Hi guys, need some help understanding Example 3 in the Blue Box in the CFAI material. For the second investor, I’m unable to figure out how to get to the numbers 1,568,627 euros in layer 1 and 431,373 euros in layer 3, or the allocation 78.43% and 21.57%. Any help is appreciated, Thanks!
Behavioral Portfolio Theory, Ex 3, Blue Box
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1. Assume the second investor puts X amount into Layer 3 and the rest in Layer 1 (2000000 - X). You can find X using the following equation :
(2M - X)(1 + 0.01) + X (1 - 0.5) = 1.8M
Note that Layer 1 is expected to yield 1% and Layer 3 -50% with 15% prob. Since the first Layer is risk-free, the above allocation should result in 1.8M, which happens to be his safety level, with 15% probability.
2. Solve the equaton for X:
X = 431,373 (21.57%)
(2M - X) = 1,568,627 (78.43%)
- The 2nd investor would get 2,067,451 with 50% prob:
1,568,627 (1 + 0.01) + 431,373 (1+0.12) = 2,067,451 (12% of return given in the vignette)
- He’d get 2,339,216 with 35% prob:
1,568,627 (1 + 0.01) + 431,373 (1+0.75) = 2,339,216 (75% of return given in the vignette)
You might wonder why the 2nd layer is not being used at all. I bet CFAI won’t draw up a question with more than one unknown
I guess that given the low return on Portfolio 2 - with 4.6% - a better return is obtained with the portfolios layers 1 and 3.
(2M - X)(1 + 0.01) + X (1 - 0.5) = 1.8M
Note that Layer 1 is expected to yield 1% and Layer 3 -50% with 15% prob. Since the first Layer is risk-free, the above allocation should result in 1.8M, which happens to be his safety level, with 15% probability.
2. Solve the equaton for X:
X = 431,373 (21.57%)
(2M - X) = 1,568,627 (78.43%)
- The 2nd investor would get 2,067,451 with 50% prob:
1,568,627 (1 + 0.01) + 431,373 (1+0.12) = 2,067,451 (12% of return given in the vignette)
- He’d get 2,339,216 with 35% prob:
1,568,627 (1 + 0.01) + 431,373 (1+0.75) = 2,339,216 (75% of return given in the vignette)
You might wonder why the 2nd layer is not being used at all. I bet CFAI won’t draw up a question with more than one unknown
I guess that given the low return on Portfolio 2 - with 4.6% - a better return is obtained with the portfolios layers 1 and 3.
1% in Layer 1 is risk free. Other two are risky.
Combined probability for Layer 2 - -3(10%) + 5(80%) + 9(10%) = 4.6% -> which is less than the aspirational level of 5% the investor decides. So come what may - he will not invest in Layer 2.
Layer 1 even though it is only 1% return - it is risk free. So he will invest in Layer 1.
Layer 3 has a combined prob adjusted return (prob * return) of 24.75% - so he will invest there as well.
Combined probability for Layer 2 - -3(10%) + 5(80%) + 9(10%) = 4.6% -> which is less than the aspirational level of 5% the investor decides. So come what may - he will not invest in Layer 2.
Layer 1 even though it is only 1% return - it is risk free. So he will invest in Layer 1.
Layer 3 has a combined prob adjusted return (prob * return) of 24.75% - so he will invest there as well.
and this explains the 1.8 Million, and separation of the 85% part on the end of the writeup.
Your job is to allocate assets among different layers that would fit with his needs and desires.
Layer 1, if you invest 1 dollar here, you will get 1% on your investment with a probability of 100%.
He invests 1,568,627 and expects a return of 1,584,313.
Layer 3,
The investment amount is 431,373. And there are three scenarios that could happen.
Scenario1: you lose 50% of 431,373. The probability of this happening is 15%. Your ending value in this layer is 215,687. Now, if you add this number to the return from Layer 1, you get 1,800,000. The chance of that stays 15%.
Scenario 2: you gain 12% on 431373, so the return is 483138. Add this number to return on layer 1 and you get 2,067,451, with a chance of 50%.
Scenario 3. The ending value is 2,339,216, with a chance of that happening being 35%.
How does 85% value come up?
“The portofolio will result in at least 2,067,451”. This means that you add the scenario 2 and scenario 3 together. 50%+35%=85%.
Your job is to allocate assets among different layers that would fit with his needs and desires.
Layer 1, if you invest 1 dollar here, you will get 1% on your investment with a probability of 100%.
He invests 1,568,627 and expects a return of 1,584,313.
Layer 3,
The investment amount is 431,373. And there are three scenarios that could happen.
Scenario1: you lose 50% of 431,373. The probability of this happening is 15%. Your ending value in this layer is 215,687. Now, if you add this number to the return from Layer 1, you get 1,800,000. The chance of that stays 15%.
Scenario 2: you gain 12% on 431373, so the return is 483138. Add this number to return on layer 1 and you get 2,067,451, with a chance of 50%.
Scenario 3. The ending value is 2,339,216, with a chance of that happening being 35%.
How does 85% value come up?
“The portofolio will result in at least 2,067,451”. This means that you add the scenario 2 and scenario 3 together. 50%+35%=85%.