# Debenture Valuation and Pricing Let’s break down each part of your question step by step. --- ## c) Amount Raised on Initial Issue of Debentures **Given:** - Total face value issued: $1,000,000 - Coupon rate: 13.00% per annum - Coupons paid semi-annually - Maturity: 7 years - Issued at par (since coupon rate equals current interest rate) **Explanation:** When debentures are issued, if the coupon rate equals the current market rate (13% in this case), the debentures are issued at their face value (par value). **Calculation:** - Amount raised = Total face value issued = **$1,000,000** --- ## d) Value of Existing Debenture After One Year ### **New Information:** - New yield (market interest rate): 12.00% per annum - Time to maturity: 6 years (after 1 year has passed) - Remaining coupon payments: 12 (since coupons are paid semi-annually) - Each coupon payment: \( \text{Face value} \times \frac{\text{coupon rate}}{2} = \$1,000,000 \times \frac{13\%}{2} = \$65,000 \) per period ### **Step-by-Step Solution:** 1. **Find the required rate per period:** - Annual market yield = 12.00% - Semi-annual yield = 12.00% / 2 = **6.00% per period** 2. **Cash flows:** - 12 coupon payments of $65,000 (every 6 months) - $1,000,000 face value repaid at end of 12th period (6 years) 3. **Present Value (PV) Formula:** - PV of coupons: \( PV_{\text{coupons}} = C \times \left[ \frac{1 - (1 + r)^{-n} }{r} \right] \) - PV of face value: \( PV_{\text{face}} = F \times (1 + r)^{-n} \) - Where: \( C = \$65,000 \) (coupon per period) \( r = .06 \) (periodic yield) \( n = 12 \) (periods) \( F = \$1,000,000 \) (face value) 4. **Plug in values:** - \( (1 + r)^{-n} = (1.06)^{-12} \approx .55839 \) - \( PV_{\text{coupons}} = 65,000 \times \frac{1 - .55839}{.06} = 65,000 \times 7.3602 \approx \$478,413 \) - \( PV_{\text{face}} = 1,000,000 \times .55839 = \$558,390 \) - **Total Value = PV of coupons + PV of face value** 5. **Total Price:** - \( \$478,413 + \$558,390 = \$1,036,803 \) **Final Answer:** **The value of the debenture in the secondary market is approximately $1,036,800.** --- ## e) Why Has the Value of the Debenture Changed? **Explanation:** - **Interest Rate vs. Coupon Rate:** The debenture was originally issued at par because its coupon rate (13%) matched market rates. After one year, market yields fell to 12%. - **Fixed Coupon Payments:** The debenture continues to pay a fixed 13% coupon, which is now higher than the new market rate (12%). - **Market Adjustment:** Investors are willing to pay **more** for this debenture, since it pays more interest than newly issued bonds with the current 12% yield. - **Pricing Mechanism:** The price of the debenture rises above its face value (premium) so that the yield to maturity for new buyers matches the current market yield. **Summary:** **The value increased because the fixed interest payments of the debenture are now more attractive in a lower interest rate environment, causing the market price to rise above par.**