4. The prices of discount bonds (all with maturity value of $1,000) maturing in years 1, 2, 3, 4, 5 are given below. Price Time to Maturity 920 1 860 2 790 3 700 4 600 5 What is the yield to maturity on a risk-free 5% bond due in 5 years (also with maturity value of $1,000)

YTM = 10.5%

Explanation:

Solution

Given that:

The cash flow related with the 5% bond are computed below:

t = 0 1 (50) 2 (50) 3 (50) 4 (50) 5 (1050)

Now,

We calculate the discount factors which is given below:

1 / 1 + r₁ = 920/1000 = 0.92

1 / (1 + r₂) ² = 860/1000 = 0.86

1 / (1 + r₃) ³ = 790/1000 = 0.79

1 / (1 + r₄) ⁴ = 700/1000 = 0.7

1 / (1 + r₅) ⁵ = 600/1000 0.6

Thus,

P₅% bond = 50 (0.92) + 50 (0.86) + 50 (0.79) + 50 (0.7) + 1050 (0.6)

=$793. 50

For the yield to maturity (YTM) is refereed to as the IRR of this bond.

Now to solve for the YTM we have teh following.

P₅% bond = 50/YTM ║ 1 - 1 / (1 + YTM) ⁵║ + 1000 / (1 + YTM) ⁵

793.5 = 50/YTM ║ 1 - 1 / (1 + YTM) ⁵║ + 1000 / (1 + YTM) ⁵

Therefore

YTM = 10.5%

Note: the present value of all coupons was computed by applying the annuity formula, also added the PV of the face value