Question

An investment fund owns $15,000,000 principal of a corporate bond whose modified duration ​is -8.3 (years). The bond's current percentage-of-par price is 108.58% (1.0858). The fund ​may sell the bond in several weeks as part of a portfolio restructuring, and is worried that ​bond yields will rise and prices decline. So it decides to hedge its risk using another bond ​whose modified ​duration is -9.0 (years), the most liquid bond available.

​These are the relevant prices today:

​​Target bond: ModDur = -8.3. ​Percentage-of-par price: 108.58% (1.0858)

​​Hedge bond: ModDur = -9.0.    Percentage-of-par price: 103.59% (1.0359)

​Three weeks later, the fund sells its bond and covers its hedge. Prices then are:

​​Target bond: Percentage-of-par price: 105.00% (1.0500)

​​Hedge bond: Percentage-of-par price: 99.89% (0.9989)

​[NOTE: for this problem remember the distinction between the principal amount of a bond ​and its value, which is principal × decimal format price.]

​a.​What is the anticipated transaction?

​b. ​What can be done to hedge this risk? (i.e. buy/sell? what? how much in principal, how ​​much in value?)

​c.​How much does the firm pay/receive when it carries out the anticipated transaction?

​d.​What does the firm do to cover the hedge position? Did the hedge transaction produce ​​a profit or a loss, and how much?

​e.​Combining the results of the anticipated transaction and the hedge, what is the ​​​effective price of the overall transaction?

Answer 1

​a.​What is the anticipated transaction?

The anticipated transaction is to sell the bond owned by the fund at a lower price than today.

​b. ​What can be done to hedge this risk? (i.e. buy/sell? what? how much in principal, how ​​much in value?)

We need to short another bond in suach a way that modified duration of the combined portfolio is zero i.e.

Market value of owned bonds x Modified duration = Market value of the hedge bond, V x modified duration of the hedge bond

Or, 15,000,000 x 108.58% x 8.3 = V x 9

Hence, V = 15,000,000 x 108.58% x 8.3 / 9 =  15,020,233.33 and the corresponding principal value =  15,020,233.33 / 103.59% = $ 14,499,694.31

Hence, the investment fund should short (sell) the hedge bonds with market value of $ 15,020,233.33 and principal value of $  14,499,694.31

​c.​How much does the firm pay/receive when it carries out the anticipated transaction?

When the anticipated transaction is carried out, the firm receives = Sale value of the owned bonds = 105% x 15,000,000 = 15,750,000

​d.​What does the firm do to cover the hedge position? Did the hedge transaction produce ​​a profit or a loss, and how much?

It buys the the hedge bonds with principal value of $ 14,499,694.31 to cover the position. Price paid = 99.89% x  14,499,694.31 = $  14,483,744.64

So, profit = Short sale value - price paid to buy = 15,020,233.33 - 14,483,744.64 = $  536,488.69

​e.​Combining the results of the anticipated transaction and the hedge, what is the ​​​effective price of the overall transaction?

Sale value of the owned bonds + profit from hedge bonds = 15,750,000 +  $ 536,488.69 = $ 16,286,488.69

Hence, effective price =  16,286,488.69 / 15,000,000 = 108.58% (1.0858)

Hence, percentage of par price = 108.58% (1.0858)

Price per unit in $ terms = Par value per bond x percentage of par = 1,000 x 108.58% = $ 1,085.77