Kellison THE-THEORY-OF-INTEREST wersja uproszczona, Matematyka finansowa

[ Pobierz całość w formacie PDF ]
THE
THEORY
OF
INTEREST
Second Edition
STEPHEN G. KELLISON
Study Notes Prepared by
Kevin Shand, FSA, FCIA
Assistant Professor
Warren Centre for Actuarial
Studies and Research
Contents
1 The Measurement of Interest 2
1.1 Introduction........................................ 2
1.2 TheAccumulationFunctionandtheAmountFunction................ 2
1.3 The Effective Rate of Interest:
i
............................ 2
1.4 SimpleInterest ...................................... 3
1.5 CompoundInterest.................................... 3
1.6 PresentValue....................................... 4
1.7 The Effective Rate of Discount:
d
............................ 5
1.8 Nominal Rate of Interest and Discount Convertible
m
th
ly:
i
(
m
)
,d
(
m
)
........ 7
1.9 Forces of Interest and Discount:
δ
n
,δ
n
......................... 9
1.10VaryingInterest ..................................... 13
1.11SummaryofResults ................................... 13
2 Solution of Problems in Interest 14
2.1 Introduction........................................ 14
2.2 ObtainingNumericalResults .............................. 14
2.3 DeterminingTimePeriods................................ 14
2.4 TheBasicProblem.................................... 15
2.5 EquationsofValue.................................... 15
2.6 UnknownTime...................................... 18
2.7 UnknownRateofInterest ................................ 21
2.8 PracticalExamples.................................... 23
3 Basic Annuities 24
3.1 Introduction........................................ 24
3.2 Annuity-Immediate.................................... 24
3.3 Annuity–Due ....................................... 28
3.4 AnnuityValuesOnAnyDate.............................. 33
3.5 Perpetuities........................................ 39
3.6 NonstandardTermsandInterestRates......................... 42
3.7 UnknownTime...................................... 42
3.8 UnknownRateofInterest ................................ 43
3.9 VaryingInterest ..................................... 47
3.10AnnuitiesNotInvolvingCompoundInterest...................... 50
4 More General Annuities 51
4.1 Introduction........................................ 51
4.2 Annuities Payable At A Different Frequency Than Interest Is Convertible . . . . . 51
4.3 Further Analysis of Annuities Payable Less Frequency Than Interest Is Convertible 52
4.4 Further Analysis of Annuities Payable More Frequency Than Interest Is Convertible 62
4.5 ContinuousAnnuities .................................. 76
4.6 BasicVaryingAnnuities ................................. 80
4.7 MoreGeneralVaryingAnnuities ............................ 93
4.8 ContinuousVaryingAnnuities.............................. 99
4.9 SummaryOfResults................................... 100
1
5 Yield Rates 101
5.1 Introduction........................................ 101
5.2 DiscountedCashFlowAnalysis............................. 101
5.3 UniquenessOfTheYieldRate ............................. 103
5.4 ReinvestmentRates ................................... 104
5.5 InterestMeasurementOfAFund............................ 106
5.6 Time-WeightedRatesOfInterest............................ 109
5.7 PortfolioMethodsandInvestmentYearMethods ................... 112
5.8 CapitalBudgeting .................................... 112
5.9 MoreGeneralBorrowing/LendingModels ....................... 112
6 Amortization Schedules and Sinking Funds 113
6.1 Introduction........................................ 113
6.2 FindingTheOutstandingLoan............................. 113
6.3 AmortizationSchedules ................................. 114
6.4 Sinking Funds . ...................................... 116
6.5 DifferingPaymentPeriodsandInterestConversionPeriods ............. 119
6.6 VaryingSeriesofPayments ............................... 125
6.7 AmortizationWithContinuousPayments ....................... 130
6.8 Step-RateAmountsOfPrincipal ............................ 130
7 Bonds and Other Securities 131
7.1 Introduction........................................ 131
7.2 TypesOfSecurities.................................... 131
7.3 PriceofABond ..................................... 132
7.4 PremiumAndDiscountPricingOfABond ...................... 135
7.5 ValuationBetweenCouponPaymentDates ...................... 140
7.6 DeterminationOfYieldRates.............................. 141
7.7 CallableBonds ...................................... 144
7.8 SerialBonds........................................ 145
7.9 SomeGeneralizations................................... 145
7.10OtherSecurities...................................... 145
7.11ValuationOfSecurities.................................. 145
8 Practical Applications 146
8.1 Introduction........................................ 146
8.2 TruthInLending..................................... 146
8.3 Real Estate Mortgages . . . ............................... 146
8.4 ApproximateMethods .................................. 146
8.5 DepreciationMethods .................................. 146
8.6 CapitalizedCost ..................................... 151
8.7 ShortSales ........................................ 152
8.8 ModernFinancialInstruments ............................. 153
9 More Advanced Financial Analysis 156
9.1 Introduction........................................ 156
9.2 AnEconomicRationaleforInterest........................... 156
9.3 DeterminantsoftheLevelofInterestRates ...................... 156
9.4 RecognitionofInflation ................................. 156
9.5 ReflectingRiskandUncertainty............................. 156
2
9.6 YieldCurves ....................................... 157
9.7 InterestRateAssumptions................................ 158
9.8 Duration.......................................... 158
9.9 Immunization....................................... 161
9.10 Matching Assets and Liabilities . . . . . ........................ 164
3
1 The Measurement of Interest
1.1 Introduction
Interest
– compensation a borrower of capital pays to a lender of capital
– lender has to be compensated since they have temporarily lost use of their capital
– interest and capital are almost always expressed in terms of money
1.2 The Accumulation Function and the Amount Function
The Financial Transaction
– an amount of money or capital (Principal) is invested for a period of time
– at the end of the investment period, a total amount (Accumulated Value) is returned
– difference between the Accumulated Value and the Principal is the Interest Earned
Accumulation Function:
a
(
t
)
–let
t
be the number of investment years (
t
≥
0), where
a
(0) = 1
– assume that
a
(
t
) is continuously increasing
–
a
(
t
) defines the pattern of accumulation for an investment of amount 1
a
(
t
)
–let
k
be the initial principal invested (
k>
0) where
A
(0) =
k
·
–
A
(
t
) is continuously increasing
–
A
(
t
) defines the Accumulated Value that amount
k
grows to in
t
years
1)
– interest earned is the difference between the Accumulated Value at the end of a period and
the Accumulated Value at the beginning of the period
−
A
(
n
−
1.3 The Effective Rate of Interest:
i
Definition
–
i
is the amount of interest earned over a one-year period when 1 is invested
–let
i
n
be the effective rate of interest earned during the
n
th period of the investment where
interest is paid at the end of the period
–
i
is also defined as the ratio of the amount of Interest Earned during the period to the
Accumulated Value at the beginning of the period
i
n
=
A
(
n
)
−
A
(
n
−
1)
=
I
n
A
(
n
1)
,
for integral
n
≥
1
A
(
n
−
1)
−
2
Amount Function:
A
(
t
)=
k
Interest Earned during the
n
th period:
I
n
=
A
(
n
)
[ Pobierz całość w formacie PDF ]
  • zanotowane.pl
  • doc.pisz.pl
  • pdf.pisz.pl
  • klobuckfatima.xlx.pl