Step 1

Define three bonds available for hedging the original portfolio. Specify values for the settlement date, maturity date, face value, and coupon rate. For simplicity, accept default values for the coupon payment periodicity (semiannual), end-of-month payment rule (rule in effect), and day-count basis (actual/actual). Also, synchronize the coupon payment structure to the maturity date (that is, no odd first or last coupon dates). Set any inputs for which defaults are accepted to empty matrices ([]) as placeholders where appropriate. The intent is to hedge against duration and convexity and constrain total portfolio price.

Settle     = '19-Aug-1999';
Maturity   = ['15-Jun-2005'; '02-Oct-2010'; '01-Mar-2025'];
Face       = [500; 1000; 250];
CouponRate = [0.07; 0.066; 0.08];

Also, specify the yield curve for each bond.

Yields = [0.06; 0.07; 0.075];

Step 2

Use Financial Toolbox™ functions to calculate the price, modified duration in years, and convexity in years of each bond.

The true price is quoted (clean price plus accrued interest).

[CleanPrice, AccruedInterest] = bndprice(Yields,CouponRate,... 
Settle, Maturity, 2, 0, [], [], [], [], [], Face);

Durations = bnddury(Yields, CouponRate, Settle, Maturity,...
2, 0, [], [], [], [], [], Face);

Convexities = bndconvy(Yields, CouponRate, Settle,... 
Maturity, 2, 0, [], [], [], [], [], Face);

Prices  =  CleanPrice + AccruedInterest

Prices =

  530.4248
  994.4065
  273.4051

Step 3

Set up and solve the system of linear equations whose solution is the weights of the new bonds in a new portfolio with the same duration and convexity as the original portfolio. In addition, scale the weights to sum to 1; that is, force them to be portfolio weights. You can then scale this unit portfolio to have the same price as the original portfolio. Recall that the original portfolio duration and convexity are 10.3181 and 157.6346, respectively. Also, note that the last row of the linear system ensures that the sum of the weights is unity.

A = [Durations'
     Convexities'
     1 1 1];

b = [ 10.3181
     157.6346
       1];

Weights = A\b

Weights =

   -0.3043
    0.7130
    0.5913

Step 4

Compute the duration and convexity of the hedge portfolio, which should now match the original portfolio.

PortfolioDuration  = Weights' * Durations
PortfolioConvexity = Weights' * Convexities

PortfolioDuration =

   10.3181

PortfolioConvexity =

  157.6346

Step 5

Finally, scale the unit portfolio to match the value of the original portfolio and find the number of bonds required to insulate against small parallel shifts in the yield curve.

PortfolioValue = 100000;
HedgeAmounts   = Weights ./ Prices * PortfolioValue

HedgeAmounts =

  -57.3716
   71.7044
  216.2653

Step 6

Compare the results.

Notice that the hedge matches the duration, convexity, and value ($100,000) of the original portfolio. If you are holding that first portfolio, you can hedge by taking a short position in the new portfolio.

Just as the approximations of the first example are appropriate only for small parallel shifts in the yield curve, the hedge portfolio is appropriate only for reducing the impact of small level changes in the term structure.