Question

Consider the following information on returns and probabilities:

  State Probability    X   Z       

Boom .25 15% 10%

Normal .60 10% 9%

Recession .15 5% 10%

What is the standard deviation for a portfolio with an investment of $6,000 in asset X and $4,000 in asset Z?

A) 6.10%

B) 8.15%

C) 1.85%

D) 3.00%

E) 5.18%

Answer 1

Investment in asset X = $ 6000

Investment in asset Z = $ 4000

Total investment = $ 6000 + $ 4000 = $ 10000

Probability in boom (P_B) = 0.25

Probability in Normal (P_N) = 0.60

Probability in Recession (P_R) = 0.15

Solution :-

weight of asset X (Wx) 6000 – 0.6 - O. 6 looco Weight of Wr) asset ? Yooo - = 0.4 10000 Portfolio deturn in Boom (PIRB) - Return of x in boom x W x lt [Return of 2 in boom & Wa] Ĉ (15%)(0.6) + (109) (0.4) i 9% + 4% = 13% of 0.13 Portfolio return in Normal (P(RN) - Return of x in Normal x W x + Return of in Normal x Wz] = (10%) (0.6) + (9%)6.4) @ 6%. + 3.6%. = 9.67. g 0,096

- Portfolio detuan in Recession (PLR) p) Return of x in recession x W x + [ Return of 2 in recession & W₂] = (5%)(0.6) + (10%)(0.6) = 3% + 4% = 7% cs 0.07 رحمام Expected return of portfolio [Elke , - 10.25 ) (PR)8) + (0.60)(PIR rv) +(0.15)/PIR?) = (0.25) (13%) + (0.60)(9.6% ) + (0.15) d7%) = 3.25% + 5.76% + 1.05% = 10.06% of 0.1006 50? = (PUR)- Epp Pe + (PR)v - Ekip) Powe + (PlRe-ER))Pe so? ₂ (0.13 - 0,1006) 0.25 +10.096-0, 1006) 0.60 +10.67-0.1006 0.15

60 Sot = 10.029430.25 +1 -0.0046) 0.60 + -0.0306) *0.15 SO = (0,00086436) 0.25 +(6.0002)16) 0.60 +(0.00093636) 0.15 SD = 0.00021609 +0.000012696 + 0.000 140454 SD = 0.00036924 so = 0.00036924 SD = 0.0192 of 1.92% for Thus, Standard deviation portfolio is 1.92%