continuously paying annuities · the set-up: unit increase in payments assume that we have compound...
TRANSCRIPT
![Page 1: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/1.jpg)
Continuously paying annuities
1 Compound interest
2 Compound interest: Increasing payments
3 General Accumulation Function
![Page 2: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/2.jpg)
Continuously paying annuities
1 Compound interest
2 Compound interest: Increasing payments
3 General Accumulation Function
![Page 3: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/3.jpg)
The Set-up: Level annuity
Assume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:
• the annuity lasts for n interest periods;
• the payments take place continuously, at a rate of 1 per interestperiod.
• an i . . . stands for the present value of the above annuity, i.e.,
an i =1− e−δn
δ
• sn i . . . stands for the accumulated value of the above annuity, i.e.,
sn i =eδn − 1
δ
![Page 4: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/4.jpg)
The Set-up: Level annuity
Assume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:
• the annuity lasts for n interest periods;
• the payments take place continuously, at a rate of 1 per interestperiod.
• an i . . . stands for the present value of the above annuity, i.e.,
an i =1− e−δn
δ
• sn i . . . stands for the accumulated value of the above annuity, i.e.,
sn i =eδn − 1
δ
![Page 5: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/5.jpg)
The Set-up: Level annuity
Assume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:
• the annuity lasts for n interest periods;
• the payments take place continuously, at a rate of 1 per interestperiod.
• an i . . . stands for the present value of the above annuity, i.e.,
an i =1− e−δn
δ
• sn i . . . stands for the accumulated value of the above annuity, i.e.,
sn i =eδn − 1
δ
![Page 6: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/6.jpg)
The Set-up: Level annuity
Assume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:
• the annuity lasts for n interest periods;
• the payments take place continuously, at a rate of 1 per interestperiod.
• an i . . . stands for the present value of the above annuity, i.e.,
an i =1− e−δn
δ
• sn i . . . stands for the accumulated value of the above annuity, i.e.,
sn i =eδn − 1
δ
![Page 7: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/7.jpg)
An Example• Find the constant force of interest δ at which
s20 = 3 · s10
⇒ Using the formula above, we get that
e20·δ − 1
δ= 3 · e
10·δ − 1
δ
So,
e20δ − 3 · e10δ + 2 = 0
Hence,
(e10δ − 1)(e10δ − 2) = 0
We reject the root δ = 0 and obtain the unique solution
δ =ln(2)
10= 0.0693
![Page 8: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/8.jpg)
An Example• Find the constant force of interest δ at which
s20 = 3 · s10
⇒ Using the formula above, we get that
e20·δ − 1
δ= 3 · e
10·δ − 1
δ
So,
e20δ − 3 · e10δ + 2 = 0
Hence,
(e10δ − 1)(e10δ − 2) = 0
We reject the root δ = 0 and obtain the unique solution
δ =ln(2)
10= 0.0693
![Page 9: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/9.jpg)
An Example• Find the constant force of interest δ at which
s20 = 3 · s10
⇒ Using the formula above, we get that
e20·δ − 1
δ= 3 · e
10·δ − 1
δ
So,
e20δ − 3 · e10δ + 2 = 0
Hence,
(e10δ − 1)(e10δ − 2) = 0
We reject the root δ = 0 and obtain the unique solution
δ =ln(2)
10= 0.0693
![Page 10: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/10.jpg)
An Example• Find the constant force of interest δ at which
s20 = 3 · s10
⇒ Using the formula above, we get that
e20·δ − 1
δ= 3 · e
10·δ − 1
δ
So,
e20δ − 3 · e10δ + 2 = 0
Hence,
(e10δ − 1)(e10δ − 2) = 0
We reject the root δ = 0 and obtain the unique solution
δ =ln(2)
10= 0.0693
![Page 11: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/11.jpg)
An Example• Find the constant force of interest δ at which
s20 = 3 · s10
⇒ Using the formula above, we get that
e20·δ − 1
δ= 3 · e
10·δ − 1
δ
So,
e20δ − 3 · e10δ + 2 = 0
Hence,
(e10δ − 1)(e10δ − 2) = 0
We reject the root δ = 0 and obtain the unique solution
δ =ln(2)
10= 0.0693
![Page 12: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/12.jpg)
Continuously paying annuities
1 Compound interest
2 Compound interest: Increasing payments
3 General Accumulation Function
![Page 13: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/13.jpg)
The Set-up: Unit increase in paymentsAssume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:• the annuity lasts for n interest periods;• the payments take place continuously, at a rate of t per interest
period at time t.• (I a)n i . . . stands for the present value of the above annuity, i.e.,
(I a)n i =an i − nvn
δ
• It is easy to see what happens by noting that
(I a)n i =
∫ n
0
t · v t dt
• (I s)n i . . . stands for the accumulated value of the above annuity,i.e.,
(I s)n i =sn i − n
δ
![Page 14: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/14.jpg)
The Set-up: Unit increase in paymentsAssume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:• the annuity lasts for n interest periods;• the payments take place continuously, at a rate of t per interest
period at time t.• (I a)n i . . . stands for the present value of the above annuity, i.e.,
(I a)n i =an i − nvn
δ
• It is easy to see what happens by noting that
(I a)n i =
∫ n
0
t · v t dt
• (I s)n i . . . stands for the accumulated value of the above annuity,i.e.,
(I s)n i =sn i − n
δ
![Page 15: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/15.jpg)
The Set-up: Unit increase in paymentsAssume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:• the annuity lasts for n interest periods;• the payments take place continuously, at a rate of t per interest
period at time t.• (I a)n i . . . stands for the present value of the above annuity, i.e.,
(I a)n i =an i − nvn
δ
• It is easy to see what happens by noting that
(I a)n i =
∫ n
0
t · v t dt
• (I s)n i . . . stands for the accumulated value of the above annuity,i.e.,
(I s)n i =sn i − n
δ
![Page 16: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/16.jpg)
The Set-up: Unit increase in paymentsAssume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:• the annuity lasts for n interest periods;• the payments take place continuously, at a rate of t per interest
period at time t.• (I a)n i . . . stands for the present value of the above annuity, i.e.,
(I a)n i =an i − nvn
δ
• It is easy to see what happens by noting that
(I a)n i =
∫ n
0
t · v t dt
• (I s)n i . . . stands for the accumulated value of the above annuity,i.e.,
(I s)n i =sn i − n
δ
![Page 17: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/17.jpg)
The Set-up: Unit increase in paymentsAssume that we have compound interest with the effective interest rateper interest period equal to i .Consider the following continuous annuity:• the annuity lasts for n interest periods;• the payments take place continuously, at a rate of t per interest
period at time t.• (I a)n i . . . stands for the present value of the above annuity, i.e.,
(I a)n i =an i − nvn
δ
• It is easy to see what happens by noting that
(I a)n i =
∫ n
0
t · v t dt
• (I s)n i . . . stands for the accumulated value of the above annuity,i.e.,
(I s)n i =sn i − n
δ
![Page 18: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/18.jpg)
Continuously paying annuities
1 Compound interest
2 Compound interest: Increasing payments
3 General Accumulation Function
![Page 19: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/19.jpg)
Unit payment stream
• Let v(t) denote the general discount function
• Let us first consider the basic continuous annuity, i.e., the annuitythat pays at the unit rate at all times.
• Then, the present value of such an annuity with length n equals∫ n
0
v(t) dt
• We still denote the above present value by an• In the special case of compound interest, the above formula collapses
to the one already familiar to us from the compound interest set-upYou can verify this through simple integration . . .
![Page 20: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/20.jpg)
Unit payment stream
• Let v(t) denote the general discount function
• Let us first consider the basic continuous annuity, i.e., the annuitythat pays at the unit rate at all times.
• Then, the present value of such an annuity with length n equals∫ n
0
v(t) dt
• We still denote the above present value by an• In the special case of compound interest, the above formula collapses
to the one already familiar to us from the compound interest set-upYou can verify this through simple integration . . .
![Page 21: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/21.jpg)
Unit payment stream
• Let v(t) denote the general discount function
• Let us first consider the basic continuous annuity, i.e., the annuitythat pays at the unit rate at all times.
• Then, the present value of such an annuity with length n equals∫ n
0
v(t) dt
• We still denote the above present value by an• In the special case of compound interest, the above formula collapses
to the one already familiar to us from the compound interest set-upYou can verify this through simple integration . . .
![Page 22: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/22.jpg)
Unit payment stream
• Let v(t) denote the general discount function
• Let us first consider the basic continuous annuity, i.e., the annuitythat pays at the unit rate at all times.
• Then, the present value of such an annuity with length n equals∫ n
0
v(t) dt
• We still denote the above present value by an• In the special case of compound interest, the above formula collapses
to the one already familiar to us from the compound interest set-upYou can verify this through simple integration . . .
![Page 23: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/23.jpg)
Unit payment stream
• Let v(t) denote the general discount function
• Let us first consider the basic continuous annuity, i.e., the annuitythat pays at the unit rate at all times.
• Then, the present value of such an annuity with length n equals∫ n
0
v(t) dt
• We still denote the above present value by an• In the special case of compound interest, the above formula collapses
to the one already familiar to us from the compound interest set-upYou can verify this through simple integration . . .
![Page 24: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/24.jpg)
Any payment stream
• Let f (t) be a continuous function which represents the rate ofpayments of a continuous annuity on the time interval [0, n]
• Then, the present value of this annuity can be obtained as∫ n
0
f (t) · v(t) dt
![Page 25: Continuously paying annuities · The Set-up: Unit increase in payments Assume that we have compound interest with the e ective interest rate per interest period equal to i. Consider](https://reader033.vdocuments.mx/reader033/viewer/2022050113/5f4a8ee833d6fa7ada392d62/html5/thumbnails/25.jpg)
Any payment stream
• Let f (t) be a continuous function which represents the rate ofpayments of a continuous annuity on the time interval [0, n]
• Then, the present value of this annuity can be obtained as∫ n
0
f (t) · v(t) dt