Financial Toolbox

Sensitivity of Bond Prices to Parallel Shifts in the Yield Curve

Often bond portfolio managers want to consider more than just the sensitivity of a portfolio's price to a small shift in the yield curve, particularly if the investment horizon is long. This example shows how MATLAB can visualize the price behavior of a portfolio of bonds over a wide range of yield curve scenarios, and as time progresses toward maturity.

This example uses the Financial Toolbox bond pricing functions to evaluate the impact of time-to-maturity and yield variation on the price of a bond portfolio. It plots the portfolio value and shows the behavior of bond prices as yield and time vary. This example M-file is ftspex3.m.

Step 1.   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.

• Settle     = '15-Jan-1995';
  Maturity   = datenum(['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];
  

Also, specify the points on the yield curve for each bond.

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

Step 2.   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;
  

Step 3.   Assume 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;
  

Step 4.   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 = datenum(D(1):2018, D(2), D(3)); % 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.

• Prices = Prices * BondAmounts;
  

Reshape the bond values to conform to the underlying evaluation grid.

• Prices = reshape(Prices, NumYields, NumTimes);
  

Step 5.   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 (i.e., $100,000).

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

Add the base portfolio value to the existing surface plot.

• hold on                  % Add the current value for reference
  basemesh = mesh(dt, dy, 100000*ones(NumYields, NumTimes));
  

Make it transparent, plot it so the price surface shows through, and draw a box around the plot.

• set(basemesh, 'facecolor', 'none');
  set(basemesh, 'edgecolor', 'm');
  set(gca, 'box', 'on');
  

Plot the x-axis using two-digit year (YY format) labels for ticks.

• dateaxis('x', 11);
  

Add axis labels and set the three-dimensional viewpoint. MATLAB produces the figure.

• xlabel('Evaluation Date (YY Format)');
  ylabel('Change in Yield');
  zlabel('Portfolio Price');
  hold off
  view(-25,25);
  
  
  

MATLAB's 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.

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