In: Accounting
Consider the following bonds: · Bond “LUKE” is a 3-year bearing coupon bond of 5% every six-months with a face value of $1,000. Coupon payments of $50 are made every 6 months. · Bond “ROUGE” is a 3-year bearing coupon bond of 12% per year with a face value of $1,000. Coupon payments of $120 are made every 12 months. · Suppose that the yield on the bond is 6% per annum with continuous compounding.
a) Calculate the bond’s price, duration & convexity.
b) Regarding TSIR: explain the meaning of duration. According to the “Liquidity Preference Theory”, which is the best option for a trader for a 3-years investment? “LUKE” or “ROUGE”? Justify your answer.
c) How would you price the bond “LUKE” if the yield curve fell down 100 basis points (continuous compounding)?
d) How much is the modified duration regarding a rate in semi-annual compounding?
(i). Calculation of Bond Price (LUKE)
a. Coupon Payments every six month = $50
T= 3 Years * 2= 6 Peiods for semi annual coupon payments
Yield on Bond = 6% P.a = 6/2= 3% Semi Annually
Bond Price= 50/(1+0.03)1+ 50/(1+0.03)2 + 50/(1+0.03)3 +50/(1+003)4+50/(1+0.03)5 +50/(1+0.03)6 +1000/(1+0.03)6
= 50*0.971 + 50*0.943 + 50*0.915 + 50*0.888 + 50*0.863 + 50*0.837 + 1000* 0.837
=48.55 + 47.15 + 45.75 + 44.40 + 43.15 + 41.85 + 837
=$ 1107.85/-
b. Bond's Duration
Peiod (X) | Cash Flow(A) | Discounting Factor(B)3% |
Disc. Cash Flow(A*B) (W) |
W*X |
1 | 50 | 0.971 | 48.55 | 48.55 |
2 | 50 | 0.943 | 47.15 | 94.3 |
3 | 50 | 0.915 | 45.75 | 137.25 |
4 | 50 | 0.888 | 44.40 | 177.6 |
5 | 50 | 0.863 | 43.15 | 215.75 |
6 | 1050 | 0.837 | 878.85 | 5273.10 |
TOTAL | 1107.85 | 5946.55 |
Duration = (WX)/W= 5946.55/1107.85= 5.37/2 =2.69 Years
c. Convexity= d2(B(r))/B*d*r2
Where d= Duration = 6
B = Bond Price = $1000
r= Interest Rate = 3%
=(6)2(1000(3%))/1000*6*(3%)2
=1080/540 = 2 Answer
Calculation of Bond Price (ROUGE)
a. Coupon Payment Annually = $120
T= 3 Years, Rate = 6%
Bond Price= 120/(1+0.06)1+ 120/(1+0.06)2+ 120/(1+0.06)3 + 1000/(1+0.04)3
= 120*0.943 +120*0.890 + 120*0.840 + 1000*0.840
=113 + 106.8 + 100.8 + 840 = $1160.60/-
b. Duration of Bond
Peiod (X) | Cash Flow(A) | Discounting Factor(B)6% |
Disc. Cash Flow(A*B) (W) |
W*X |
1 | 120 | 0.943 | 113 | 113 |
2 | 120 | 0.890 | 106.8 | 213.6 |
3 | 120 | 0.840 | 100.8 | 302.4 |
3 | 1000 | 0.840 | 840 | 2520 |
TOTAL | 1160.6 | 3149 |
Duration = (WX)/W= 3149/1160.6= 2.71 Years
c.
Convexity= d2(B(r))/B*d*r2
Where d= Duration = 3
B = Bond Price = $1000
r= Interest Rate = 6%
=(3)2(1000(6%))/1000*3*(6%)2
=540/360 = 1.5 Answer
(ii) Meaning of Duration:- As The Term Suggests, it is expressed in the form of a number of years and measures a bond sensitivity to change in interest rates. To be specific, it measures the change in the market value of security due to a 1% change in interest rate. Usually the higher duration, the more volatility in the prices. In other words it is the number of years required to get the present value of future payments from a Security/bond.
Liquidity Preference Theory is a model that suggests that an investor should demand a higher rate of interest or premium on securities with long term maturities that carry greater risk because all factors being equal, investors prefer cash or other highly liquid holdings. There are basically three motives to determine demand for liquidity
As per the above Explanation Best option for Trader is LUKE Bond as per liquidity preference theory it is more liquid as compared to rouge because coupon payments are being made semi-annually rather than annually.
(ii) LUKE Bond Price If Yield Curve fell down by 100 basis points (i.e 1%) 6%-1% = 5%
Coupon Payments every six month = $50
T= 3 Years * 2= 6 Periods for semi-annual coupon payments
Yield on Bond = 5% P.a = 5/2= 2.5% Semi-Annually
Bond Price= 50/(1+0.025)1+ 50/(1+0.025)2 + 50/(1+0.025)3 +50/(1+025)4+50/(1+0.025)5 +50/(1+0.025)6 +1000/(1+0.025)6
= 50*0.976 + 50*0.952 + 50*0.928 + 50*0.906 + 50*0.884 + 50*0.862 + 1000* 0.862
=48.8+47.6+46.4+45.3+44.2+43.1+862
=$ 1137.4/- Answer
=$ 1107.85/-
(iv) Modified Duration in case of Semi-Annual Compounding
Formula = Macauley Duration/(1+ YTM)
where Macauley duration =2.69 years
YTM= 6/2 =3%
n= 3*2= 6 periods
=2.69/(1+0.03)= 2.61 Years Answer