Question

6. At what annual rate would $1,000 have to be invested in order to grow to $4,046 in 10 years?
   

Answer 1

Solution:-

Whenever an investor analyzes at a potential investment opportunity, he looks at the compounded annual growth rate (CAGR) offered by the investment. Higher the CAGR the better the investment and vice-versa.

In simple words, CAGR can be defined as the annual return that investment generates on its value at the beginning of the year. Let's say that an investment gets doubled in 5 years. So, total return on initial principal is 100% in 5 years. However, we can't say that the annual return is 20%. This is because we must consider that we reinvested every year's return back into the investment which increased our principal every year and we must calculate annual returns taking that reinvestment into account. This is why instead of simply dividing the total return by no. of periods, we calculate CAGR. The formula for CAGr is as follows:

Future value = Present value*(1+r)ⁿ

where,

Future value= Maturity value of an investment

Present value= Initial investment

r= Annual rate of return

n= number of periods

Thus, putting the values provided in the question, we get as follows:

4,046 = 1,000*(1+r)¹⁰

r = (4.046)^(1/10) - 1

r= 15%

Thus, the annual rate at which $1,000 should be invested in order to grow to $4,046 in 10 years is 15%.

To summarise for understanding purposes, this is how the impact of CAGR looks like during an investment cycle:-

Year	Opening principal for the year	Return for the year (Opening principal*CAGR)	Closing principal for the year (Opening principal + return)
1	1,000	1,000*15%= 150	1,150
2	1,150	1,150*15%= 172.5	1,322.5
3	1,322.5	1,322.5*15%= 198.375	1,520.88
4	1,520.88	1,520.88*15%= 228.13	1,749.01
5	1,749.01	1,749.01*15%= 262.35	2,011.36
6	2,011.36	2,011.36*15%= 301.70	2,313.06
7	2,313.06	2,313.06*15%= 346.96	2,660.02
8	2,660.02	2,660.02*15%= 399	3,059.02
9	3,059.02	3,059.02*15%= 458.85	3,517.87
10	3,517.87	3,517.87*15%= 527.68	4,046