# Step-by-Step Solutions ## Exhibit Data Summary **Exhibit 1 (Bonds):** | Maturity | Face Value | Coupon Rate | Yield to Maturity (YTM) | |------------------|------------|-------------|-------------------------| | 30 June 2029 | $100.00 | 4.70% p.a. | 5.26% p.a. | | 15 August 2031 | $100.00 | 4.50% p.a. | 5.68% p.a. | - Coupons paid every 15 August and 15 February. **Exhibit 2 (Preference Shares):** | Par Value | Dividend Rate | Yield | |-----------|--------------|---------| | $10.00 | 6.75% p.a. | 7.44% | **Exhibit 3 (Ordinary Shares Forecast):** | Year Ended 30 June | Forecast EPS Growth | Payout Ratio | |--------------------|--------------------|--------------| | 2026 | 16.% | 55% | | 2027 | 11.% | 60% | | 2028 | 7.% | 65% | | 2029 onwards | 4.% | 70% | --- ## 1. Theoretical Market Price of the 30 June 2029 Bond (as at 30 June 2025) ### Key Data - **Maturity date:** 30 June 2029 - **Settlement date:** 30 June 2025 - **Face value:** $100.00 - **Coupon rate:** 4.70% p.a. (paid semi-annually) - **YTM:** 5.26% p.a. (compounded semi-annually) - **Coupons paid:** 15 August, 15 February ### A. Calculate Cash Flows #### Coupon Amount \[ \text{Coupon per period} = \frac{4.70\% \times \$100}{2} = \$2.35 \] #### Number of Periods - **First coupon after 30 June 2025:** 15 August 2025 - **Final coupon/maturity:** 30 June 2029 - **Count the number of coupon periods:** From 15 August 2025 to 30 June 2029 = 4 years, 10.5 months ≈ 8 coupon payments (every six months): | Coupon # | Date | Time from 30 June 2025 (years) | |----------|----------------|-------------------------------| | 1 | 15 Aug 2025 | .1233 | | 2 | 15 Feb 2026 | .6247 | | 3 | 15 Aug 2026 | 1.1233 | | 4 | 15 Feb 2027 | 1.6247 | | 5 | 15 Aug 2027 | 2.1233 | | 6 | 15 Feb 2028 | 2.6247 | | 7 | 15 Aug 2028 | 3.1233 | | 8 | 15 Feb 2029 | 3.6247 | | Final | 30 Jun 2029 | 4. | But the final payment is not on a coupon date. Since maturity is 30 June 2029, the last period is shorter than 6 months (from 15 Feb 2029 to 30 June 2029 is 135 days). #### Calculate Time Fractions (per AOFM conventions) - **Days between 15 Feb 2029 and 30 June 2029**: 135 days - **Standard semi-annual period**: 182 days (Feb-Aug or Aug-Feb) - **Fraction for final period**: 135 / 182 = .7418 So, for the present value calculation, treat the final payment as a fractional period. ### B. Yield Per Period \[ \text{YTM per period} = \frac{5.26\%}{2} = 2.63\% \] ### C. Calculate Present Value Let’s define: - \( C = \$2.35 \) (coupon) - \( F = \$100 \) (face value) - \( r = .0263 \) (periodic YTM) - Number of full periods: 7 (up to 15 Feb 2029) - Last period: .7418 of a half-year #### Present Value Formula \[ \text{PV} = \sum_{i=1}^{7} \frac{C}{(1+r)^i} + \frac{C}{(1+r)^{7.7418}} + \frac{F}{(1+r)^{7.7418}} \] #### Calculate Each Term - For \( i = 1 \) to \( 7 \): standard semiannual periods - Last coupon and principal discounted 7.7418 periods #### Calculation Table | Period | Date | n (periods) | Payment | |--------|----------------|-------------|----------------| | 1 | 15 Aug 2025 | 1 | \$2.35 | | 2 | 15 Feb 2026 | 2 | \$2.35 | | 3 | 15 Aug 2026 | 3 | \$2.35 | | 4 | 15 Feb 2027 | 4 | \$2.35 | | 5 | 15 Aug 2027 | 5 | \$2.35 | | 6 | 15 Feb 2028 | 6 | \$2.35 | | 7 | 15 Aug 2028 | 7 | \$2.35 | | 8 | 30 Jun 2029 | 7.7418 | \$2.35 + \$100 | #### Discount Factors - \( (1.0263)^1 = 1.0263 \) - \( (1.0263)^2 = 1.0532 \) - \( (1.0263)^3 = 1.0807 \) - \( (1.0263)^4 = 1.1088 \) - \( (1.0263)^5 = 1.1375 \) - \( (1.0263)^6 = 1.1668 \) - \( (1.0263)^7 = 1.1968 \) - \( (1.0263)^{7.7418} \approx e^{7.7418 \cdot \ln(1.0263)} \approx e^{.2016} \approx 1.2229 \) #### Present Value Calculation \[ PV = \sum_{i=1}^{7} \frac{2.35}{(1.0263)^i} + \frac{102.35}{1.2229} \] Calculate each term: - \( \frac{2.35}{1.0263} = 2.29 \) - \( \frac{2.35}{1.0532} = 2.23 \) - \( \frac{2.35}{1.0807} = 2.17 \) - \( \frac{2.35}{1.1088} = 2.12 \) - \( \frac{2.35}{1.1375} = 2.07 \) - \( \frac{2.35}{1.1668} = 2.02 \) - \( \frac{2.35}{1.1968} = 1.96 \) - \( \frac{102.35}{1.2229} = 83.74 \) Add all: \[ 2.29 + 2.23 + 2.17 + 2.12 + 2.07 + 2.02 + 1.96 + 83.74 = \boxed{98.60} \] **Theoretical Market Price = \$98.60** --- ## 2. Theoretical Market Price of the 15 August 2031 Bond (as at 30 June 2025) ### Key Data - **Maturity:** 15 August 2031 - **Face value:** $100.00 - **Coupon rate:** 4.50% p.a. (\$2.25 semiannual) - **YTM:** 5.68% p.a. (\$2.84 semiannual) - **Settlement date:** 30 June 2025 ### A. Coupon Schedule First coupon after 30 June 2025: 15 August 2025. From 15 August 2025 to 15 August 2031: 12 years, so 12*2 = 24 half-year periods, but since settlement is 30 June 2025, the first period is a short first coupon. - **Number of full periods:** From 15 Aug 2025 to 15 Aug 2031 = 12 years = 24 periods. - **But settlement is 46 days before first coupon (30 June to 15 Aug).** #### Day count fraction for first period: - 30 June to 15 August = 46 days - Standard period = 182 days - **Fraction:** 46 / 182 = .2527 - Time to each payment (\(n\)): | Coupon # | Date | n (periods) | |----------|----------------|-------------| | 1 | 15 Aug 2025 | .2527 | | 2 | 15 Feb 2026 | 1.2527 | | ... | ... | ... | | 13 | 15 Aug 2031 | 12.2527 | So, total periods = 12.2527. ### B. Yield Per Period \[ \text{YTM per period} = \frac{5.68\%}{2} = 2.84\% \] ### C. Present Value Formula \[ PV = \sum_{i=1}^{12} \frac{2.25}{(1.0284)^{i-1 + .2527}} + \frac{100 + 2.25}{(1.0284)^{12.2527}} \] #### Calculate Each Term - Use \( (1.0284)^{n} \) for each period. For simplicity, let's calculate for the first and last periods: - \( (1.0284)^{.2527} = e^{.2527 \cdot \ln(1.0284)} \approx e^{.00709} \approx 1.0071 \) - \( (1.0284)^{1.2527} = 1.0284 \times 1.0071 = 1.0356 \) - \( (1.0284)^{12.2527} = e^{12.2527 \cdot \ln(1.0284)} \approx e^{.3426} \approx 1.4083 \) Now, - First coupon: \( \frac{2.25}{1.0071} = 2.236 \) - Second: \( \frac{2.25}{1.0356} = 2.173 \) - Continue for 12 coupons - Final: \( \frac{102.25}{1.4083} = 72.64 \) Sum coupons (approximate): - 2.236 + 2.173 + ... (each coupon decreases slightly due to discounting) Let's estimate sum of coupons (assume average present value per coupon is about 1.9, for 12 payments): - \( 1.9 \times 12 = 22.8 \) - Add final payment: 72.64 \[ PV \approx 22.8 + 72.64 = \boxed{95.44} \] **Theoretical Market Price = \$95.44** --- ## 3. Theoretical Market Price of a Preference Share (as at 30 June 2025) ### Key Data - **Par value:** $10.00 - **Dividend:** 6.75% p.a. ($.675 per share) - **Yield:** 7.44% p.a. ### Formula \[ \text{Price} = \frac{\text{Dividend}}{\text{Yield}} \] \[ \text{Price} = \frac{.675}{.0744} = \$9.08 \] **Theoretical Market Price = \$9.08** --- ## 4. Theoretical Market Price of an Ordinary Share (as at 30 June 2025) ### Data - **Forecast EPS and payout ratios:** See Exhibit 3 - **Cost of equity:** 10.5% ### Step 1: Forecast Dividends Let \( E_ \) = EPS in 2025 (unknown, but not needed for relative valuation). Assume current year is 2025, so forecast starts from 2026. #### Dividend for Year \( t \): \[ \text{Dividend}_t = \text{EPS}_t \times \text{Payout Ratio}_t \] #### EPS Growth Rates: - 2026: 16.% - 2027: 11.% - 2028: 7.% - 2029+: 4.% #### Payout Ratio: - 2026: 55% - 2027: 60% - 2028: 65% - 2029+: