利息论四.ppt

利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金引言引言债务偿还的问题是利息论中的一大类问题，债务偿还的问题是利息论中的一大类问题，是每个个人和家庭都需要面对的问题。是每个个人和家庭都需要面对的问题。债务的偿还主要有三种方式债务的偿还主要有三种方式:1.1.满期偿还满期偿还:到期一次性偿还本金和利息到期一次性偿还本金和利息2.2.分期偿还（分期偿还（amortization):amortization):按一定的时间间按一定的时间间隔隔,分期偿还本金和利息分期偿还本金和利息3.3.偿债基金（偿债基金（sinking fundsinking fund）:每期支付利息每期支付利息,并按期另存一笔钱并按期另存一笔钱,建立基金建立基金,使基金到期积使基金到期积累值恰好等于贷款本金。累值恰好等于贷款本金。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金4-14-1 未偿还贷款余额未偿还贷款余额 在分期偿还中有在分期偿还中有4 4个基本量个基本量L L：贷款本金（贷款金额）：贷款本金（贷款金额）n n：还款期限：还款期限i i：贷款利率：贷款利率R R：分期偿还方式中每次还款金额：分期偿还方式中每次还款金额利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金定义：定义：为为k k时刻的未偿还贷款余额时刻的未偿还贷款余额（outstanding balance just after kth outstanding balance just after kth payment).payment).计算计算k k时刻的未偿还贷款余额有两种方法，一时刻的未偿还贷款余额有两种方法，一种是将来法，一种是过去法。种是将来法，一种是过去法。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金将来法（将来法（by prospective method)by prospective method)：过去法（过去法（by retrospective method)by retrospective method)：利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金可以证明：可以证明：依据：依据：所有付款的现值所有付款的现值=贷款金额贷款金额 把两边都积累到把两边都积累到t t时刻时刻 所有付款在所有付款在t t时刻的当前值时刻的当前值=贷款金额在贷款金额在t t时刻的积累值时刻的积累值 过去付款在过去付款在t t时刻的积累值时刻的积累值+将来付款在将来付款在t t时刻的现值时刻的现值=贷款金额在贷款金额在t t时刻的积累值时刻的积累值 利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.1.14.1.1 A loan of 100000 is being repaid by 15 A loan of 100000 is being repaid by 15 equal annual installments at the end of each equal annual installments at the end of each year at 6%interest effective annually.year at 6%interest effective annually.Immediately after the 8th pmt,the loan Immediately after the 8th pmt,the loan renegotiated as follows:renegotiated as follows:(1):The borrower will made seven annual pmt of(1):The borrower will made seven annual pmt of k to repay the loan with the first pmt made k to repay the loan with the first pmt made three years from the date of renegotiation.three years from the date of renegotiation.(2):The interest rate is changed to 7.5%(2):The interest rate is changed to 7.5%effective annually.effective annually.Calculate k.Calculate k.利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金4-24-2分期偿还表分期偿还表 表中我们要计算如下量：表中我们要计算如下量：分别对应的是每期的还款额，每期支付中的利息，分别对应的是每期的还款额，每期支付中的利息，每期支付中的本金，以及每个时刻的未偿还余额。每期支付中的本金，以及每个时刻的未偿还余额。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金 模型：每期还款模型：每期还款1 1，每年利率为，每年利率为i i。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金 几个重要公式：几个重要公式：利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.2.14.2.1 L=380,n=3,i=0.1 L=380,n=3,i=0.1试列出分期偿还明细表。试列出分期偿还明细表。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.2.24.2.2 Mark takes out a 30-year loan on Mark takes out a 30-year loan on Jan.1,1992 for 20000 at an annual effective Jan.1,1992 for 20000 at an annual effective rate of 5%.Pmts are made at the end of each rate of 5%.Pmts are made at the end of each year.year.On Jan.1,2002,Mark takes out a 20-On Jan.1,2002,Mark takes out a 20-year loan for 10000 at an annual effective year loan for 10000 at an annual effective rate of 7%.Pmts are also made at the end of rate of 7%.Pmts are also made at the end of each year.each year.Calculate the principal of the year Calculate the principal of the year 2002 on both loans.2002 on both loans.利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金4-34-3偿还频率与计息频率不一致的分期偿还表偿还频率与计息频率不一致的分期偿还表 这里仅先考虑每期偿还额相等的情形。这里仅先考虑每期偿还额相等的情形。有两种模型，一种是一个支付期有有两种模型，一种是一个支付期有k k个计息期，个计息期，一种是一个计息期有一种是一个计息期有m m个支付期。个支付期。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金 模型一模型一：一个支付期有：一个支付期有k k个计息期，总共有个计息期，总共有n n个个计息期，每次还款额为计息期，每次还款额为1 1。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.3.14.3.1 A loan of L is to be repaid with 40 A loan of L is to be repaid with 40 pmts of 100 at the end of each quarter.pmts of 100 at the end of each quarter.Interest on the loan is charged at a nominal Interest on the loan is charged at a nominal rate i,convertible monthly.The outstanding rate i,convertible monthly.The outstanding balance immediately after the 8th and 24balance immediately after the 8th and 24thth pmts are 2308.15 and 1345.50.Calculate the pmts are 2308.15 and 1345.50.Calculate the amount of interest repaid in the 15th pmt.amount of interest repaid in the 15th pmt.利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金模型模型二二：一个计息期中有：一个计息期中有m m个支付期，共有个支付期，共有n n个计个计息期，每一个支付期上还款息期，每一个支付期上还款1/m1/m。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.3.24.3.2 A loan is being amortized by mean of A loan is being amortized by mean of level at an annual effective rate of 8%.The level at an annual effective rate of 8%.The amount of principal repaid in the 12th pmt amount of principal repaid in the 12th pmt is 1000 and the amount of principal repaid is 1000 and the amount of principal repaid in the tth pmt is 3700.Calculate t.in the tth pmt is 3700.Calculate t.利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金作业：作业：2 2，7 7，9,119,11利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金4-44-4变额偿还支付变额偿还支付贷款总额贷款总额L L，期限，期限n n，利率，利率i i，每次偿还额为，每次偿还额为R Rk k，建立分，建立分期偿还表。期偿还表。THANK YOUSUCCESS2024/4/18 周四19可编辑利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.4.14.4.1 A loan of 100000 is repaid with A loan of 100000 is repaid with unequal annual pmts at the end of each year unequal annual pmts at the end of each year for 30 years.Each of the first 29 pmts is for 30 years.Each of the first 29 pmts is equal to two times the amount of the equal to two times the amount of the interest then.The final pmt repays the interest then.The final pmt repays the remaining loan balance at that time.remaining loan balance at that time.Interest is charged at an annual effective Interest is charged at an annual effective rate of 8%.Calculate the amount of the rate of 8%.Calculate the amount of the final loan pmt.final loan pmt.利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.4.24.4.2 Don takes out a 10-year loan of L,Don takes out a 10-year loan of L,which he repays with annual pmts at the end which he repays with annual pmts at the end of each year using amortization method.of each year using amortization method.Interest on the loan is charged at an annual Interest on the loan is charged at an annual effective rate i.Don repays the loan with a effective rate i.Don repays the loan with a decreasing series of pmts.He repays 1000 in decreasing series of pmts.He repays 1000 in the year one,900 in the year two,the year one,900 in the year two,and 100 and 100 in the year ten.The amount of principal in the year ten.The amount of principal repaid in year three is equal to 600.repaid in year three is equal to 600.Calculate L.Calculate L.利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.4.34.4.3 等额本金偿还方式，此种偿还方式中，每等额本金偿还方式，此种偿还方式中，每次偿还的本金相同。见书上例次偿还的本金相同。见书上例4-124-12。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金4-54-5 连续偿还的分期偿还表连续偿还的分期偿还表 期限期限n n，每个时刻上的支付率为，每个时刻上的支付率为1 1。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金 推广：若在推广：若在t t时刻的支付率为时刻的支付率为R Rt t利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.5.14.5.1 书上例书上例4-154-15。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金4-64-6 偿债基金（偿债基金（sinking fund method)sinking fund method)偿债基金是另一种重要的债务偿还方式。偿债基金是另一种重要的债务偿还方式。偿债基金方式是要求借款者定期地支付当期偿债基金方式是要求借款者定期地支付当期的利息，还必须定期地向偿债基金付款，以保证的利息，还必须定期地向偿债基金付款，以保证偿债基金能够在贷款期限末积累到贷款本金。偿债基金能够在贷款期限末积累到贷款本金。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金L:L:贷款总额贷款总额n n：贷款期限：贷款期限i i：贷款利率：贷款利率j j：偿债基金利率：偿债基金利率D D：偿债基金每期存款（：偿债基金每期存款（periodic sinking fund deposit)periodic sinking fund deposit)P P：借款人每期支付：借款人每期支付(periodic outlay by borrower)(periodic outlay by borrower)利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金若若i=j,i=j,偿债基金表偿债基金表4-54-5利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金更进一步地，我们先分析更进一步地，我们先分析P P1 1，关于，关于P P利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.6.14.6.1贷款额为贷款额为10000,10000,贷款利率是贷款利率是8%,8%,贷款期限是贷款期限是1010年年,比较下列比较下列4 4种贷款方式种贷款方式:(1)(1)分期偿还法分期偿还法(2)(2)偿债基金利率为偿债基金利率为6%6%(3)(3)偿债基金利率为偿债基金利率为8%8%(4)(4)偿债基金利率为偿债基金利率为10%10%利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金2 2，关于，关于NBNBt t，NPNPt t，NINIt t，它们分别表示，它们分别表示t t时刻的净余额，净本金以时刻的净余额，净本金以及净利息。及净利息。利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.6.2 4.6.2 贷款额为贷款额为380,380,贷款利率是贷款利率是10%,10%,偿债基金利偿债基金利率是率是7%,7%,贷款期限是贷款期限是3 3年年,列出偿债基金法的明细表列出偿债基金法的明细表.利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.6.3 4.6.3 Ran takes out a 10-year loan for 4600.Ran takes out a 10-year loan for 4600.Interest on the loan is charged at 9%annual Interest on the loan is charged at 9%annual effective rate.Ran repay the loan by making effective rate.Ran repay the loan by making 10 annual pmts at the end of each year 10 annual pmts at the end of each year consisting of the interest on the loan and a consisting of the interest on the loan and a sinking fund deposit.The total pmt each year sinking fund deposit.The total pmt each year is 743.The amount is the sinking fund will is 743.The amount is the sinking fund will exactly pay off the loan at the end of the 10 exactly pay off the loan at the end of the 10 years.During the first 9 years of the loan,years.During the first 9 years of the loan,the sinking fund annual effective rate is the sinking fund annual effective rate is 7.75%,after year 9,the sinking fund annual 7.75%,after year 9,the sinking fund annual effective rate is k.Calculate k.(5%)effective rate is k.Calculate k.(5%)利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金例例4.6.4 4.6.4 Jason and Mark each take out a 17-year Jason and Mark each take out a 17-year loan of L.Jason repays his loan using the loan of L.Jason repays his loan using the amortization method at an annual effective amortization method at an annual effective rate of i.He makes an annual pmt of 500 at rate of i.He makes an annual pmt of 500 at the end of each year.the end of each year.Mark repays the loan using the sinking Mark repays the loan using the sinking fund method.He repays interest annually also fund method.He repays interest annually also at an annual effective rate i.In addition,at an annual effective rate i.In addition,he makes level annual deposit at the end of he makes level annual deposit at the end of each year for 17 years into a sinking fund.each year for 17 years into a sinking fund.The annual effective rate of the sinking fund The annual effective rate of the sinking fund is 4.62%and she pays off the loan after 17 is 4.62%and she pays off the loan after 17 years.Markyears.Marks total pmt each year is equal s total pmt each year is equal to 10%of the original loan amount.Calculate to 10%of the original loan amount.Calculate L.(i=8.4665%,L=4422)L.(i=8.4665%,L=4422)3.偿还频率与计息频率不一致的偿债基金法偿还频率与计息频率不一致的偿债基金法四种频率四种频率(1)(1)贷款利息支付频率贷款利息支付频率(2)(2)贷款计息频率贷款计息频率(3)(3)偿债基金存款频率偿债基金存款频率(4)(4)偿债基金计息频率偿债基金计息频率例例4-18,4-194-18,4-19利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金4.变动偿还系列各期利息支付不变,偿债基金每次存款改变.设借款人各次还款为R1,R2,Rn.贷款L,则第k次存入偿债基金的金额为Rk-iL,故有例4-20,4-21利息论讲义利息论讲义第四章第四章 分期偿还与偿债基金分期偿还与偿债基金作业：作业：13,15,17,23,28,30,33,3613,15,17,23,28,30,33,36THANK YOUSUCCESS2024/4/18 周四38可编辑