{"id":1214,"date":"2014-06-28T17:37:59","date_gmt":"2014-06-28T22:37:59","guid":{"rendered":"http:\/\/fortmarinus.com\/blog\/?p=1214"},"modified":"2014-06-28T22:36:18","modified_gmt":"2014-06-29T03:36:18","slug":"how-to-handle-change-and-cagr-for-negative-numbers","status":"publish","type":"post","link":"http:\/\/fortmarinus.com\/blog\/1214\/","title":{"rendered":"How to handle Percent Change and CAGR for negative numbers"},"content":{"rendered":"

Sometimes finance deals with negative quantities that become less negative over time. For example, consider a profit\/(loss) of ($50M) in year 1 that becomes a profit\/(loss) of only ($1M) in year 4. If we apply the traditional formulas for Percent Change and Compound Annual Growth Rate (CAGR)<\/a>, we find that the results do not align with common-sense interpretation. Beginning with % Change, the usual formula is:<\/p>\n

\\%\\triangle=\\frac{F-I}{I}=\\frac{F}{I}-1,<\/span> plugging in our example values: \\%\\triangle=\\frac{-1--50}{-50} = -98\\%<\/a><\/p>\n

Common-sense says our profit is increasing<\/em>, therefore we expect +98%. Using an absolute value in the denominator adjusts the formula in such a way that is consistent with the common-sense interpretation.<\/p>\n

\\%\\triangle_{ADJ} = \\frac{F-I}{abs(I)},<\/span> plugging in our example values: \\%\\triangle_{ADJ} = \\frac{-1--50}{abs(-50)}=+98\\%<\/a><\/p>\n

_______________________________________________________________________________________<\/span><\/p>\n

Now, the usual formula for CAGR is:<\/p>\n

CAGR=\\left (\\frac{F}{I} \\right )^{\\frac{1}{time}}-1,<\/span> plugging in our example values: CAGR=\\left (\\frac{-1}{-50} \\right )^{\\frac{1}{3}}-1=-73\\%<\/a><\/p>\n

Again, the common sense interpretation expects a positive<\/em>\u00a0growth rate since profit is increasing. We can not, however, simply reverse the sign as with % change. Let us re-write CAGR to illustrate the solution. This form is identical to the usual formula. Re-arranging it in this way allows us to see that % Change is embedded in the formula:<\/p>\n

CAGR=\\left (\\frac{F}{I}-1+1 \\right )^{\\frac{1}{time}}-1=\\left (\\%\\triangle+1 \\right )^{\\frac{1}{time}}-1<\/p>\n

If we replace % Change with Adjusted % Change, we will have an Adjusted CAGR that yields the growth rate consistent with the common-sense interpretation:<\/p>\n

CAGR_{ADJ}=\\left (\\%\\triangle_{ADJ}+1 \\right )^{\\frac{1}{t}}-1=\\left (\\frac{F-I}{abs(I)}+1\\right )^{\\frac{1}{t}} -1=\\left (\\frac{F-I+abs(I))}{abs(I)} \\right )^{\\frac{1}{t}}-1<\/p>\n

Therefore,<\/p>\n

CAGR_{ADJ}=\\left (\\frac{F-I+abs(I))}{abs(I)} \\right )^{\\frac{1}{time}}-1,<\/span> plugging in our example values: CAGR_{ADJ}=\\left (\\frac{-1--50+abs(-50))}{abs(-50)} \\right )^{\\frac{1}{3}}-1=+26\\%<\/a><\/p>\n

_______________________________________________________________________________________<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Sometimes finance deals with negative quantities that become less negative over time. For example, consider a profit\/(loss) of ($50M) in year 1 that becomes a profit\/(loss) of only ($1M) in year 4. If we apply the traditional formulas for Percent Change and Compound Annual Growth Rate (CAGR), we find that the results do not align […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[9],"tags":[],"_links":{"self":[{"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/posts\/1214"}],"collection":[{"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/comments?post=1214"}],"version-history":[{"count":76,"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/posts\/1214\/revisions"}],"predecessor-version":[{"id":1466,"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/posts\/1214\/revisions\/1466"}],"wp:attachment":[{"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/media?parent=1214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/categories?post=1214"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/fortmarinus.com\/blog\/wp-json\/wp\/v2\/tags?post=1214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}