A continuous payment of annuity, 100 yuan in the first year, 120 yuan in the second year, 140 yuan in the third year, and so on until 380 yuan in the 15th year. Assuming the actual annual interest rate is 5%, what is the present value of the annuity
\ (80 \ times \ frac {1- \ left (\ frac {1} {1 + 0.05} \ right) ^ {15}} {\ log (1 +0.05)} + 20 \ times \ frac {\ frac {1- \ frac {1} {(1 + 0.05) ^ {15}}} {0.05 \ times \ frac {1} {1 + 0.05}}-15 \ left (\ frac {1} {1 + 0.05} \ right) ^ {15}} {\ log (1 + 0.05)} \)
#wolfram
80*((1-(1/(1+0.05))^15)/ln(1+0.05))+20*(((1-(1/(1+0.05)^15))/(0.05*(1/(1+0.05)))-(15*(1/(1+0.05))^15))/ln(1+0.05)
(1-(1/(1+0.05)^15))/(0.05*(1/(1+0.05)))
A continuous payment of annuity, 200 yuan in the first year, and 15 yuan less than the previous year every year, until the final payment of 50 yuan. Assuming an annual effective interest rate of 8%, please calculate the final value of the annuity at the end of the 15th year.
\ (\ left (11 \ times \ frac {(1 + 0.08) ^ {11} -1} {\ log (1 + 0.08)} + 15 \ times \ right. \ left. \ frac {11 (1 + 0.08 ) ^ {11}-\ frac {(1 + 0.08) ^ {11} -1} {0.08}} {\ log (1 + 0.08)} \ right) (1 + 0.08) ^ {4} \)
#wolfram
(11*((1+0.08)^11-1)/ln(1+0.08)+15*(11*(1+0.08)^11-((1+0.08)^11-1)/0.08)/ln(1+0.08))*(1+0.08)^4
The payment rate of a 10-year continuous annuity at time t is \ (ρ (t) = 2t + 1 \) , assuming that the interest force is \ (δ (t) = 0.3 + 0.2t \) . Please calculate the present value of the annuity at time 0
(\ left. \ Int_ {0} ^ {10} (2 t + 1) \ exp [-\ int_ {0} ^ {t} (0.3 + 0.2s) ds \ right] dt \)
A cash flow is continuously paid from time 5 to time 10, and the payment rate at time t is \ (ρ (t) = t ^ 2 + 2t \) The interest force from time 0 to time 8 is \ (δ (t) = 0.002t + 0.01 \) , the interest force from time 8 to time 10 is \ (δ (t) = 0.0006t ^ 2 + 0.001t \) . Please calculate the final value of the cash flow at the end of the 10th year.
\ (a (5) = \ exp \ left (\ int_ {0} ^ {5} 0.002 t + 0.01 dt \ right) \)
\ (\ int_ {5} ^ {8} \ left (t ^ {2} +2 t \ right) \ exp \ left (0.144-0.001 t ^ {2} -0.01 t \ right) dt = 173.387 \)
int_5^8 {(t^2+2t)*exp(0.144-0.001*t^2-0.01*t)}dt
\(\int_{8}^{10}\left(t^{2}+2 t\right) \exp \left(0.205-0.0005 t^{2}-0.0002 t^{3}\right)d t=201.349\)
(int_8^10 {(t^2+2t)*exp(0.205-0.0005*t^2-0.0002*t^3)}dt
173.387*201.349*1.07788415=37630.33591591755145
(Equal series annuity) Set an annuity of n periods, the annual effective interest rate is \ (i \) , pay at time \ (k \) \ (x_k = x_1 + (k-1) ∆ \) , where, \ ( 1 ⩽ k ⩽ n \) .
(1) Please give the expression of the present value of the annuity.
(2) According to the formula given in (1), calculate the present value of the annuity at time 0 and the time at 12 when \ (n = 11, x_1 = 350, ∆ = 50, i = 5 \% \) end value.
= \ (v x_ {1} + \ left (x_ {1} + \ Delta \ right) v ^ {2} + \ left (x_ {1} +2 \ Delta \ right) v ^ {3} + \ dots . + v ^ {n} \ left (x_ {1} + (n-1) \ Delta \ right) \)
\ (= x \ cdot \ frac {v \ left (1-v ^ {n} \ right) } {1-v} + \ frac {\ Delta} {1-v} \ left (\ frac {v \ left (1-v ^ {n} \ right)} {1-v} -v- (n- 1) v ^ {n + 1} \ right) \)
A 10-year financial product, the product meets the following conditions:
(1) Each year can get 10,000 yuan, these payments are calculated at the annual effective interest rate \ (5 \% \) ;
(2) At the end of each year The interest is calculated at the annual effective interest rate of \ (4 \% \) .
If the annual rate of return of the financial product is \ (5 \% \) , please calculate the current selling price of the financial product .
((((1-(1/(1+0.04))^10)/(0.04*(1/(1+0.04)))*(1+0.04)^10-10)/0.04*500+10^5))/(1+0.05)^10
#80471.38
(\(\frac{\frac{1-\left(\frac{1}{1+0.04}\right)^{10}}{0.04 \times \frac{1}{1+0.04}}(1+0.04)^{10}-10}{0.04} \times 500+10^{5}\)）/(1+0.05)^10