Bond Prices and Yield Curve Parallel Shifts

This example uses Financial Toolbox™ bond pricing functions to evaluate the impact of time-to-maturity and yield variation on the price of a bond portfolio. Also, this example shows how to visualize the price behavior of a portfolio of bonds over a wide range of yield curve scenarios, and as time progresses toward maturity.

Specify values for the settlement date, maturity date, face value, coupon rate, and coupon payment periodicity of a four-bond portfolio. For simplicity, accept default values for the end-of-month payment rule (rule in effect) and day-count basis (actual/actual). Also, synchronize the coupon payment structure to the maturity date (no odd first or last coupon dates). Any inputs for which defaults are accepted are set to empty matrices ([]) as placeholders where appropriate. Also, specify the points on the yield curve for each bond.

Settle     = datetime(1995,1,15);
Maturity   = datetime( [2020, 4, 3;...
                        2025, 5,14;...
                        2019, 6, 9;...
                        2019, 2,25])

Maturity = 4x1 datetime
   03-Apr-2020
   14-May-2025
   09-Jun-2019
   25-Feb-2019

Face       = [1000; 1000; 1000; 1000];
CouponRate = [0; 0.05; 0; 0.055];
Periods    = [0; 2; 0; 2];

Yields = [0.078; 0.09; 0.075; 0.085];

Use Financial Toolbox functions to calculate the true bond prices as the sum of the quoted price plus accrued interest.

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

Prices  =  CleanPrice + AccruedInterest

Prices = 4×1

  145.2452
  594.7757
  165.8949
  715.7584

Assume that the value of each bond is $25,000, and determine the quantity of each bond such that the portfolio value is $100,000.

BondAmounts = 25000 ./ Prices;

Compute the portfolio price for a rolling series of settlement dates over a range of yields. The evaluation dates occur annually on January 15, beginning on 15-Jan-1995 (settlement) and extending out to 15-Jan-2018. Thus, this step evaluates portfolio price on a grid of time of progression (dT) and interest rates (dY).

dy = -0.05:0.005:0.05;               % Yield changes

D  = datevec(Settle);                % Get date components
dt = datetime(year(Settle):2018, month(Settle), day(Settle)); % Get evaluation dates

[dT, dY]  =  meshgrid(dt, dy); % Create grid

NumTimes  =  length(dt);       % Number of time steps
NumYields =  length(dy);       % Number of yield changes
NumBonds  =  length(Maturity); % Number of bonds

% Preallocate vector
Prices = zeros(NumTimes*NumYields, NumBonds);

Now that the grid and price vectors have been created, compute the price of each bond in the portfolio on the grid one bond at a time.

for i = 1:NumBonds

   [CleanPrice, AccruedInterest] = bndprice(Yields(i)+... 
   dY(:), CouponRate(i), dT(:), Maturity(i), Periods(i),...
   [], [], [], [], [], [], Face(i));

   Prices(:,i) = CleanPrice + AccruedInterest;

end

Scale the bond prices by the quantity of bonds and reshape the bond values to conform to the underlying evaluation grid.

Prices = Prices * BondAmounts;
Prices = reshape(Prices, NumYields, NumTimes);

Plot the price of the portfolio as a function of settlement date and a range of yields, and as a function of the change in yield (dY). This plot illustrates the interest-rate sensitivity of the portfolio as time progresses (dT), under a range of interest-rate scenarios. With the following graphics commands, you can visualize the three-dimensional surface relative to the current portfolio value (that is, $100,000).

figure                   % Open a new figure window
surf(dt, dy, Prices)     % Draw the surface

hold on                  % Add the current value for reference
basemesh = mesh(dt, dy, 100000*ones(NumYields, NumTimes));
set(basemesh, 'facecolor', 'none');
set(basemesh, 'edgecolor', 'm');
set(gca, 'box', 'on');

xlim(datetime([1993,2020],1,1))
xlabel('Evaluation Date');
ylabel('Change in Yield');
zlabel('Portfolio Price');
hold off
view(-25,25);

MATLAB® three-dimensional graphics allow you to visualize the interest-rate risk experienced by a bond portfolio over time. This example assumed parallel shifts in the term structure, but it might similarly have allowed other components to vary, such as the level and slope.