Question

PMG Inc. has an asset portfolio that consists of $130 million of 15-year, 5.5 percent coupon, $1,000 bonds with annual coupon payments that sell at par.

a-1. What will be the bonds’ new prices when market yields change immediately by ± 0.10 percent?
a-2. What will be the new prices if market yields change immediately by ± 2.00 percent?
b-1. The duration of these bonds is 10.5896 years. What are the predicted bond prices in each of the four cases using the duration rule?
b-2. What is the amount of error between the duration prediction and the actual market values?

Answer 1

Solution a1) Face Value (FV) of the bond = $1000

Coupon rate = 5.5%

Coupon amount = Coupon rate*Face Value = 5.5%*1000 = $55

Number of years (Nper) = 15

Since the bond is selling at par, thus, yield-to-maturity (YTM) of the bond = Coupon rate of the bond

Hence, YTM of the bond = 5.5%

Case 1) When YTM of the bond decreases by 0.10%

Thus, New YTM = 5.5% - 0.1% = 5.4%

The price of the bond is calculated using the PV function in the Excel = PV(Rate, Nper, PMT, FV, Type)

= PV(5.4%, 15, 55, 1000,0)

= -$1,010.10

(Negative sign indicates the cash outflow)

New market price of the bond = $1,010.10

Case 2) When YTM of the bond increases by 0.10%

Thus, New YTM = 5.5% + 0.1% = 5.6%

The price of the bond is calculated using the PV function in the Excel = PV(Rate, Nper, PMT, FV, Type)

= PV(5.6%, 15, 55, 1000,0)

= -$990.03

(Negative sign indicates the cash outflow)

New market price of the bond = $990.03

Solution a-2) Case 1) When YTM of the bond decreases by 2.0%

Thus, New YTM = 5.5% - 2% = 3.5%

The price of the bond is calculated using the PV function in the Excel = PV(Rate, Nper, PMT, FV, Type)

= PV(3.5%, 15, 55, 1000,0)

= -$1,230.35

(Negative sign indicates the cash outflow)

New market price of the bond = $1,230.35

Case 2) When YTM of the bond increases by 2.0%

Thus, New YTM = 5.5% + 2.0% = 7.5%

The price of the bond is calculated using the PV function in the Excel = PV(Rate, Nper, PMT, FV, Type)

= PV(7.5%, 15, 55, 1000,0)

= -$823.46

(Negative sign indicates the cash outflow)

New market price of the bond = $823.46

Solution b-1) The duration of these bonds = 10.5896 years.

Change in Price (P) = -D x [(Change in the rate R)/(1+R)] x P

where R = YTM

and P = Price

Case 1) Change in YTM = +0.10%

Change in Price (P) = - 10.5896*(0.10%)*1000/(1+5.5%)

= -10.5896/(1.055)

= -10.03754

Hence, New Price = 1000 - 10.03754 = $989.96

Case 2) Change in YTM = -0.10%

Change in Price (P) = - 10.5896*(-0.10%)*1000/(1+5.5%)

= 10.5896/(1.055)

= 10.03754

Hence, New Price = 1000 + 10.03754 = $1,010.59

Case 3) Change in YTM = +2%

Change in Price (P) = - 10.5896*(2%)*1000/(1+5.5%)

= -211.792/(1.055)

= -200.75071

Hence, New Price = 1000 - 200.75071 = $799.24929 = $799.25

Case 4) Change in YTM = -2%

Change in Price (P) = - 10.5896*(-2%)*1000/(1+5.5%)

= 211.792/(1.055)

= 200.75071

Hence, New Price = 1000 + 200.75071 = $1,200.75071 = $1,200.75

Solution b-2)

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