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Fix SimpleThenCompounded NPV for long first coupon period #2473

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Fix SimpleThenCompounded NPV for long first coupon period #2473

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91 changes: 76 additions & 15 deletions ql/cashflows/cashflows.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -618,6 +618,9 @@ namespace QuantLib {
Time t = 0.0;
Date lastDate = npvDate;
const DayCounter& dc = y.dayCounter();
bool stc = (y.compounding() == SimpleThenCompounded);
DiscountFactor discount = 1.0;
bool firstCoupon = stc;
for (const auto& i : leg) {
if (i->hasOccurred(settlementDate, includeSettlementDateFlows))
continue;
Expand All @@ -627,8 +630,22 @@ namespace QuantLib {
c = 0.0;
}

t += getStepwiseDiscountTime(i, dc, npvDate, lastDate);
DiscountFactor B = y.discountFactor(t);
Time dt = getStepwiseDiscountTime(i, dc, npvDate, lastDate);
t += dt;
DiscountFactor B;
if (stc) {
DiscountFactor b;
if (firstCoupon) {
b = 1.0 / (1.0 + y.rate() * dt);
firstCoupon = false;
} else {
b = y.discountFactor(dt);
}
discount *= b;
B = discount;
} else {
B = y.discountFactor(t);
}
P += c * B;
dPdy += t * c * B;

Expand Down Expand Up @@ -660,6 +677,10 @@ namespace QuantLib {
Natural N = y.frequency();
Date lastDate = npvDate;
const DayCounter& dc = y.dayCounter();
bool stc = (y.compounding() == SimpleThenCompounded);
DiscountFactor discount = 1.0;
Real alpha = 0.0;
bool firstCoupon = stc;
for (const auto& i : leg) {
if (i->hasOccurred(settlementDate, includeSettlementDateFlows))
continue;
Expand All @@ -669,8 +690,24 @@ namespace QuantLib {
c = 0.0;
}

t += getStepwiseDiscountTime(i, dc, npvDate, lastDate);
DiscountFactor B = y.discountFactor(t);
Time dt = getStepwiseDiscountTime(i, dc, npvDate, lastDate);
t += dt;
DiscountFactor B;
if (stc) {
DiscountFactor b;
if (firstCoupon) {
b = 1.0 / (1.0 + r * dt);
alpha = dt * b;
firstCoupon = false;
} else {
b = y.discountFactor(dt);
alpha += dt / (1.0 + r / N);
}
discount *= b;
B = discount;
} else {
B = y.discountFactor(t);
}
P += c * B;
switch (y.compounding()) {
case Simple:
Expand All @@ -683,10 +720,7 @@ namespace QuantLib {
dPdy -= c * B * t;
break;
case SimpleThenCompounded:
if (t<=1.0/N)
dPdy -= c * B*B * t;
else
dPdy -= c * t * B/(1+r/N);
dPdy -= c * B * alpha;
break;
case CompoundedThenSimple:
if (t>1.0/N)
Expand Down Expand Up @@ -833,6 +867,7 @@ namespace QuantLib {
DiscountFactor discount = 1.0;
Date lastDate = npvDate;
const DayCounter& dc = y.dayCounter();
bool firstCoupon = (y.compounding() == SimpleThenCompounded);
for (const auto& i : leg) {
if (i->hasOccurred(settlementDate, includeSettlementDateFlows))
continue;
Expand All @@ -842,7 +877,14 @@ namespace QuantLib {
amount = 0.0;
}

DiscountFactor b = y.discountFactor(getStepwiseDiscountTime(i, dc, npvDate, lastDate));
Time dt = getStepwiseDiscountTime(i, dc, npvDate, lastDate);
DiscountFactor b;
if (firstCoupon) {
b = 1.0 / (1.0 + y.rate() * dt);
firstCoupon = false;
} else {
b = y.discountFactor(dt);
}
discount *= b;
lastDate = i->date();

Expand Down Expand Up @@ -992,6 +1034,10 @@ namespace QuantLib {
Rate r = y.rate();
Natural N = y.frequency();
Date lastDate = npvDate;
bool stc = (y.compounding() == SimpleThenCompounded);
DiscountFactor discount = 1.0;
Real alpha = 0.0, beta = 0.0;
bool firstCoupon = stc;
for (const auto& i : leg) {
if (i->hasOccurred(settlementDate, includeSettlementDateFlows))
continue;
Expand All @@ -1001,8 +1047,26 @@ namespace QuantLib {
c = 0.0;
}

t += getStepwiseDiscountTime(i, dc, npvDate, lastDate);
DiscountFactor B = y.discountFactor(t);
Time dt = getStepwiseDiscountTime(i, dc, npvDate, lastDate);
t += dt;
DiscountFactor B;
if (stc) {
DiscountFactor b;
if (firstCoupon) {
b = 1.0 / (1.0 + r * dt);
alpha = dt * b;
beta = -dt * dt * b * b;
firstCoupon = false;
} else {
b = y.discountFactor(dt);
alpha += dt / (1.0 + r / N);
beta -= dt / (N * (1.0 + r / N) * (1.0 + r / N));
}
discount *= b;
B = discount;
} else {
B = y.discountFactor(t);
}
P += c * B;
switch (y.compounding()) {
case Simple:
Expand All @@ -1015,10 +1079,7 @@ namespace QuantLib {
d2Pdy2 += c * B*t*t;
break;
case SimpleThenCompounded:
if (t<=1.0/N)
d2Pdy2 += c * 2.0*B*B*B*t*t;
else
d2Pdy2 += c * B*t*(N*t+1)/(N*(1+r/N)*(1+r/N));
d2Pdy2 += c * B * (alpha*alpha - beta);
break;
case CompoundedThenSimple:
if (t>1.0/N)
Expand Down
54 changes: 54 additions & 0 deletions test-suite/bonds.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -1892,6 +1892,60 @@ BOOST_AUTO_TEST_CASE(testFixingConvention) {
BOOST_CHECK_EQUAL(couponF->fixingDate(), expectedFollowing);
}

BOOST_AUTO_TEST_CASE(testSimpleThenCompoundedLongFirstPeriod) {

BOOST_TEST_MESSAGE("Testing SimpleThenCompounded with a long first coupon period...");

// A semiannual bond with an 8-month first coupon period. The ICMA
// specification requires simple discounting for the first period
// regardless of its length; the time-magnitude threshold in
// InterestRate::compoundFactor() produces the wrong result in this case.

Date settlement(15, January, 2023);
Settings::instance().evaluationDate() = settlement;

// Backward schedule with an explicit first coupon date produces a long
// (8-month) first stub: Jan 15 -> Sep 15 -> Mar 15 -> ... -> Mar 15, 2026
Schedule sch(settlement,
Date(15, March, 2026),
Period(Semiannual),
NullCalendar(),
Unadjusted, Unadjusted,
DateGeneration::Backward,
false,
Date(15, September, 2023));

DayCounter dc = Thirty360(Thirty360::BondBasis);
FixedRateBond bond(0, 100.0, sch, std::vector<Rate>(1, 0.06), dc);

InterestRate yield(0.05, dc, SimpleThenCompounded, Semiannual);

// Expected dirty price computed from the ICMA formula:
// B1 = 1/(1 + 0.05*8/12) = 30/31 (simple for the long first period)
// B_k = B1 * (1/1.025)^(k-1) (compound for subsequent periods)
// cash flows: 4, 3, 3, 3, 3, 103
Real expectedPrice = 102.8931;
Real tolerance = 1.0e-4;

Real calculated = BondFunctions::dirtyPrice(bond, yield, settlement);
if (std::fabs(calculated - expectedPrice) > tolerance)
BOOST_ERROR("failed to reproduce dirty price for long-first-period bond"
<< std::fixed << std::setprecision(6)
<< "\n calculated: " << calculated
<< "\n expected: " << expectedPrice
<< "\n tolerance: " << tolerance);

// Yield roundtrip: the implied yield must recover the input.
Bond::Price price{calculated, Bond::Price::Dirty};
Rate impliedYield = BondFunctions::yield(bond, price,
dc, SimpleThenCompounded, Semiannual,
settlement, 1.0e-10, 100, 0.05);
if (std::fabs(impliedYield - yield.rate()) > 1.0e-7)
BOOST_ERROR("yield roundtrip failed for long-first-period bond"
<< "\n recovered: " << impliedYield
<< "\n expected: " << yield.rate());
}

BOOST_AUTO_TEST_SUITE_END()

BOOST_AUTO_TEST_SUITE_END()
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