Question

Your portfolio contains 40% of Bond I, 20% of Bond II, 20% of Bond Ill and 20% of Bond IV. Details of the four bonds are given below: $613.91 10-year zero coupon government bond, par value $1000, current price = II. 10-year zero coupon corporate bond, par value $1000, default premium= 2% III. 5 year 15 % coupon corporate bond, par value $1000, annual coupon payments, default premium 9% and YTM for similar government bond is 6% IV. 5 year 15% government coupon bond, par value $1000, annual coupon payments, YTM=6% a) Find the price of Bond II, III and IV, respectively (6 marks) b) Find the Macaulay's duration of Bond I, II, II and IV, respectively (6 marks) c) What is the duration of your portfolio by using your answers in the part (b)? (2.5 marks) d) If you forecast that the yield curve will shift upwards in the near future, how can you adjust your portfolio to minimize the effect on your portfolio? (2.5 marks) e) What is the convexity of Bond 1? If Bond I's yield increases by 1%, what is the price of Bond I based on duration-with-convexity rule? (6 marks) f) Briefly explain how duration and convexity affect your bond investments. (2 marks)

Answer 1

A) it is given that for 10 year zero coupon bond by government is trading at $613.91

Say yiled Of this bond is r

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1000 613.91 (1+r)10

1000 613.91 (1+r)10

r = 5%

For bond 2, it is given that there is a default premium of 2%. Therefore yiled. Should be 5%+2% =7%

Therefore price of bond 2 =

1000 = 508.349 (1 7%) 10

Bond 3 is a 5 year annual coupon. Paying bond with coupon rate = 15%. It is given that it is having default premium of 9% and a similar government bond is having ytm. Of 6 %. Therefore ytm of bond 3 = 6%+9% =15%.ytm is equal to coupon rate. therefore bond will be btrading at par value 1000.

Bond 4 is government bond with ytm = 6% and annual coupon rate =15%

Therefore price =​​​​​​

150 150 150 150 1150 1379.113 + (1 6% )3 (1+6%)4 (16%) (16%)2 (16%)5

B. Macaulays duration of zero coupon bonds will be equal to the tenure of the bond

Therefore Macaulays duration for bond 1 and 2 = 10 years

For bond 3 and 4, Macaulays duration is calculated As follows.

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BOND 3 time*PV of payment/(sum of Coupon PV of payment PVs) Year Payment dicount factor 150.00 1 0.869565217 130.4347826 0.130434783 150.00 113.4215501 2 0.756143667 0.2268431 150.00 98.62743486 3 0.657516232 0.295882305 150.00 0.571753246 4 85.76298684 0.343051947 5 1150.00 0.497176735 571.7532456 2.858766228 3.854978363 Total 1000 Bond 4 time PV of payment/(sum of Coupon Payment dicount factor PV of payment PVs) Year 141.509434 0.10260904 1 150.00 0.943396226 2 150.00 0.88999644 133.499466 0.193601962 150.00 125.9428925 0.27396504 3 0.839619283 4 150.00 0.792093663 118.8140495 0.344610113 5 1150.00 0.747258173 859.3468988 3.115578855 4.03036501 Total 379.112741

The number given in bold is Macaulays duration.

C) MACAULAYS duration of the portofio is the weighted average of the durations of the bonds

Here we have 0.4*10 + (0.2*10) +(0.2*3.855)+(0.2*4.03) =7.577

D) if yield curve is going to shift upwards, prices are going to fall. To minimize the effect on our portfolio we should increase the weight of bonds with low duration And decrease the weight of bonds with high duration. Price changes are proportional to duration.