YTM 与零息债券的零利率不同，以及 QuantLib Python 中贴现的结算日问题

Question

我使用以下代码使用零息债券和固定息票债券引导了收益率曲线；

# Importing Libraries:
# The code imports necessary libraries:
# pandas for data manipulation, matplotlib.pyplot for plotting, and QuantLib (ql) for quantitative finance calculations.
import pandas as pd
import matplotlib.pyplot as plt
# Use the QuantLib or ORE Libraries
import QuantLib as ql

# Setting Evaluation Date:
# Sets the evaluation date to May 31, 2023, for all subsequent calculations.
today = ql.Date(21, ql.November, 2023)
ql.Settings.instance().evaluationDate = today

# Calendar and Day Count:
# Creates a calendar object for South Africa and specifies the day-count convention (Actual/365 Fixed)
calendar = ql.NullCalendar()
day_count = ql.Actual365Fixed()

# Settlement Days:
zero_coupon_settlement_days = 4
coupon_bond_settlement_days = 3

# Face Value
faceAmount = 100

data = [
    ('11-09-2023', '11-12-2023', 0, 99.524, zero_coupon_settlement_days),
    ('11-09-2023', '11-03-2024', 0, 96.539, zero_coupon_settlement_days),
    ('11-09-2023', '10-06-2024', 0, 93.552, zero_coupon_settlement_days),
    ('11-09-2023', '09-09-2024', 0, 89.510, zero_coupon_settlement_days),
    ('22-08-2022', '22-08-2024', 9.0, 96.406933, coupon_bond_settlement_days),
    ('27-06-2022', '27-06-2025', 10.0, 88.567570, coupon_bond_settlement_days),
    ('27-06-2022', '27-06-2027', 11.0, 71.363073, coupon_bond_settlement_days),
    ('22-08-2022', '22-08-2029', 12.0, 62.911623, coupon_bond_settlement_days),
    ('27-06-2022', '27-06-2032', 13.0, 55.976845, coupon_bond_settlement_days),
    ('22-08-2022', '22-08-2037', 14.0, 52.656596, coupon_bond_settlement_days)]

helpers = []

for issue_date, maturity, coupon, price, settlement_days in data:
    price = ql.QuoteHandle(ql.SimpleQuote(price))
    today = ql.Date(21, ql.November, 2023)
    issue_date = ql.Date(issue_date, '%d-%m-%Y')
    maturity = ql.Date(maturity, '%d-%m-%Y')
    schedule = ql.Schedule(today, maturity, ql.Period(ql.Semiannual), calendar, ql.DateGeneration.Backward,
                           ql.Following, ql.DateGeneration.Backward, False)
    helper = ql.FixedRateBondHelper(price, settlement_days, faceAmount, schedule, [coupon / 100], day_count,
                                    False)
    helpers.append(helper)

curve = ql.PiecewiseCubicZero(today, helpers, day_count)

# Enable Extrapolation:
# This line enables extrapolation for the yield curve.
# Extrapolation allows the curve to provide interest rates or rates beyond the observed data points,
# which can be useful for pricing or risk management purposes.
curve.enableExtrapolation()

# Zero Rate and Discount Rate Calculation:
# Calculates and prints the zero rate and discount rate at a specific
# future date (May 28, 2048) using the constructed yield curve.
date = ql.Date(11, ql.December, 2023)
zero_rate = curve.zeroRate(date, day_count, ql.Annual).rate()
forward_rate = curve.forwardRate(date, date + ql.Period(1, ql.Years), day_count, ql.Annual).rate()
discount_rate = curve.discount(date)
print("Zero rate as at 28.05.2048: " + str(round(zero_rate*100, 4)) + str("%"))
print("Forward rate as at 28.05.2048: " + str(round(forward_rate*100, 4)) + str("%"))
print("Discount factor as at 28.05.2048: " + str(round(discount_rate, 4)))

# Print the Zero Rates, Forward Rates and Discount Factors at node dates
# print(pd.DataFrame(curve.nodes()))
node_data = {'Date': [],
             'Zero Rates': [],
             'Forward Rates': [],
             'Discount Factors': []}

for dt in curve.dates():
    node_data['Date'].append(dt)
    node_data['Zero Rates'].append(curve.zeroRate(dt, day_count, ql.Annual).rate())
    node_data['Forward Rates'].append(curve.forwardRate(dt, dt + ql.Period(1, ql.Years), day_count, ql.Annual).rate())
    node_data['Discount Factors'].append(curve.discount(dt))

node_dataframe = pd.DataFrame(node_data)

print(node_dataframe)

node_dataframe.to_excel('NodeRates.xlsx')

# Printing Daily Zero Rates:
# Prints the daily zero rates from the current date (May 31, 2023) to a maturity date that is 30
# years later. It calculates and prints the zero rates for each year using the constructed yield curve.
maturity_date = calendar.advance(today, ql.Period(1, ql.Years))
current_date = today
while current_date <= maturity_date:
    zero_rate = curve.zeroRate(current_date, day_count, ql.Annual).rate()
    print(f"Date: {current_date}, Zero Rate: {zero_rate}")
    current_date = calendar.advance(current_date, ql.Period(1, ql.Years))

# Creating Curve Data for Plotting:
# Creates lists of curve dates, zero rates, and forward rates for plotting.
# It calculates both zero rates and forward rates for each year up to 25 years from the current date.
                'Zero Rate': [],
                'Discount Factor': [],
                'Clean Price': [],
                'Dirty Price': []}

# Calculate bond prices and yields
for issue_date, maturity, coupon, price, settlement_days in data:
    price = ql.QuoteHandle(ql.SimpleQuote(price))
    today = ql.Date(21, ql.November, 2023)
    issue_date = ql.Date(issue_date, '%d-%m-%Y')
    maturity = ql.Date(maturity, '%d-%m-%Y')
    schedule = ql.Schedule(today, maturity, ql.Period(ql.Semiannual), calendar, ql.DateGeneration.Backward,
                           ql.Following, ql.DateGeneration.Backward, False)
    bondEngine = ql.DiscountingBondEngine(ql.YieldTermStructureHandle(curve))
    bond = ql.FixedRateBond(settlement_days, faceAmount, schedule, [coupon / 100], day_count)
    bond.setPricingEngine(bondEngine)

    # Calculate bond yield, clean price, and dirty price
    bondYield = bond.bondYield(day_count, ql.Compounded, ql.Annual)
    bondCleanPrice = bond.cleanPrice()
    bondDirtyPrice = bond.dirtyPrice()
    zero_rate = curve.zeroRate(maturity, day_count, ql.Annual).rate()
    discount_factor = curve.discount(maturity)

    # Append the results to the DataFrame
    bond_results['Issue Date'].append(issue_date)
    bond_results['Maturity Date'].append(maturity)
    bond_results['Coupon Rate'].append(coupon)
    bond_results['Price'].append(price.value())
    bond_results['Settlement Days'].append(settlement_days)
    bond_results['Yield'].append(bondYield)
    bond_results['Zero Rate'].append(zero_rate)
    bond_results['Discount Factor'].append(discount_factor)
    bond_results['Clean Price'].append(bondCleanPrice)
    bond_results['Dirty Price'].append(bondDirtyPrice)

# Create a DataFrame from the bond results
bond_results_df = pd.DataFrame(bond_results)

# Print the results
print(bond_results_df)

bond_results_df.to_excel('BondResults.xlsx')

我有以下问题或疑问； (i) 我获得的前 4 只零息债券的到期收益率与零（或即期）利率略有不同。我的预期是，前 4 只零息债券的到期收益率 (YTM) 和零利率应该相同。是什么导致了这些细微的差异？我该如何解决这个问题？ (ii) 在尝试通过简单地使用到期收益率（应类似于零利率）和应计期间贴现面值 100 来手动计算前 4 个零息债券的价格时，我注意到我需要调整结算日 (T+4)，本例中零息债券为 4 天。我对结算日的理解是，在这种情况下，债券在到期后 4 天结算，因此这将使应计期（或贴现期）增加 4 天。令人惊讶的是，为了使零息债券达到相同的价格，我注意到我将应计期减少了 4 天而不是增加。我是否遗漏了什么或者这不正确。

最终结果将在 BondResults.xlsx 工作簿中

Answer 1

4个结算日意味着，如果我今天从您那里购买债券，我不会立即收到，而是在4个工作日内收到。正如您所看到的，这会缩短折扣期；您需要的折扣不是从到期日到今天，而是从到期日到今天起 4 天。