利息理论

Rsk = 1 =⇒ R = Ran =⇒ an s
k
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
1 sk
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.2 Differing payment and interest conversion period
Approach 2 develop algebraic expressions Attention Please: the algebraic developments are different between: the annuities payable less frequently than interest is convertible (Section 4.3) the annuities payable more frequently than interest is convertible (Section 4.4)
Annuity-due
1 1 1 ... R R] [R R ... R R] [R R ... ... 0 k 2k 1 ··· ··· [R R ... R R] ... n-k n
SC百度文库OOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
present value: 1 + v k + v 2k + ... + v n−k = accumulated value: sn an (1 + i)n = ak ak
present value: v k + v 2k + ... + v k ·k = accumulated value: sn an (1 + i)n = sk sk
Return SCHOOL OF INSURANCE AND ECONOMICS–Li Yan THE THEORY OF INTEREST
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.2 Differing payment and interest conversion period
Approach 1 Step 1: find the rate of interest, convertible at the same frequency as payments are made Step 2: using the techniques discussed in Chapter 3.
Example (4.3) At what annual effective rate of interest will payments of $100 at the end of every quarter accumulate to $2500 at the end of five years?
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.2 Differing payment and interest conversion period
Annuities for which the payment period and the interest conversion period differ and the payments are level.
Compare SCHOOL OF INSURANCE AND ECONOMICS–Li Yan THE THEORY OF INTEREST
an ak
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
n
an sk
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
Alternate argument:
1 1 [R R ... R R] [R R ... R R] ... ... 0 k 2k ··· ··· 1 [R R ... R R] ... n-k n 1
Example (4.2) A loan of $3000 is to be repaid with quarterly installment at the end of each quarter for five years. If the rate of interest charged on the loan is 10% convertible semiannually, find the amount of each quarterly payment.
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
Other considerations present value of a perpetuity-immediate v +v
k 2k
vk 1 a∞ + ... = = = 1 − vk isk sk 1 a∞ = iak ak
present value of a perpetuity-due 1 + v k + v 2k + ... =
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan THE THEORY OF INTEREST
Return to Example 4.4
Chapter 4
More general annuities
§4.2 Differing payment and interest conversion period
Alternate argument:
1 1 [R R ... R R] [R R ... R R] ... ... 0 k 2k ··· ··· 1 [R R ... R R] ... n-k n 1
Rsk = 1 =⇒ R =
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
Annuity-immediate
1 ... ... 0 k ... ... 2k 1 ··· ··· n-k 1 ... ... n 1
k: the number of interest conversion periods in one period n: the term of the annuity measured in interest conversion periods
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
annuity-immediate annuity-due other considerations
Return to Example 4.6
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.2 Differing payment and interest conversion period
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
At a given force of interest δ: (1 + i) = e δ , v = e −δ , ∴ e −δnk = v nk , e δnk = (1 + i)nk Then present value = e −kδ + e −2kδ + ... + e −nkδ
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan THE THEORY OF INTEREST
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
1 sk
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
Alternate argument:
1 1 [R R ... R R] [R R ... R R] ... ... 0 k 2k ··· ··· 1 [R R ... R R] ... n-k n 1
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan
THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.3 Annuities payable less frequently than interest is convertible
SCHOOL OF INSURANCE AND ECONOMICS–Li Yan THE THEORY OF INTEREST
Chapter 4
More general annuities
§4.2 Differing payment and interest conversion period
Example (4.1) Find the accumulated value at the end of four years of an investment fund in which $100 is deposited at the beginning of each quarter for the first two years and $200 is deposited at the beginning of each quarter for the second two years, if the fund earns 12% convertible monthly.