Question

Consider a 2-year $4000 bond that's redeemable at par and pays semi-annual coupons at a rate of c2) 8%. 70. (a) Suppose that the yield rate is 4% compounded annually. Determine: The purchase price of the bond. P = $ %3D The bond's duration to 3 decimals. D: years %3| Note: Use the purchase price to the closest cent in your duration calculation. (b) Suppose that the yield rate is 4% compounded semi-annually. Determine: The purchase price of the bond. P = $ The bond's duration to 3 decimals. D = years Note: Use the purchase price to the closest cent in your duration calculation. (C) Suppose that the yield rate is 4% compounded continuously. Determine: The purchase price of the bond. P = $ %3D The bond's duration to 3 decimals. D = years %3D

Answer 1

Solution:

Here Cash flows are the Coupon for the first 3 periods and for the fourth period its coupon plus Value at maturity.

Coupon = Coupon rate * Value of the bond

= 8% * 4000 = 320 { here 8% rate is assumed at semi-annually }

For fourth year, it is Coupon + Value of bond

= 320+4000 = 4320

==>Here, Discount Factor for option a and b = 1/(1+r)^n

Example: for option (a) for year 1 = 1/(1+0.02)^1 = 0.98039 approximately

{ here n is period, and rate is semi-annually}

==> Proportion = Present Value of cashflow at n / Price of the bond

==> Weighted Average = Proportion*Period

(a) Here rate is 2%. as in question it is given 4% annually, but we are calculating semi-annually.

Semi-annually rate = Annual rate / 2

= 4% / 2 = 2%

Period 1 2 B D=B*C F = AE Cash Flow Discount Factor Present Value of Cashflows Proportion Weighted average 320 0.980392157 313.7254902 0.064 0.06385 320 0.961168781 307.57401 0.063 0.12519 320 0.942322335 301.5431471 0.061 0.18410 4320 0.923845426 3991.01224 0.812 3.24878 4913.854888 3.62191 4 Price Value of the Bond Bond Duration 4913.855 3.622

(b) Here Yield rate is 4% semi-annually

Period 1 2 D = B C F=APE Cash Flow Discount Factor Present Value of Cashflows Proportion Weighted average 320 0.961538462 307.6923077 0.067 0.06717 320 0.924556213 295.8579882 0.065 0.12917 3200 .888996359 284.4788348 0.062 0.18631 4320 0.854804191 3692.754105 0.806 3.22456 4580.783236 3.60721 3 4 Price Value of the Bond Bond Duration 4580.783 3.607

(c) Here Discount Factor = e^(-r*Period)

This is done as the rate is compounded continuously.

Here r is 4%

А Period 2 3 4 в D=B*C F = A*E Cash Flow Discount Factor Present Value of Cashflows Proportion Weighted average 320 0.960789439 307.4526205 0.067 0.06731 320 0.923116346 295.3972308 0.065 0.12934 320 0.886920437 283.8145397 0.062 0.18640 4320 0.852143789 3681.261168 0.806 3.22357 4567.925559 3.60661 Price Value of the Bond Bond Duration 4567.926 3.607