Question

A court has ordered Security Enterprises to pay 200000 in two years and 500000 in five years. In order to meet this important liability, they wish to invest in a combination of two-year 10% par-value bonds with annual coupons and five-year zero-coupon bonds. Each of these is sold to yield an annual effective yield of 4%. how much of each type of bond should be purchased so that the present value and duration conditions of Redington immunization are satisfied at the current 4% rate? what is the face value of the 2 year bond purchased? 5 year bond? Answers: 2 year= $179,736.12, FV= $161,463.94 5 year= $416,138.68,

Answer 1

Liability = 200,000 in two years and 500,000 in five years.

Discount rate = current yield = y = 4%

PV and duration of liabilities

time, t	Liability, L	PV of L = L / (1 + y)t = L / (1 + 4%)t	PV x t
2	200,000.00	184,911.24	369,822.49
5	500,000.00	410,963.55	2,054,817.77
Total	700,000.00	595,874.80	2,424,640.25

Hence, PV of liabilities = 595,874.80 and duration of liabilities = 2,424,640.25 / 595,874.80 =  4.0690 years

Let A and B be the number of 2 year coupon bond and 5 year zero coupon bond respectively.

2 year coupon bond is 10% annual coupon; let's assume it's face (par) value to be $ 1,000

It's PV (current price) and duration is as shown below:

time, t	Payment, P	PV of P = P / (1 + y)t = P / (1 + 4%)t	PV x t
1	100.00*	96.15	96.15
2	1,100.00**	1,017.01	2,034.02
Total	1,200.00	1,113.17	2,130.18

* Annual coupon = 10% x Par value = 10% x 1,000 = 100

**Annual coupon + Par value repayment = 100 + 1,000 = 1,100

PV_2yearCB = 1,113.17

Duration = D_2yearCB = 2,130.18 / 1,113.17 =  1.9136 years

For 5 year zero coupon bond (ZCB), PV_5yearZCB = 1,000 / (1 + y)⁵ = 1,000 / (1 + 4%)⁵ =  821.93

and Duration = D_5yearZCB = 5 years

Hence, to match PV: A x PV_2yearCB + B x PV_5yearZCB = A x 1,113.17 + B x 821.93 = PV of Liabilities = 595,874.80

Or, 1,113.17A + 821.93B = 595,874.80 --------Equation (1)

Duration of portfolio = sum of market value proportion weighted duration of each security

Market value proportion of 2 year coupon bond = A x 1,113.17 / (A x 1,113.17 + B x 821.93) = A x 1,113.17 / 595,874.80 = 0.00186812A

Market value proportion of 5 year zero coupon bond = B x 821.83 / (A x 1,113.17 + B x 821.93) = B x 821.83 / 595,874.80 = 0.001379362B

To match duration:

0.00186812A x D_2yearCB + 0.001379362B x D_yearZCB = 0.00186812A x 1.9136 + 0.001379362B x 5 = 0.003575A + 0.0068968B = Duration of Liabilities = 4.0690

Or, 0.003575A + 0.0068968B = 4.0690 ------------ Equation (2)

0.0068968 x Equation (1) - 821.93 x Equation (2) gives:

0.0068968 x 1,113.17A - 821.93 x 0.003575A = 4.74A = 0.0068968 x 595,874.80 - 821.93 x 4.0690 = 765.18

Or, A =  765.18 / 4.74 =161.463940

From equation (1): B = (595,874.80 - 1,113.17A) / 821.93 = 506.2963324

So your final answer:

Value of 2 year coupon bond in market value terms = A x PV_2yearCB = 161.463940 x 1,113.17 =  179,736.12

Face value of 2 year coupon bond = A x Face value per bond = A x 1000 = 161.463940 x 1000 =  161,463.94

Value of 5 year ZCB in market value terms = B x PV_5yearZCB = 506.2963324 x 821.93 =  416,138.68

Face value of 5 year zero coupon bond = B x Face value per bond = 506.2963324 x 1000 = 506,296.33