Question

Bonds payable ($1M face) due in 10 years, paying interest at 10%, is issued to yield 8%. What is the issue price?

explanation appreciated

Answer 1

ISSUE PRICE OF BOND: $ 1,134,198

• The price of a bond is the present value of all future cash flows from it discounted at a required rate of return. This required rate of return is the YTM (Yield to Maturity).
• YTM is the yield the bond-holder earns provided he holds the bond till maturity.
• In this question YTM = 8%
• Generally, the cash flows from a bond are periodic coupon payments and the redemption of face value on maturity.
• Coupon is always paid on face value: $1,000,000 × 10% = $100,000

Time period Present Value Cash flow ($) Factor @ 8% $ 100,000 0.92593 100,000 0.85734 100,000 0.79383 100,000 0.73503 100,000 0.68058 100,000 0.63017 100,000 0.58349 $ 100,000 0.54027 100,000 0.50025 $ 1,100,000 X 0.46319 PRICE OF THE BOND Present Value (PV) of Cash flow ($) 92,593.00 = $ 85,734.00 79,383.00 = $ 73,503.00 $ 68,058.00 = $ 63,017.00 58,349.00 54,027.00 = $ 50,025.00 = $ 509,509.00 $ 1,134,198.00 10

• Year 10 cash flow = coupon + face value redemption = $100,000 + $1,000,000 = $1,100,000

1 PV factor = (1+r)n Where, • r = rate of return • n = number of years (or periods)

Year PV Factor @ 8% computation PV factor (rounded to 5 decimal places) 0.92593 1 (1 + 0.08)1 0.85734 (1 + 0.08) 0.79383 (1 + 0.08)3 0.73503 (1 + 0.08) 0.68058 (1 + 0.08)5 0.63017 (1 + 0.08) 0.58349 (1 + 0.08) 0.54027 (1 + 0.08) 0.50025 (1 + 0.08) 0.46319 (1 + 0.08)10

Alternatively, the present value of all coupon payments which are in the nature of annuity can be found as follows:

• Present value of annuity = Annuity × PVAF(10 years, 8%) = $100,000 × 6.71008 = $ 671,008
• Present value of face value redeemed = $1,000,000 × 0.46319 = $ 463,190
• Price of the bond = $ 671,008 + $ 463,190 = $1,134,198

PVAF: Present Value Annuity Factor

PVAF -1-(1+r)-n r= interest rate per period = 8% n = number of periods = 10 years PVAF - 1-(1 + 0.08)-10 -= 6.71008 0.08