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# Compound Annual Growth Rate McKinsey PST Questions

Hi,

How would you approach those two excersises below? I am preparing alone (without a tutor and math/ GMAT prep. background) so I found those tasks rather difficult for me..

I've found suggestion here: https://www.preplounge.com/en/consulting-forum/math-prep-cagr-without-excel-any-tips-750#a1460 to use Taylor series - but that approach seems not to be applicable for all the cases (i.e. if the (1+x)^a=V is (1+(-x))^a - decreasing by 1 p.p. is not helpful at all)..

I am looking for several generic&fast approaches to cover all possible cases (i.e. solutions are too close to each other, growth rate is negative, etc.)

1. The Revenue in Y0 = 11.64. We project annual growth to be 7%. Which of the foll. is the closest estimate of the projected Revenue in Y5?

• a) 15.2
• b) 16.3
• c) 17.4
• d) 18.5

My approach:

1) 11.64(1+0.07)^5 = y

then either:

2) use binomial formula: (1+x)^5 = 1+5x+10x^2+10x^3+5x^4+x^5 => x = 1.403

or

3) (1+0.07)^5 = 1.07^4 * 1.07 = 1.15*1.15*1.07=1.323*1.07 = 1.42

4) y = 11.64*1.4 = 16.3 => b)

2. Number of cows 3 years ago - 2 150 000, now - 500 000. Assuming cows declining at a constant rate, which of the foll. is the closest estimate of the annual % drop in the number?

• a) 20%
• b) 25%
• c) 40%
• d) 55%

My approach:

1) 2 150 000 (1-x)^3 = 500 000

(1-x)^3 = 0.232

2) x=20%:

0.8^3 = 0.64*0.8 = 0.512 => wrong

3) x=25%: too close to 20% => wrong

4) x=40%:

0.6^3 = 0.36*0.6 = 0.216 ~ 0.232

5) x = 55%:

0.45^3 = 0.20*0.45 = 0.09 => wrong.

6) c)

Hi,

How would you approach those two excersises below? I am preparing alone (without a tutor and math/ GMAT prep. background) so I found those tasks rather difficult for me..

I've found suggestion here: https://www.preplounge.com/en/consulting-forum/math-prep-cagr-without-excel-any-tips-750#a1460 to use Taylor series - but that approach seems not to be applicable for all the cases (i.e. if the (1+x)^a=V is (1+(-x))^a - decreasing by 1 p.p. is not helpful at all)..

I am looking for several generic&fast approaches to cover all possible cases (i.e. solutions are too close to each other, growth rate is negative, etc.)

1. The Revenue in Y0 = 11.64. We project annual growth to be 7%. Which of the foll. is the closest estimate of the projected Revenue in Y5?

• a) 15.2
• b) 16.3
• c) 17.4
• d) 18.5

My approach:

1) 11.64(1+0.07)^5 = y

then either:

2) use binomial formula: (1+x)^5 = 1+5x+10x^2+10x^3+5x^4+x^5 => x = 1.403

or

3) (1+0.07)^5 = 1.07^4 * 1.07 = 1.15*1.15*1.07=1.323*1.07 = 1.42

4) y = 11.64*1.4 = 16.3 => b)

2. Number of cows 3 years ago - 2 150 000, now - 500 000. Assuming cows declining at a constant rate, which of the foll. is the closest estimate of the annual % drop in the number?

• a) 20%
• b) 25%
• c) 40%
• d) 55%

My approach:

1) 2 150 000 (1-x)^3 = 500 000

(1-x)^3 = 0.232

2) x=20%:

0.8^3 = 0.64*0.8 = 0.512 => wrong

3) x=25%: too close to 20% => wrong

4) x=40%:

0.6^3 = 0.36*0.6 = 0.216 ~ 0.232

5) x = 55%:

0.45^3 = 0.20*0.45 = 0.09 => wrong.

6) c)

• Date ascending
• Date descending

Hi,

whenever you are required to solve a compound interest problem you can go with two different logic:

1. When you want to know how much will value X will be in Y years given Z% growth each year:

1.1. Logic: You should multiply Y with Z that will be the simple growth and then add a compound margin of around 5-10%points (but only in case that Y*Z is small, ie. less or equal 45).

1.2. Example: What will be 100 in 5 years given 7% growth each year. 5*7=35 and then I would add around 5-7%, thus 40-42%. Therefore, around 140-142. The exact answer is 140.25.

2. When you want to know in how many years will be X double given its growth rate of Y%.

2.1. Logic: Rule of 72 says that you should divide 72 by the growth factor which would result the number of years the original value would double.

2.2. Example: In what years will 100 double given its 7% growth each year. 72/7 ~ 10. Little more than 10 years. Exact result: in 10th yearend it will be 196.7, in 11th yearend it will be 210.5. Thus little more than 10 years.

2.3. Additionally, there are even more than this one rule, go to a videostreaming platform and search for Flessibilita. He has a 15 minutes of video on compound margin.

Levente

Hi,

whenever you are required to solve a compound interest problem you can go with two different logic:

1. When you want to know how much will value X will be in Y years given Z% growth each year:

1.1. Logic: You should multiply Y with Z that will be the simple growth and then add a compound margin of around 5-10%points (but only in case that Y*Z is small, ie. less or equal 45).

1.2. Example: What will be 100 in 5 years given 7% growth each year. 5*7=35 and then I would add around 5-7%, thus 40-42%. Therefore, around 140-142. The exact answer is 140.25.

2. When you want to know in how many years will be X double given its growth rate of Y%.

2.1. Logic: Rule of 72 says that you should divide 72 by the growth factor which would result the number of years the original value would double.

2.2. Example: In what years will 100 double given its 7% growth each year. 72/7 ~ 10. Little more than 10 years. Exact result: in 10th yearend it will be 196.7, in 11th yearend it will be 210.5. Thus little more than 10 years.

2.3. Additionally, there are even more than this one rule, go to a videostreaming platform and search for Flessibilita. He has a 15 minutes of video on compound margin.

Levente

(edited)

Hi Alex,

in general in these questions there are three options:

• Type 1: The proposed solution numbers are not close. You can use simple interest to eliminate some options and plug in the remaining proposed solutions
• Type 2: The proposed solution numbers are close, the interest rate is low. You can use the Taylor series
• Type 3: The proposed solution numbers are close, the interest rate is high. No shortcuts in this case, you have to do the full computation. It could be better to skip the questions in time-constrained tests, as solution may be pretty long to compute

The definition of low interest rate depends on the period of time considered for compounding; most of the time interests till 15% are fine for Taylor series.

Question 1

This is a Type 2 as for the definition above as solutions are close and interest rate is low. You can use Taylor series till the third element. Main requirement is to know by art the Taylor series:

1+a*x+0,5*a*(a-1)*x^2

where a= years, x=interest

1. Apply Taylor series till step 3 to find the growth approximation:
• 1+a*x+0,5*a*(a-1)*x^2
• 1+5*0,07+0,5*5*4*0,07^2
• 1+0,35+10*0,0049
• 1,35+0,049
• 1,4
2. Multiply by initial value: 11,64*1,4= 16,29
3. Results will be slightly higher than what you found, thus b

Question 2

This is a Type 1 as for the definition above. Numbers are pretty different so you can use the elimination process using simple interest first.

1. Find simple interest. Actual interest will be higher in this case, as for the compound effect, since it's related to negative interest. Thus (1650/2150)/3=26%. This excludes a and b
2. You cannot use Taylor series here, as the interest rate is too high. Thus you can just plug in the remaining numbers and verify the solution using the formula 2150*(1-x)^3=500, which will show 0.4 is the solution. Thus
• 2150*0,6^3 = 2150*0,216 = 464
• 2150*0,45^3 = 2150*0,09 = 196

Hope this helps,

Francesco

Hi Alex,

in general in these questions there are three options:

• Type 1: The proposed solution numbers are not close. You can use simple interest to eliminate some options and plug in the remaining proposed solutions
• Type 2: The proposed solution numbers are close, the interest rate is low. You can use the Taylor series
• Type 3: The proposed solution numbers are close, the interest rate is high. No shortcuts in this case, you have to do the full computation. It could be better to skip the questions in time-constrained tests, as solution may be pretty long to compute

The definition of low interest rate depends on the period of time considered for compounding; most of the time interests till 15% are fine for Taylor series.

Question 1

This is a Type 2 as for the definition above as solutions are close and interest rate is low. You can use Taylor series till the third element. Main requirement is to know by art the Taylor series:

1+a*x+0,5*a*(a-1)*x^2

where a= years, x=interest

1. Apply Taylor series till step 3 to find the growth approximation:
• 1+a*x+0,5*a*(a-1)*x^2
• 1+5*0,07+0,5*5*4*0,07^2
• 1+0,35+10*0,0049
• 1,35+0,049
• 1,4
2. Multiply by initial value: 11,64*1,4= 16,29
3. Results will be slightly higher than what you found, thus b

Question 2

This is a Type 1 as for the definition above. Numbers are pretty different so you can use the elimination process using simple interest first.

1. Find simple interest. Actual interest will be higher in this case, as for the compound effect, since it's related to negative interest. Thus (1650/2150)/3=26%. This excludes a and b
2. You cannot use Taylor series here, as the interest rate is too high. Thus you can just plug in the remaining numbers and verify the solution using the formula 2150*(1-x)^3=500, which will show 0.4 is the solution. Thus
• 2150*0,6^3 = 2150*0,216 = 464
• 2150*0,45^3 = 2150*0,09 = 196

Hope this helps,

Francesco

(edited)

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