CFA Exam Question of the Day

Rationale:
To maintain the continuity of a price-weighted index following a stock split, the divisor must be adjusted so that the index level remains unchanged immediately after the event. First, calculate the index value prior to the stock split: Initial Sum of Prices = $120 (X) + $80 (Y) + $40 (Z) = $240 Initial Index Value = Initial Sum of Prices / Initial Divisor = $240 / 3.0 = 80 Next, determine the new prices after the stock split. Stock X undergoes a 2-for-1 split, so its price becomes $120 / 2 = $60. The prices for Y and Z remain unchanged. New Sum of Prices = $60 (X) + $80 (Y) + $40 (Z) = $180 To find the new divisor, set the new sum of prices equal to the original index value and solve for the new divisor: New Index Value = New Sum of Prices / New Divisor Since the index value must remain 80: 80 = $180 / New Divisor New Divisor = $180 / 80 = 2.250 Using the initial divisor (3.000) would be incorrect because it fails to adjust for the stock split. If the divisor remained 3.0, the index value would incorrectly drop from 80 to $180 / 3.0 = 60, implying a significant market decline when only a mechanical adjustment is needed. The value of 60.000 represents the index value if the divisor were not adjusted after the split ($180 / 3.0 = 60.000). This would incorrectly reflect a market downturn rather than a necessary technical adjustment to maintain index continuity. The value of 1.333 might arise from an incorrect calculation such as dividing the initial sum of prices by the new sum of prices ($240 / $180), or some other misapplication of ratios, rather than correctly determining the divisor needed to preserve the index level.

Rationale:
The Option-Adjusted Spread (OAS) is a measure of the yield spread that remains after accounting for the value of any embedded options in a bond. For a callable bond, the embedded call option benefits the issuer, not the bondholder. Therefore, to determine the spread attributable solely to the bond's credit risk and liquidity (i.e., the OAS), the value of this option must be subtracted from the bond's Z-spread. The Z-spread itself accounts for the bond's cash flows being discounted at the Treasury spot rate curve but does not explicitly adjust for the optionality. In this case, the bond's Z-spread is given as 340 basis points, and the fair value of the embedded call option is 25 basis points. OAS = Z-spread - Value of Call Option OAS = 340 bps - 25 bps = 315 basis points. Using the Z-spread directly as the OAS is incorrect because the Z-spread does not adjust for the embedded call option. It represents the spread over the entire Treasury spot rate curve that equates the present value of the bond's cash flows to its market price, but without isolating the cost of the option. Adding the option value to the Z-spread would be appropriate for a putable bond, where the embedded option benefits the bondholder. For a callable bond, the call option benefits the issuer, meaning the bondholder receives a lower effective spread once that optionality is removed. Other incorrect subtractions from the Z-spread would not reflect the actual value of the embedded option provided. The calculation must specifically use the given Z-spread and the estimated fair value of the call option. The other information provided, such as YTM, Treasury yield, Modified Duration, and Convexity, is relevant for other analyses but not directly for calculating the OAS from the Z-spread and option value.

Rationale:
Under IFRS, available-for-sale (AFS) securities are reported on the balance sheet at fair value, with unrealized gains and losses recognized in other comprehensive income (OCI) rather than profit or loss. These unrealized gains and losses accumulate in accumulated other comprehensive income (AOCI) within equity. Since these unrealized gains will eventually be subject to tax when the securities are sold, they create a temporary difference between the accounting carrying value and the tax base, necessitating the recognition of a deferred tax liability. The fair value of the AFS segment increased by 150 basis points (1.5%) on a carrying value of $1.5 billion. This results in an increase in the AFS securities asset of $1.5 billion * 0.015 = $22.5 million. This unrealized gain creates a deferred tax liability (DTL) at the statutory tax rate of 30%. The increase in DTL is $22.5 million * 0.30 = $6.75 million. The net-of-tax unrealized gain is recognized in OCI and accumulates in AOCI. Therefore, Accumulated Other Comprehensive Income increases by $22.5 million - $6.75 million = $15.75 million. This correctly reflects the increase in assets, the corresponding increase in liabilities (DTL), and the increase in equity (AOCI), maintaining the balance sheet equation. The option suggesting an increase in Retained Earnings is incorrect because unrealized gains and losses on AFS securities are recognized in OCI and accumulated in AOCI, not in retained earnings. Retained earnings are affected by net income, which includes realized gains and losses. The option indicating that Accumulated Other Comprehensive Income increases by the gross amount of $22.5 million and Deferred Tax Liabilities are unchanged is incorrect. While AOCI is the correct account for the unrealized gain, it must be reported net of its tax effect. The unrealized gain creates a taxable temporary difference, requiring the recognition of a deferred tax liability. The option proposing an increase in Deferred Tax Assets is incorrect. An unrealized gain on AFS securities represents a future taxable amount when the gain is realized. This gives rise to a deferred tax *liability*, not a deferred tax asset. A deferred tax asset would arise from a future deductible amount.

Rationale:
The formulation of null and alternative hypotheses is fundamentally driven by the claim being tested. The alternative hypothesis ($H_a$) represents the specific claim that the researcher is trying to find evidence for, while the null hypothesis ($H_0$) represents the status quo, the absence of the claimed effect, and always includes the condition of equality. In this scenario, the model developers assert that the new model's average daily excess return *significantly exceeds* that of the benchmark. This phrasing clearly indicates a directional claim, specifically that the return is *greater than* the benchmark's 2.5 bps. Therefore, the alternative hypothesis must be a 'greater than' inequality, $H_a: \mu_M > 2.5 \text{ bps}$. Consequently, the null hypothesis, which must encompass all possibilities not covered by the alternative and include the equality, becomes $H_0: \mu_M \le 2.5 \text{ bps}$. One incorrect formulation places a strict 'greater than' inequality in the null hypothesis. The null hypothesis must always contain the equality sign, representing the condition under which we assume no significant effect or difference, and against which we gather evidence to potentially reject it. It cannot specify a strict inequality. Another incorrect formulation presents a two-tailed test. This type of test is appropriate when the claim is simply that a parameter is *different from* a specific value, without specifying a direction (e.g., better or worse). However, the developers' claim is specifically that the new model's return *exceeds* the benchmark, indicating a clear directional hypothesis (a one-tailed test) rather than a non-directional difference. A third incorrect formulation represents a left-tailed test. This would be appropriate if the claim were that the model's return is *less than* the benchmark. Since the developers assert that the model's return *exceeds* the benchmark, a right-tailed test is required, making this directional formulation inappropriate for the stated claim.

Rationale:
The portfolio standard deviation must be calculated using the portfolio variance formula, which includes the covariance between the assets. First, convert basis points to decimals: Asset A's standard deviation (σ_A) is 0.18, and Asset B's standard deviation (σ_B) is 0.12. The weights are w_A = 0.60 and w_B = 0.40, and the correlation (ρ_AB) is +0.40. The expected returns and risk-free rate are irrelevant for calculating the portfolio's standard deviation. The portfolio variance (σ_P^2) is calculated as: σ_P^2 = w_A^2 σ_A^2 + w_B^2 σ_B^2 + 2 w_A w_B ρ_AB σ_A σ_B σ_P^2 = (0.60)^2 (0.18)^2 + (0.40)^2 (0.12)^2 + 2 (0.60)(0.40)(0.40)(0.18)(0.12) σ_P^2 = (0.36)(0.0324) + (0.16)(0.0144) + 2 (0.24)(0.40)(0.0216) σ_P^2 = 0.011664 + 0.002304 + 0.0041472 σ_P^2 = 0.0181152 The portfolio standard deviation (σ_P) is the square root of the portfolio variance: σ_P = √0.0181152 ≈ 0.134606 Converting this to basis points (0.134606 * 10,000) yields approximately 1346 basis points. The option of 1560 basis points results from simply taking a weighted average of the individual standard deviations (0.60 * 0.18 + 0.40 * 0.12 = 0.108 + 0.048 = 0.156), which implicitly assumes perfect positive correlation and ignores the diversification benefits of a correlation less than +1.0. The option of 1182 basis points results from assuming a zero correlation coefficient (ρ_AB = 0), thereby ignoring the positive covariance term in the portfolio variance calculation. In this case, σ_P = √((0.60)^2 (0.18)^2 + (0.40)^2 (0.12)^2) = √(0.011664 + 0.002304) = √0.013968 ≈ 0.118186. The option of 991 basis points results from incorrectly subtracting the covariance term instead of adding it (as if the correlation were negative), which would lead to an underestimation of risk given the positive correlation. In this case, σ_P = √((0.60)^2 (0.18)^2 + (0.40)^2 (0.12)^2 - 2 (0.60)(0.40)(0.40)(0.18)(0.12)) = √(0.011664 + 0.002304 - 0.0041472) = √0.0098208 ≈ 0.09909.