy the output to http://www.pastebin.com so we can inspect whats going on in the query.
  [1]: https://i.sstatic.net/OWDCi.jpg