from part i) We have the following information t F(t-1,t) B(0,t) 0 - - 1 2% 98.02 2 4% 94.18 3 3% 91.39 4 5% 86.94 t is time B(s,t) is price of zero coupon bond (ZCB) at time s maturing at time t for 100 nominal F(s,t) is forward rate from time s to t At time 0, We have a portfolio of 1000 nominal ZCB maturing in 2 yrs, 2000 nominal ZCB maturing in 4 yrs At time 1, forward rates have changed to: t F(t-1,t) 1 - 2 5% 3 4% 4 6% in part ii) we are asked to calculate loss if investor sells the bond at time 1 Value of portfolio at time 0 under original rates = 1000/100*B(0,2) + 2000/100*B(0,4) = 10*94.18 + 20*86.94 = 2680.6 Value of portfolio at time 1 under original rates = 1000*exp(-F(1,2)) + 2000*exp(-F(1,2) - F(2,3) - F(3,4) = 1000*exp(-0.04) + 2000*exp(-0.04 - 0.03 - 0.05) = 2734.63 Value of portfolio at time 1 under changed rates = 1000*exp(-F(1,2)) + 2000*exp(-F(1,2) - F(2,3) - F(3,4) = 1000*exp(-0.05) + 2000*exp(-0.05 - 0.04 - 0.06) = 2672.6 Solution for loss is given as Value of portfolio at time 0 under original rates - value of portfolio at time 1 under changed rates = 2680.6 - 2672.6 = 8 My doubt: Loss should be Value of portfolio at time 1 under original rates - value of portfolio at time 1 under changed rates = 2734.63 - 2672.6 = 63.03 The rate for year 1 has not changed. So, the worth of the portfolio has reduced due to change in forward rates for years 2 through 4. Hence that reduction in value is loss. As part of solution, they have calculated loss as purchase price - sale price, but these two prices relate to different time periods and need to be brought to the same time period by accumulating purchase price to time 1 to find loss at time 1? The concept of present value has been ignored completely. Kindly clarify if my approach is correct. Thanks in advance!
Bonds are purchased at time zero and then sold at time one. The two data items of importance are the values of the bonds at those two timestamps. I don't think it makes sense to calculate the value of the portfolio at time 1 under the original rates because at time 1 the rates have changed. It would be like saying, if my one-year old assumptions had actually played out in reality then this is the loss I would have made. The investor is concerned only with the actual loss.
