Question

consider the following information about three stocks:

state of economy	probability of state of economy	rate of return if state occurs - stock a	rate of return if state occurs - stock b	rate of return if state occurs - stock c
normal	.58	.21	.19	.17
bust	.24	-.05	-.38	-.46

a-1. if your portfolio is invested 35 percent each in a and b and 30 percent in c, what is the portfolio expected return?
note: do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

a-2. what is the variance?
note: do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

a-3. what is the standard deviation?
note: do not round intermediate calculations and enter your answer to 5 decimal places, e.g., .16161.

b. if the expected t-bill rate is 4.10 percent, what is the expected risk premium on the portfolio?
note: do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

c-1. if the expected inflation rate is 3.70 percent, what are the approximate and exact expected real returns on the portfolio?
note: do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.

c-2. what are the approximate and exact expected real risk premiums on the portfolio?
note: do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.

(table and final answer labels repeated below, but main problem text above)

Explanation:

Step1: Calculate Portfolio Weights

Weights: \( w_A = 0.35 \), \( w_B = 0.35 \), \( w_C = 0.30 \) (Wait, no—wait, the problem says 35% each in A and B, 30% in C. So \( w_A = 0.35 \), \( w_B = 0.35 \), \( w_C = 0.30 \).

Step2: Calculate State Returns

First, find portfolio return for each state:
- **Boom (Prob = 0.26):**
\( R_{Boom} = 0.35 \times 0.26 + 0.35 \times 0.38 + 0.30 \times 0.55 \)
\( = 0.091 + 0.133 + 0.165 = 0.389 \) (38.9%)
- **Normal (Prob = 0.58):**
\( R_{Normal} = 0.35 \times 0.21 + 0.35 \times 0.19 + 0.30 \times 0.17 \)
\( = 0.0735 + 0.0665 + 0.051 = 0.191 \) (19.1%)
- **Bust (Prob = 0.16? Wait, no—probabilities: Boom=0.26, Normal=0.58, Bust=1 - 0.26 - 0.58 = 0.16? Wait, the table says:
State of Economy: Boom (Prob 0.26), Normal (0.58), Bust (0.16? Wait, original table: "Probability of State of Economy: Boom.26, Normal.58, Bust.16? Wait, the user's image: "Probability of State of Economy: Boom.26, Normal.58, Bust.16? Wait, the numbers under Bust: -24, -05, -46? Wait, no, the table:

Wait, let's re-express the table:

| State | Probability | Stock A | Stock B | Stock C |
|-------------|-------------|---------|---------|---------|
| Boom | 0.26 | 0.26 | 0.38 | 0.55 |
| Normal | 0.58 | 0.21 | 0.19 | 0.17 |
| Bust | 0.16? Wait, 0.26 + 0.58 = 0.84, so Bust is 1 - 0.84 = 0.16? Wait, the image shows "Bust" with -24, -05, -46? Wait, no—Stock A: Boom=0.26, Normal=0.21, Bust=-0.24? Wait, maybe the returns are decimal percentages? Wait, the numbers: Stock A: Boom.26 (26%), Normal.21 (21%), Bust-.24 (-24%); Stock B: Boom.38 (38%), Normal.19 (19%), Bust-.05 (-5%); Stock C: Boom.55 (55%), Normal.17 (17%), Bust-.46 (-46%). Probabilities: Boom=0.26, Normal=0.58, Bust=1 - 0.26 - 0.58 = 0.16? Wait, 0.26 + 0.58 = 0.84, so Bust is 0.16. Yes.

So recalculate portfolio returns:

- **Boom (p=0.26):**
\( R_B = 0.35 \times 0.26 + 0.35 \times 0.38 + 0.30 \times 0.55 \)
\( = 0.091 + 0.133 + 0.165 = 0.389 \) (38.9%)
- **Normal (p=0.58):**
\( R_N = 0.35 \times 0.21 + 0.35 \times 0.19 + 0.30 \times 0.17 \)
\( = 0.0735 + 0.0665 + 0.051 = 0.191 \) (19.1%)
- **Bust (p=0.16):**
\( R_{Bu} = 0.35 \times (-0.24) + 0.35 \times (-0.05) + 0.30 \times (-0.46) \)
\( = -0.084 - 0.0175 - 0.138 = -0.2395 \) (-23.95%)

Step3: Calculate Expected Return (\( E[R] \))

\( E[R] = p_{Boom}R_{Boom} + p_{Normal}R_{Normal} + p_{Bust}R_{Bust} \)
\( = 0.26 \times 0.389 + 0.58 \times 0.191 + 0.16 \times (-0.2395) \)
Calculate each term:
- \( 0.26 \times 0.389 = 0.10114 \)
- \( 0.58 \times 0.191 = 0.11078 \)
- \( 0.16 \times (-0.2395) = -0.03832 \)
Sum: \( 0.10114 + 0.11078 - 0.03832 = 0.1736 \) (17.36%)

Step4: Calculate Variance (\( \sigma^2 \))

Variance is \( E[(R - E[R])^2] \).
First, find \( (R - E[R])^2 \) for each state:
- **Boom:** \( (0.389 - 0.1736)^2 = (0.2154)^2 = 0.046397 \)
- **Normal:** \( (0.191 - 0.1736)^2 = (0.0174)^2 = 0.00030276 \)
- **Bust:** \( (-0.2395 - 0.1736)^2 = (-0.4131)^2 = 0.170652 \)

Now multiply by probabilities and sum:
\( \sigma^2 = 0.26 \times 0.046397 + 0.58 \times 0.00030276 + 0.16 \times 0.170652 \)
Calculate each term:
- \( 0.26 \times 0.046397 \approx 0.012063 \)
- \( 0.58 \times 0.00030276 \approx 0.0001756 \)
- \( 0.16 \times 0.170652 \approx 0.027304 \)
Sum: \( 0.012063 + 0.0001756 + 0.027304 \approx 0.0395426 \) (3.95426% squared, or 0.0395426 in decimal terms)

Step5: Calculate Standard Deviation (\( \sigma \))

\( \sigma = \sqrt{\text{Variance}} = \sqrt{0.0395426} \approx 0.19885 \) (19.89% when rounded to 2 decimals? Wait, the problem says round to 2 decimal places for percent, or 5 decimal places for decimal? Wait, a-3 says "round to 5 decimal places". Wait, variance (a-2) is percent squared? Wait, no—returns are in decimals (e.g., 0.26 is 26%). So variance is in (decimal)^2, standard deviation in decimal.

Wait, let's redo variance with decimals:

From Step4, variance \( \sigma^2 \approx 0.0395426 \). Then standard deviation \( \sigma = \sqrt{0.0395426} \approx 0.19885 \) (so 0.19885, which is 19.89% when expressed as a percent, but as a decimal, 0.19885, rounded to 5 decimal places: 0.19885 (already 5 decimals? Wait, 0.19885 is five decimal places: 0.19885. Wait, 0.0395426 square root: let's calculate more accurately. \( \sqrt{0.0395426} \approx 0.198853 \), so to 5 decimal places: 0.19885.

Step6: Expected Risk Premium (Part b)

Risk premium = Expected return - Risk-free rate (T-bill rate = 4.10% = 0.041).
\( \text{Risk Premium} = 0.1736 - 0.041 = 0.1326 \) (13.26%)

Step7: Real Returns (Part c)

Inflation rate = 3.70% = 0.037.

Approximate Real Return (c-1 Approx):

Approximate real return = Nominal return - Inflation = \( 0.1736 - 0.037 = 0.1366 \) (13.66%)? Wait, no—wait, approximate real return formula: \( r \approx R - i \), where \( R \) is nominal, \( i \) is inflation.

Exact Real Return (c-1 Exact):

Exact real return: \( r = \frac{1 + R}{1 + i} - 1 = \frac{1 + 0.1736}{1 + 0.037} - 1 = \frac{1.1736}{1.037} - 1 \approx 1.1317 - 1 = 0.1317 \) (13.17%)

Risk Premium (Real):

Approximate real risk premium: Nominal risk premium - Inflation = \( 0.1326 - 0.037 = 0.0956 \) (9.56%)
Exact real risk premium: \( \frac{1 + \text{Nominal Risk Premium}}{1 + i} - 1 = \frac{1 + 0.1326}{1 + 0.037} - 1 = \frac{1.1326}{1.037} - 1 \approx 1.0922 - 1 = 0.0922 \) (9.22%)

But let's focus on part a-1 (Expected Return) first, as a sample:

a-1. Portfolio Expected Return:

As calculated in Step3: \( E[R] = 0.1736 \), so 17.36% (rounded to 2 decimal places).

Answers (Sample for a-1):

a-1. Expected Return: 17.36%
a-2. Variance: 0.03954 (or 3.954% squared, but as decimal variance: ~0.03954)
a-3. Standard Deviation: 0.1989 (or 19.89%)
b. Risk Premium: 13.26%
c-1. Approx Real Return: 13.66%, Exact Real Return: 13.17%
c-2. Approx Real Risk Premium: 9.56%, Exact Real Risk Premium: 9.22%

(Note: The calculations above are based on correcting the probability of Bust: 1 - 0.26 - 0.58 = 0.16. If the Bust probability was misread, adjust accordingly. For example, if Bust probability is 0.16 (as 0.26 + 0.58 + 0.16 = 1), then the calculations hold. If there was a typo and Bust is 0.16, then proceed. If the original problem has Bust probability as 0.16, these are correct.)

For the purpose of answering a-1 (Portfolio Expected Return):

Answer:

s (Sample for a-1):
a-1. Expected Return: 17.36%
a-2. Variance: 0.03954 (or 3.954% squared, but as decimal variance: ~0.03954)
a-3. Standard Deviation: 0.1989 (or 19.89%)
b. Risk Premium: 13.26%
c-1. Approx Real Return: 13.66%, Exact Real Return: 13.17%
c-2. Approx Real Risk Premium: 9.56%, Exact Real Risk Premium: 9.22%

(Note: The calculations above are based on correcting the probability of Bust: 1 - 0.26 - 0.58 = 0.16. If the Bust probability was misread, adjust accordingly. For example, if Bust probability is 0.16 (as 0.26 + 0.58 + 0.16 = 1), then the calculations hold. If there was a typo and Bust is 0.16, then proceed. If the original problem has Bust probability as 0.16, these are correct.)

For the purpose of answering a-1 (Portfolio Expected Return):