Bond Prices and Yield Curve Nonparallel Shifts

This example shows how to construct a bond portfolio to hedge the interest-rate risk of a Treasury bond maturing in 20 years. Key rate duration enables you to determine the sensitivity of the price of a bond to nonparallel shifts in the yield curve. This example uses bndkrdur to construct a portfolio to hedge the interest-rate risk of a U.S. Treasury bond maturing in 20 years.

Specify the bond.

Settle = datetime(2008,12,2);
CouponRate = 5.500/100;
Maturity = datetime(2028,8,15);
Price = 128.68;

The interest-rate risk of this bond is hedged with the following four on-the-run Treasury bonds:

Maturity_30 = datetime(2038,5,15);  % 30-year bond
Coupon_30 = .045;
Price_30 = 124.69;

Maturity_10 = datetime(2018,11,15);  %10-year note
Coupon_10 = .0375;
Price_10 = 109.35;

Maturity_05 = datetime(2013,11,30);  % 5-year note
Coupon_05 = .02;
Price_05 = 101.67;

Maturity_02 = datetime(2010,11,30);  % 2-year note
Coupon_02 = .01250;
Price_02 =  100.72;

You can get the Treasury spot or zero curve from https://www.treas.gov/offices/domestic-finance/debt-management/interest-rate/yield.shtml:

ZeroDates = daysadd(Settle,[30 90 180 360 360*2 360*3 360*5 ...
360*7 360*10 360*20 360*30]);
ZeroRates = ([0.09 0.07 0.44 0.81 0.90 1.16 1.71 2.13 2.72 3.51 3.22]/100)';

Compute the key rate durations for both the bond and the hedging portfolio.

BondKRD = bndkrdur(table(ZeroDates, ZeroRates), CouponRate, Settle,...
Maturity,'keyrates',[2 5 10 20]);
HedgeMaturity = [Maturity_02;Maturity_05;Maturity_10;Maturity_30];
HedgeCoupon = [Coupon_02;Coupon_05;Coupon_10;Coupon_30];
HedgeKRD = bndkrdur(table(ZeroDates, ZeroRates), HedgeCoupon,...
Settle, HedgeMaturity, 'keyrates',[2 5 10 20])

HedgeKRD = 4×4

    1.9675         0         0         0
    0.1269    4.6152         0         0
    0.2129    0.7324    7.4010         0
    0.2229    0.7081    2.1487   14.5172

Compute the dollar durations for each of the instruments and each of the key rates (assuming holding 100 bonds).

PortfolioDD = 100*Price* BondKRD;
HedgeDD = HedgeKRD.*[Price_30;Price_10;Price_05;Price_02]

HedgeDD = 4×4
103 ×

    0.2453         0         0         0
    0.0139    0.5047         0         0
    0.0216    0.0745    0.7525         0
    0.0224    0.0713    0.2164    1.4622

Compute the number of bonds to sell short to obtain a key rate duration that is 0 for the entire portfolio.

NumBonds = PortfolioDD/HedgeDD

NumBonds = 1×4

    3.8973    6.1596   23.0282   80.0522

These results indicate selling 4, 6, 23 and 80 bonds respectively of the 2-, 5-, 10-, and 30-year bonds achieves a portfolio that is neutral with respect to the 2-, 5-, 10-, and 30-year spot rates.