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

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