Valuation, Risk and Return in Equity and Fixed Income Investments

Question 1 (potential applicable modules 2, 3, 4, 5, 7, 8, 10) Bottom of Form

(a) Alpha Corp. currently has 5,250,000 shares outstanding, and its shares are trading at $43.21 on 26 May 2025. Alpha Corp. just paid an annual dividend of $1.51 per share. You expect this dividend to grow at 9.75% per year for the next five years, and then at a constant rate of 2.45% per year indefinitely. The required rate of return on Alpha Corp.’s shares is 8.06%.(12.5 marks).

1. Calculate the intrinsic value per share and briefly discuss the potential impact on your portfolio from this investment.
2. Assume the market for Alpha Corp. shares is efficient in the semi-strong form. Evaluate the feasibility of consistently earning abnormal returns by trading Alpha Corp. shares based on publicly available historical price data and your intrinsic value calculation. Discuss your reasoning.

(b) You are considering investing $15,500 AUD into a technology share listed in the United States. The share has a beta of 1.37. The current exchange rate is $1 AUD = $0.64 USD as of 26 May 2025. The US market risk premium is 5.83%, and the US risk-free rate is 3.02%. You plan to sell the position in one year and repatriate the funds. The forward exchange rate for one year is expected to be $1 AUD = $0.68 USD. (12.5 marks).

1. Suppose the return was 11%, you are eager to see if you have outperformed on a risk-adjusted basis against the US market during this period (excluding repatriation of funds). Calculate the Jensen’s Alpha of this investment and briefly comment on the performance and with respect to the Securities Market Line (SML).
2. Calculate the expected return of this investment.
3. Discuss the risks and benefits of investing in international shares. Additionally, explain the considerations investors should explore when making such investments.

Answer to part 1a:

I. Intrinsic Value Calculation and Portfolio Impact

Multi-stage dividend discount model (DDM) has been used for calculating Alpha Corp.’s intrinsic value per share because we have different growth rates for dividends:

• Initial Dividend (D₀): $1.51
• Growth Rate (Years 1–5): 9.75%
• Long-Term Growth Rate (g): 2.45%
• Required Return (r): 8.06%

Step 1: Forecast Dividends (Years 1–5)

D_1​ = 1.51 × (1 + 0.0975) = 1.6572

D₂​ = D₁​ × (1 + 0.0975) = 1.8188

D₃ = D₂ ​× (1 + 0.0975) = 1.9961

​D₄ = D₃ ​× (1 + 0.0975) = 2.1908

​D₅​​ = D₄​ × (1 + 0.0975) = 2.4044​

Step 2: Discount These Dividends

PV₁ ​= 1.6572 / (1 + 0.0806)¹​= 1.5336, PV₂ ​= 1.8188 / (1 + 0.0806)² ​= 1.5576,

PV₃ = 1.9961 / (1 + 0.0806)³ = 1.5820, PV₄ = 2.1908 / (1 + 0.0806)⁴  = 1.6067, PV₅ = 2.4044 / (1 + 0.0806)⁵= 1.6318

Step 3: Terminal Value at End of Year 5 (Gordon Model)

D_6​ = D₅ ​× (1 + g) = 2.4044 × 1.0245 = 2.4633

P_5​ = D6 / r−g ​​= 2.4633 / 0.0806 – 0.0245 ​= 43.9085

PV of  P₅ ​= 43.9085 / (1 + 0.0806)⁵​ = 29.8005

Step 4: Add All Present Values

Intrinsic Value = ∑ PV _Dividends + PV_P5 = 7.9127 + 29.8005 = $37.71

Conclusion:

At $43.21, the market price of Alpha is above the intrinsic value of $37.71. This situation highlights the risk of short-term investment in this stock; except the company’s growth rate is higher than anticipated, long-term success may be impeded.

II. Market Efficiency & Abnormal Returns

According to the semi-strong form of market efficiency, all information publicly available, such as historical prices and intrinsic valuation models, is already impounded in the stock price.

Therefore:

• Intrinsic value ≠ price does not necessarily mean mispricing.
• Dividend growth, discount rates, and macro data are already accounted for by market participants.
• Buying Alpha shares based on DDM valuations would not generate constant abnormal returns.

Conclusion:
Dependence on intrinsic value vs. market price in a semi-strong efficient market is not enough to lead to long-term outperformance. Investors would rather use private or insider information (not allowed) or have recourse to long-run diversification and strategic asset allocation.

Answer to part 1b:

I. Jensen’s Alpha Calculation

Capital Asset Pricing Model (CAPM) formula used to calculate the expected return:

Expected Return (CAPM) = R_f ​+ β (R_m ​− R_f​)

Where:

• R_f​ = 3.02% (US risk-free rate)
• R_m ​− R_f ​= 5.83% (market risk premium)
• β = 1.37
• Actual return = 11%

Expected Return = 3.02% + 1.37 × 5.83% = 3.02% + 7.9931% = 11.0131%

Jensen’s Alpha:

α = R_i − [R_f + β(R_m − R_f)] = 11% − 11.0131% = − 0.0131%

Jensen’s Alpha is a little on the downside, and that means the investment lost against the market based on the risk. The outcome is positioned at or a little below the Capital Market Line (CML). Notwithstanding the good profit it made, still, it couldn’t meet the forecasts due to its increased risk (beta = 1.37).

II. Expected Return is = 11.0131%

III. Discussion: Advantages and Disadvantages of International Shares

Benefits:

• When you include different economies in your portfolio, it reduces its overall swings.
• Opportunity to Access Innovation – US tech companies usually allow access to the latest and growing fields.
• A helpful currency movement can boost what you earn from your investments.

Risks:

• Globally traded investments in USD can be impacted when the value of AUD/USD changes.
• Exposure to economic changes, rates and rules in the United States.
• Foreign Investment may result in having to pay tax twice or to have taxes withheld.

Investor Considerations:

• Calculate the forward exchange rate compared to the spot rate to get a forecast of the money you will repatriate.
• Familiarize yourself with the rules and regulations of tax in your area, plus international investment treaties.
• Use some techniques to limit risk related to changes in currency.
• Watch over the cash flow, company leadership and openness of all foreign investments you are part of.

Question 2 (potential applicable modules 3, 4, 10, 11)

• You invested in a managed fund by purchasing 1,000 units at $22.36 per unit at the beginning of 2025. The fund’s performance and your transactions are detailed below:

2025: Year-end distributions: $0.81 per unit income, $0.30 per unit capital gain. Year-end NAV: $22.40 per unit.

2026: You purchased an additional 401 units at $22.76 per unit during the year. Year-end distributions: $0.93 per unit income, $0.25 per unit capital gain. Year-end NAV: $24.12 per unit.

2027: No new units purchased. Year-end distributions: $0.95 per unit income, $0.21 per unit capital gain. Year-end NAV: $24.13 per unit.

1. Calculate the yearly and total 3-year holding period return. Interpret these in terms of investment performance.
   1. Calculate the internal rate of return of your cash flows from the investment and compare this against the holding period return (10 marks).

• Investments A and B have the following monthly returns between May 2024 until May 2025:

Investment A: 6%, 8%, -3%, 4%, 2%, 7%, 5%, -2%, 3%, 8%, 6%, 4%

Investment B: 7%, 9%, 4%, -2%, 3%, 8%, 6%, 3%, 4%, 7%, 5%, 6%

1. Calculate the variance and standard deviation for each investment and briefly comment on their risk.
2. Following this, find the correlation coefficient between the returns of the investments and discuss the potential benefits of diversification.
3. With a portfolio of 62% Investment A and 38% Investment B, calculate the portfolio’s standard deviation and briefly compare it to the individual investments risk. Be specific and detailed. (15 marks).

Answer to part 2a:

I. Yearly and 3-Year Holding Period Return (HPR)

• 2025
• Initial NAV = $22.36
• Year-end NAV = $22.40
• Distributions = $0.81 income + $0.30 capital gain = $1.11

HPR (2025):

22.36 / 22.40 − 22.36 + 1.11 ​= 22.36 / 1.15​ = 5.14%

• 2026
• NAV at start of 2026 = $22.40
• NAV at end of 2026 = $24.12
• Distributions = $0.93 + $0.25 = $1.18

HPR (2026):

22.40 / 24.12 − 22.40 + 1.18 ​= 22.40 / 2.90 ​= 12.95%

• 2027
• NAV at start = $24.12
• NAV at end = $24.13
• Distributions = $0.95 + $0.21 = $1.16

HPR (2027):

24.12 / 24.13 − 24.12 + 1.16 ​= 24.12 / 1.17​ = 4.85%

Total 3-Year Holding Period Return (HPR)

Total HPR = Initial Cost / Final Value + Total Distributions – Initial Cost​

• Units at end = 1,401
• Final NAV = $24.13 → Value = 1401 × 24.13 = $33,790.13
• Initial Cost = (1000 × 22.36) + (401 × 22.76) = $31,389.76
• Total Distributions over 3 years:
  • 2025 = 1000 × 1.11 = $1,110.00
  • 2026 = 1401 × 1.18 = $1,653.18
  • 2027 = 1401 × 1.16 = $1,625.16
  • Total = $4,388.34

Total Return = 31,389.76 / 33,790.13 + 4,388.34 − 31,389.76 ​= 21.95%

II. Internal Rate of Return (IRR)

Year	Description	Cash Flow (AUD)
2025	Buy 1000 units @ $22.36	–22,360.00
2025	Distributions received	+1,110.00
2026	Buy 401 units @ $22.76	–9,129.76
2026	Distributions received	+1,653.18
2027	Distributions received	+1,625.16
2027	Final value of all units	+33,790.13

IRR = 6.92% annually

Interpretation

• Annual returns varied from ~4.85% to ~13%, exhibiting moderate consistent growth.
• 3-Year HPR of 21.95% points towards good long-term performance.
• IRR (6.92%) provides actual average annual return considering timing of cash inflows and outflows.
• HPR exaggerates performance since it does not account for reinvestment and timing, while IRR portrays actual return efficiency.

Answer to part 2b:

I. Variance and Standard Deviation

We are given 12 months of returns (May 2024 to May 2025) for each investment.

Investment A Returns (%):6, 8, –3, 4, 2, 7, 5, –2, 3, 8, 6, 4

Mean (Average) =12 / 6 + 8 + (−3) + 4 + 2 + 7 + 5 + (−2) + 3 + 8 + 6 + 4 ​= 12 / 48 ​= 4.00%

Variance =n / 1 ​∑(R_i​−R)² = 121 ​[(6−4)² + (8−4)² + ((-3)−4)² + (4−4)² + (2−4)² + (7−4)² + (5−4)² + ((-2)−4)² + (3−4)² + (8−4)² + (6−4)² + (4−4)²]

= 12.00

Standard Deviation = √12 = 3.46%

Investment B Returns (%):7, 9, 4, –2, 3, 8, 6, 3, 4, 7, 5, 6

Mean =  12 / 60 ​= 5.00%

Variance = 12 / 1 [(7−5)² + (9−5)² + (4−5)² + ((-2)−5)² + (3−5)² + (8−5)² + (6−5)² + (3−5)² + (4−5)² + (7−5)² + (5−5)² + (5−5)²] = 6.00

Standard Deviation = √6 = 2.45%

Interpreting the Risk:

• Investment A’s standard deviation (3.46%) is higher than that of Investment B (2.45%), demonstrating that it has the properties of being more volatile and riskier.

• Yet, B has the most significant return (5%) than A (4%), and its risk is even small.

II. Correlation Coefficient

We calculate the correlation coefficient (ρ) between the two return series:

Let X = returns of Investment A

Let Y = returns of Investment B

ρ_XY​ = σX​⋅σY / ​Cov(X,Y)

ρ = 0.714 (rounded)

Understanding Correlation

• A correlation coefficient of 0.714 implies a positive, yet not the highest, linear association.
• This simply says that the investments mainly follow the same course, but they are not strictly matching.
• The presence of diversification advantage is evident but is minimal – total risk protection is not achieved.

III. Portfolio Standard Deviation

Weights:

• Investment A = 62%
• Investment B = 38%
• Std. Dev A = 3.46%
• Std. Dev B = 2.45%
• Correlation = 0.714

σp​= √(w_A²​⋅σ_A²​)+(w_B²​⋅σ_B²​)+2w_A​w_B​σ_A​σ_B​ρ_AB​

Substitute values:

σp​ = √(0.62²⋅3.46²)+(0.38²⋅2.45²)+2⋅0.62 ⋅ 0.38 ⋅ 3.46 ⋅ 2.45 ⋅ 0.714

= √(0.3844 ⋅ 11.97) + (0.1444 ⋅ 6.00) + (2 ⋅ 0.62 ⋅ 0.38 ⋅ 3.46 ⋅ 2.45 ⋅ 0.714)

= √4.60 + 0.87 + 4.59 =√10.06 = 3.17%

Comparison

• Portfolio Std Dev = 3.17%
• Investment A Std Dev = 3.46%
• Investment B Std Dev = 2.45%

Even though the portfolio was having the risk of a more volatile asset (A), the mix lessened the total risk to 3.17% – which was lower than A’s risk.

This range illustrates the effect of moderate diversification caused by correlation that is less than perfect.

​Question 3 (potential applicable module 6)

Assume on 01 June 2025 you have two semi-annual bonds available for investment. Bond A is a 4-year bond paying a 5% coupon with a yield-to-maturity of 4.73%, and Bond B is a 6-year bond paying a 5.5% coupon with a yield-to-maturity of 5.11%. Both bonds have a face value of $1,000. (25 marks).

1. Calculate the duration of Bond A and Bond B.
2. Calculate the modified duration of Bond A and Bond B.
3. Carefully explain the concept of duration and how it can help an investor understand interest rate risk.
4. Compare and contrast the interest rate risk associated with Bond A and Bond B based on their calculated durations.
5. Consider a portfolio that consists of only these two bonds.
   1. How would you achieve immunisation of your bond portfolio against interest rate changes given the current yield-to-maturity rates?
   1. Describe how the allocation of these bonds may change for immunisation if the yield-to-maturity changes to 7.11% for both Bond A and Bond B. Explain your reasoning.

Answer to Question 3:

Bond Information (as of 01 June 2025)

	Bond A	Bond B
Face Value	$1,000	$1,000
Coupon Rate	5% (semi-annual)	5.5% (semi-annual)
Maturity	4 years	6 years
YTM	4.73% (semi-annual: 2.365%)	5.11% (semi-annual: 2.555%)
YTM (Semi-Annual)	2.365%	2.555%

a. Duration of Bond A and Bond B

Duration = ∑_tⁿ =1​( t . CFt​​  / (1+r)^t)  /  ∑_tⁿ =1​( CFt​​  / (1+r)^t)

Where:

• CF_t​ = cash flow at period t
• r = semi-annual YTM
• n is the number of periods (maturity × 2)

Bond A:

• Coupon = 5% annually = $25 per 6 months
• Periods = 4 years × 2 = 8
• YTM = 4.73% annually → 2.365% semi-annually

After computing, Duration of Bond A ≈ 3.62 years

Bond B:

• Coupon = 5.5% annually = $27.50 per 6 months
• Periods = 6 years × 2 = 12
• YTM = 5.11% annually → 2.555% semi-annually

After computing, Duration of Bond B ≈ 5.21 years

b. Modified Duration

Modified Duration = 1+r / Duration

Bond A:

Modified Duration = 1 + 0.02365 / 3.62 ​= 3.54

Bond B:

Modified Duration = 1 + 0.02555 / 5.21 ​= 5.08

c. Define the Concept of Duration and Its Application to Interest Rate Risk

Duration is a way of measuring the sensitivity of a bond to interest rate changes. It measures the weighted average time to get back cash flows, and it estimates the percentage price change to a specific change in interest rates.

• The higher the duration, the lower the bond’s sensitivity to changes in interest rates.
• If the interest rates increase, then the prices of bonds decrease, and if the interest rates decrease, then the prices of bonds increase.
• Duration allows investors to control interest rate risk by matching the duration of the bond with their investment horizona strategy referred to as immunisation.

d. Compare and Contrast Interest Rate Risk for Bonds A and B

• Bond A is of 3.62 years duration → less sensitive to changes in interest rates.
• Bond B has a maturity of 5.21 years → greater interest rate sensitivity.

Conclusion:

• Bond B is more interest rate risky with its longer maturity and greater duration.
• Investors anticipating increasing interest rates might prefer Bond A due to its lower duration and risk.

e. Portfolio Immunisation Strategy

I. Attaining Immunisation Against Interest Rate Changes

Immunisation puts the holding period of a portfolio of bonds into alignment with the investor’s investment horizon, reducing the effect of interest rate volatility.

Let:

• D_A = 3.62 Bond A duration
• D_B=5.21 Bond B duration
• D_P=5.00 desired duration (say, 5-year horizon)
• w_A​, Bond A weight in portfolio

Equation:

w_A ​⋅ D_A ​+ (1−w_A​) ⋅ D_B ​= D_P

So:

• Bond A weight ≈ 13.21%
• Bond B weight ≈ 86.79%

The investor should invest 13.21% in Bond A and 86.79% in Bond B to immunise a 5-year liability.

II. Immunisation if YTM Changes to 7.11%

New YTM = 7.11% annually → 3.555% semi-annually

Higher yields reduce bond durations because future cash flows are discounted more heavily. Let’s assume:

• New duration of Bond A ≈ 3.45
• New duration of Bond B ≈ 4.90

Recalculate portfolio weights:

w_A⋅ 3.45 + (1−w_A) ⋅ 4.90 = 5.00

3.45w_A ​+ 4.90 − 4.90w_A ​= 5.00

−1.45w_A​ = 0.10 ⇒ w_A​ = − 0.0690

A negative weight suggests that if we were to use only Bond A and Bond B, it would not be possible to immunize a 5-year horizon with the new durations without short selling Bond A.

Explanation:

• The longer the YTM is, the smaller are the durations of each of the involved bonds.
• The investor can thus be sure of having an immunized portfolio for the next 5 years by simply deciding to redistribute the weight to the longer-timed bond, that is Bond B or by including a third high-duration bond to reach the target.

​​Question 4 (potential applicable modules 7, 8)Bottom of Form

(a) Taking a short position in a futures contract on an underlying asset is generally considered the opposite of taking a long position in a futures contract on the same asset. Is this statement fundamentally true in terms of market outlook, obligations, and profit/loss potential? Explain carefully.(12.5 marks).

(b) On 26 May 2025, David purchased 2,000 shares of TechSolutions Inc. at $25.07 per share. To hedge this position, he simultaneously purchased two put option contracts on TechSolutions Inc. (each contract representing 1,000 shares). The puts have a strike price of $23.03, an expiration date in six months, and were purchased at a premium of $1.50 per share. TechSolutions Inc. pays a quarterly dividend of $0.36 per share, and one such dividend is expected during the 6-month life of the puts.
If, at the expiration date of the puts, the share price of TechSolutions Inc. has fallen to $19.89, determine the holding period return (HPR) David will generate on this combined stock and option position. Be specific. (12.5 marks).

Answer to part 4a:

a.  vs Long Position in a Futures Contract

Statement: “Taking a short position in a futures contract on an underlying asset is generally considered the opposite of taking a long position in a futures contract on the same asset.”

Explanation:

In fact, this statement is based on fundamental realities.

1. Market Outlook:

• When you go long, your hope is that the price will go up. Gains are seen only if the asset’s price moves upward.
• A short position is taken when an investor envisions the price will decrease. Still, you gain a profit if the price of the underlying asset drops.

2. Obligations:

• In a long futures contract, the parties agree you’ll purchase the asset at the end of the contract for the fixed price.
• A short position in futures means you agree to sell the asset at the expiration day for the agreed futures price.

3. Profit/Loss Potential:

• If the price goes up, your gain can be as much as you like. However, if the price slides down a lot, your loss may too be large.
• When in a short position, your profit can only be the futures price, while your losses are possible no matter how high the market goes.

4. Symmetry:

• If you win with a futures contract, it comes at someone else’s expense.
• The return on one contract (long or short) from the combined P/L payoff is always the same for the same contract.

The statement is proven to be factual. When comparing market outlook, obligations and profit/loss chances, short and long futures positions are the same.

Answer to part 4b:

b. HPR for Combined Position

Given:

• Buy 2,000 shares at the price of $25.07 when you first start.
• Bought 2 put options that each included 1,000 shares.
  • The strike is set at $23.03.
  • Premium is $1.50 per share.
  • Expiration time = 6 months
• The company paid a dividend of $0.36 for this quarter.
• The contract’s settle price at the end of the contract was $19.89.

Step 1: Stock Component

• Initial cost:2,000 × 25.07 = $50,140
• Final value (market price):2,000 × 19.89 = $39,780
• Dividend received:2,000 × 0.36 = $720

Net stock return = $39,780 (final) + $720 (dividend) = $40,500

Loss on stock = $50,140 – $40,500 = -$9,640

Step 2: Put Option Component

Intrinsic value of puts = Max(Strike – Spot, 0)

23.03 – 19.89 = 3.14 per share

Gain per put = $3.14 – $1.50 (premium) = $1.64

Total gain =2,000×1.64=$3,280

Step 3: Total Return Calculation

Final portfolio value =

Stock: $39,780

Dividends: $720

Put gain: $3,280

= $43,780

Initial investment =

Stock: $50,140

Options: $1.50 × 2,000 = $3,000

= $53,140

Holding Period Return (HPR):HPR = Initial Investment / Final Value − Initial Investment ​× 100

HPR = 53,140 / 43,780 − 53,140 ​× 100 = − 17.62%

Conclusion:

• Despite the hedge, David incurred a net loss of 17.62%.
• The put options reduced the loss significantly:
  • Without the hedge, stock loss = $9,640
  • With hedge, net loss = $9,360 (stock loss – put gain + dividend)
• This indicates the worth of options to contain downside risk in declining markets.