SHOW FORMULA & WORK ! ! ! Question 4 Suppose an investor plans to buy some Zero

Question

SHOW FORMULA & WORK ! ! !

Question 4 Suppose an investor plans to buy some Zero Coupon Bonds with (i) 2 years to maturity and with s2 = 0.02 and (ii) 4 years to maturity with s4-0.03. (a) What is the price, and the cost, of each bond? (b) What is the 2-year discrete forward rate after 2 years? (c) What is the (discrete) modified duration of each bond? (d) If he buys 10 of the 4 year ZC Bonds and N of the 2 year bonds, how large should N be so that the modified duration of the portfolio is less than 3 years

Explanation / Answer

Zero Coupon Bond

(a) Price and Cost of each bond

          Price of Bond=       F/(1+r)^n

          Suppose F=$100

(b) 2-year discrete forward rate after 2 years

          r2 = 0.02   ,   r4   =   0.03

           Formula      (1 + F2,2)^2 =   (1 + r4^4)   /    (1   +   r2^2)

                             (1 + F2,2) ^2 =   (1 + 0.03^4)   /    (1   +   0.02^2)

                             (1 + F2,2) ^2 =   1.1255 /     1.0404

                             (1 + F2,2) ^2 =    1.0818

                             (1 + F2,2)      =    1.04009

                                   F2,2       =     0.04009 =   4.009%

(c)    Modified duration of each bond

modified duration = 1/(1+yield/k)[1 x pvcf1 + 2 x pvcf2 +...+n x pvcfn / k x Price
Where:
k = the number of periods: two for semi-annual, 12 for monthly and so on.
n = the number of periods to maturity
yield = YTM of the bond
pvcf = the present value of cash flows discounted at the yield to maturity.

The bracket part of the equation was developed by Frederick Macaulay in 1938 and is referred to as Macaulay Duration.

So Modified duration = Macaulay's Duration/ (1 + yield/k)

                                       = (100 X 2%) / (1 + 0.02)                       =1.96

                                       = [100+ (100 X 2%) ] / (1 + 0.02)^2       =98.07         

          Macaulay's Duration = (1.96   X 1) + (98.07 X 2)   /    1.96 + 98.07

                                        = 1.96 +    196.153   /100.03

                                        =   1.981

          Modified Duration   =    1.981   /   (1 + 0.02)

                                      =    1.94

                                     = (100 X 3%) / (1 + 0.03)                       =2.91

                                       = [100+ (100 X 3%) ] / (1 + 0.03)^2         =97.08

         = [100+100 (100 X 3%) ] / (1 + 0.03)^3    = 185.73

                                      = [100+100 +100 (100 X 3%) ] / (1 + 0.03)^4 = 254.44

       Macaulay's Duration = (2.91X1) + (97.08 X 2) +(185.73 X 3) + (194.16 X 4) / 2.91 + 97.08 + 185.73 + 254.44

                                        = 2.91 + 194.16 + 557.19 + 1017.76 / 540.16

                                        = 3.281

          Modified Duration   =    3.281 /   (1 + 0.03)

                                      =    3.18

(d)   An Investor should buys 5 of the 2 year bonds

                                      = (10 X   2)   /    4

                                      =   5 bonds

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