The coupon rate and market price for the 10-year US Treasury bond are 2.50% and

Question

The coupon rate and market price for the 10-year US Treasury bond are 2.50% and 96.3828 respectively. Note, the price is expressed as a percentage of par (like other bonds). If par is $1000, then this bond is selling for $963.828.

Q1 Assume that this bond will mature in precisely 10 years, pay coupons semi-annually, and has a par value of $1000. Determine the yield to maturity for this bond.

Q2 Compute the duration of this bond and use it to estimate the new value of the bond if rates were to suddenly decline by 0.80%.

Q3 Calculate the bond's value directly (using the present value approach) assuming that rates declined 0.80% from the yield to maturity you estimated in the first question.

Q4 Compare your answers to Questions 2 and 3. Explain the source of any difference. Which is more correct

Explanation / Answer

Question 1)

The yield to maturity can be calculated with the use of Rate function/formula of EXCEL/Financial Calculator. The function/formula for Rate is Rate(Nper,PMT,-PV,FV) where Nper = Maturity Period, PMT = Interest Amount, PV = Present Value and FV = Face Value

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Here, Nper = 10*2 = 20, PMT = 1,000*2.50%*1/2 = $12.50, PV = $963.828 and FV = $1,000 (we use 2, since interest payments are semi-annual)

Using these values in the above function/formula for Rate, we get,

Yield to Maturity = Rate(20,12.50,-963.828,1000)*2 = 2.92%

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Question 2)

The duration of the bond can be calculated with the use of following table:

Duration of the Bond = Total of Present Value of Cash Flow/Current Bond Price = 14,092.16/963.828 = 14.62

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The new value of the bond can be calculated with the use of following formula:

New Value of the Bond = -Duration of the Bond*-% Decline in Rates*Current Selling Price + Current Selling Price

The first part of the equation represents the increase in bond value as a result of decline in market rates.

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New Value of the Bond = -14.62*-.80%*963.828 + 963.828 = $1,076.56

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Question 3)

The value of the bond can be calculated with the use of Present Value function/formula in EXCEL/Financial Calculator. The function/formula for calculating Present Value is PV(Rate,Nper,PMT,FV) where Rate = YTM, Nper = Period, PMT = Interest Amount and FV = Face Value

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Here, Rate = 2.92% - .80% = 2.12%, Nper = 10*2 = 20, PMT = 12.50 and FV = 1,000

Using these values in the above function/formula for Present Value, we get,

Value of Bond = PV(2.12%,20,12.50,1000) = $859.38

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Question 4)

The difference can be on account of the period taken into consideration in the calculation. In case of question 2), we took the maturity period as 14.62, while incase of question 3), we will calculate the value of the bond taking the maturity period as 20 years. Going by the principle of inverse relationship between bond price and yield, a decrease in yield should result in an increase in price (which is demonstrated in question 2), and therefore, the bond's value as per question 2) is more correct.

Period (A) Cash Flow (B) A*B Present Value of $1 at 2.50% (C) Present Value of Cash Flow (A*B*C) 1 12.5 12.5 0.976 12.20 2 12.5 25 0.952 23.80 3 12.5 37.5 0.929 34.82 4 12.5 50 0.906 45.30 5 12.5 62.5 0.884 55.24 6 12.5 75 0.862 64.67 7 12.5 87.5 0.841 73.61 8 12.5 100 0.821 82.07 9 12.5 112.5 0.801 90.08 10 12.5 125 0.781 97.65 11 12.5 137.5 0.762 104.79 12 12.5 150 0.744 111.53 13 12.5 162.5 0.725 117.88 14 12.5 175 0.708 123.85 15 12.5 187.5 0.690 129.46 16 12.5 200 0.674 134.72 17 12.5 212.5 0.657 139.65 18 12.5 225 0.641 144.26 19 12.5 237.5 0.626 148.56 20 1,012.5 20,250 0.610 12,357.99 Total $14,092.16

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