Portfolio A consists of a $20 million dollar position in a one-year zero-coupon bond (with a face value of $100) and a $60 million dollar position in a 10-year zero-coupon bond (with a face value of $100). Portfolio B consists of a $50 million dollar position in a 5.95-year zero-coupon bond (with a face value of $100). The current yield on all bonds is 10% per annum. (a) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same. (b) Explain part (a) in words (approximately one or two sentences max).

From the below calculations, the change in value of both portfolios is 0.60%

I first of all, determine the present value of both portfolios,incorporate increase of 10 basis points,then compared the change in values.

Explanation:

The applicable formula in this scenario is given FV=PV*e^(r*N)

FV is the future value is the bonds' face value repayable on redemption

PV is our unknown

e has a constant value of 2.7182818

r is the yield given as 10%

N is the number of years the investment will last

Using the formula above the present value of each portfolio can be determined.

The formula can be rewritten as PV=FV*e^-(r*N)

PV of portfolio A=20*e^-(10%*1)+60*e^-(10%*10)

                           =(20*2.7182818^-10%)+(60*2.7182818^-100%)

                            =$40.17 million

A 10 basis points increase  which is 0.1% ,in yield will give the below present value of portfolio A:

PV of  portfolio A=20*2.7182818^-(10.1%*1)+60*2.7182818^-(10.1%*10)

                            =(20*2.7182818^-10.1%)+(60*2.7172818^-101%)

                             =$39.93 million

Hence percentage change in the value of portfolio A is given  below:

percentage change=(40.17 -39.93)/40.17

                                =0.60%

The present value of portfolio using yield of 10% and 10.1% van also be determined in the same manner

PV of portfolio B=50*2.7182818^-(10%*5.95)

                            =50*2.712818^-59.5%

                            =$27.58 million

With a yield of 10.1%

PV of portfolio B=50*2.7182818^-(10.1%*5.95)

                           =50*2.7182818^-60.095 %

                           =$27.41 million

percentage change in portfolio B=(27.58-27.41)/27.58

                                                       =0.60%

y=mx+b is the right slope of velocity carry the 4 over