Welcome back! In my previous post, we learned about the term structure of interest rates in the context of bond pricing. Given the term structure, we saw how to calculate the price of a fixed-rate bond by calculating the net present value (NPV) of the bond’s future cash flows, consisting of its annual coupon payments plus the final premium repayment. The bond’s price, if you recall, was calculated using a constant interest rate, which, as stated in my previous post, is equivalent to a flat interest rate term structure.

Now, let’s assume that we know the price of a bond or, more generally, the present value of a sequence of future cash flows and we would like to know what interest rate was used to arrive at the NPV. This rate of return is commonly referred to as the investment’s ‘internal rate of return’ (IRR) or, when applied to a bond, the bond’s ‘yield to maturity’ (YTM).

Let’s look at the cash flows from the bond example we’ve been considering thus far. The bond has a face value of 100, pays annual coupons of 5% and has a price (NPV) of 105.66. The bond’s cash flows, in order are, 5.0, 5.0 and 105.0. Therefore, to compute the yield to maturity of the bond, we need to solve for the interest rate, r, in the following equation:

**105.66 = 5/(1+r) + 5/(1+r)^2 + 105/(1+r)^3**

It turns out that there is no analytical, algebraic method to determine the value of r. As such, the equation must be solved numerically, using a solver method. In the example code that follows, I use the Bisection class, one of several solver classes supported by QuantLib to back out the bond’s yield to maturity. Other solvers available in QuantLib include: Brent, Secant, Ridder, Newton and FalsePosition. Each one of these solver classes will find the zero, or root, of a function by varying a single input to the function. The solver will terminate its search when the return value of the function is less than or equal to the *accuracy* parameter, which is defined to be .000001 in the code below.

The function passed to the Bisection solver in this example takes two forms. The first is a *functor* class, which is a class with operator() overridden. The second is a lambda expression, which is a type of closure introduced in the latest C++ standard, C++ 11.

The code to calculate the bond’s yield to maturity is as follows:

#### #include <cstdlib>

#include <iostream>

#define BOOST_AUTO_TEST_MAIN

#include <boost/test/unit_test.hpp>

#include <boost/detail/lightweight_test.hpp>

#include <ql/quantlib.hpp>

#include <vector>

#include <ql/instruments/bonds/fixedratebond.hpp>

#### namespace {

using namespace QuantLib;class IRRSolver {

public:

explicit IRRSolver(const Leg& cashFlows, Real npv): _cashFlows(cashFlows),_npv(npv){};

Real operator() (const Rate& rate) const {

InterestRate interestRate(rate, ActualActual(ActualActual::Bond), Compounded, Annual);

return CashFlows::npv(_cashFlows, interestRate, false) – _npv;

}

private:

const Real _npv;

const Leg& _cashFlows;

};

#### BOOST_AUTO_TEST_CASE(testCalculateBondYieldToMaturity) {

Calendar calendar = UnitedStates(UnitedStates::GovernmentBond);

const Natural settlementDays = 3;

Date today = Date::todaysDate();

Date issueDate = today;

Date terminationDate = issueDate + Period(3, Years);

Rate rate = .03;

#### InterestRate couponRate(.05, ActualActual(ActualActual::Bond), Compounded, Annual);

Real faceValue = 100.0;

std::vector<Rate> coupons(3, couponRate);

Schedule schedule(issueDate, terminationDate, Period(Annual), calendar,

Unadjusted, Unadjusted, DateGeneration::Backward, false);

FixedRateBond fixedRateBond(settlementDays, faceValue, schedule, coupons);

boost::shared_ptr<YieldTermStructure> flatForwardRates(new FlatForward(issueDate, rate, ActualActual(ActualActual::Bond), Compounded, Annual));

Handle<YieldTermStructure> flatTermStructure(flatForwardRates);

boost::shared_ptr<PricingEngine> bondEngine(new DiscountingBondEngine(flatTermStructure));

#### fixedRateBond.setPricingEngine(bondEngine);

Real npv = fixedRateBond.NPV();

std::cout << “NPV of bond is: ” << npv << std::endl;

#### //solve for yield to maturity using bisection solver

Bisection bisection;

Real accuracy=0.000001, guess=.10;

Real min=.0025, max=.15;

#### //invoke bisection solver with IRRSolver functor

Real irr = bisection.solve(IRRSolver(fixedRateBond.cashflows(), npv),

accuracy, guess, min, max);

#### std::cout << “Bond yield to maturity (IRR) is: ” << irr << std::endl;

#### //invoke bisection solver with C++ 11 lambda expression

irr = bisection.solve([&] (const Rate& rate){ return CashFlows::npv(fixedRateBond.cashflows(), InterestRate(rate, ActualActual(ActualActual::Bond), Compounded, Annual), false) – npv;},

accuracy, guess, min, max);

#### std::cout << “Bond yield to maturity (IRR) is: ” << irr << std::endl;

}

}

The output of this code when run is:

NPV of bond is: 105.657

Bond yield to maturity (IRR) is: 0.0300004

Bond yield to maturity (IRR) is: 0.0300004

As you can see both IRR solver methods produce the same result, 3%, which is indeed the interest rate of the bond.

I hope you enjoyed this post. Please feel free to leave comments or questions that you might have. In the next installment of this series, I’ll examine how to calculate the sensitivity of a bond’s price to a change in the level of interest rates. Until then, have fun with QuantLib!

Dear Mick,

I really enjoy your code and I am trying to learn more about QuantLib.

I copied the code in Visual Studio Express 2012 and I am getting an error message for the line:

std::vector coupons(3, couponRate);

I tried to correct the vector definition by declaring it double (just a guess):

std::vector coupons(3, couponRate);

but then I am getting an error message for the line:

FixedRateBond fixedRateBond(…)

Kind regards,

Ricky

Oops… I did not add the right code.

I change the vector to:

std::vector coupons(3, couponRate);

vector coupons(3, couponRate);

I hope it shows up correctly this time. It omitted the part after vector twice before.

Hello,

The way you explain is amazing!

I am also interested in knowing about the theory of various engines which have been implemented in QuantLib. Where can I get this information?

regards,

Jagmohan