Question

An investor wishes to be sure she has $20 million in 15 months’ time. At present, 1-year and 2-year zero-coupon bonds are priced to yield 9.7%. The investor sets up a bond portfolio using the duration-matching principle. Three months after setting up the portfolio, the yields on both bonds increase to 10.2% and then remain at that level for a further 12 months. Assume that all months are of equal length, that all bonds have a par value of $100, and that investors may trade any number of bonds, including fractions of bonds.  Show that the investor will achieve her target sum.  Be precise about the timing and amounts of any transactions required.  Show your calculations.  

Answer 1

(a) Calculate the prices today of the one-year zero-coupon bond and the two-year zero-coupon bond.

P0,1 = $100/1.097 = $91.157703

P0,2 = $100/(1.097)^2 = $83.097268

(b) Amount that the investor need to invest today

Amount invested today = $20,000,000/(1.097)^1.25 = $20,000,000/1.122685942 = $17,814,421

(c) Amount invested today in the one-year bond and two-year bond

w1*1 + w2*2 = 1.25

w1+w2 = 1

which solves to give w1 = 0.75 and w2 = 0.25.

Therefore, amounts invested today are:

in the 1-year bond: 0.75*$17,814,421 = $13,360,816;

in the 2-year bond: 0.25*$17,814,421 = $4,453,605

(d) Number of 1-year bonds = $13,360,816/$91.157703 = 146,568.15

Number of 2-year bonds = $4,453,605/$83.097268 = 53,595.08

(e) After three months, yields increase and bond prices fall. These amounts can be calculated but are not needed. After a further 9 months we reach the maturity date of the one-year bonds (that is, t = 1). The cash received is 146,568.15 × $100 = $14,656,815. This cash is then used to buy more one-year bonds. The price of a one-year bond at t= 1 (maturing at t= 2) is:

P1,2 = $100/1.102 = $90.744102

Hence, the number of these bonds bought is:

$14,656,815/$90.744102 = 161,518.10,

bringing the total number of such bonds held to

53,595.08 + 161,518.10 = 215,113.18

These bonds are sold on the target date (t = 1.25).

At target date, bond prices of both 1-year zero coupon bonds and 2-year zero coupon bonds are:

Bond price (at t = 1.25) = 100/(1+10.2%)^0.75 = $92.9745

Cash received after selling 215,113.18 zero coupon bonds = 215,113.18*$92.9745 = $20,000,038