Question

Financial Mathematics: Nonlevel annuities using geometric

Let i = 7% and v = 1+ i In each part, show annuity payments on a timeline to describe the given sums as the value of that annuity at a certain point in time. In other words, visualize the sums as present/future value of certain annuities. Then compute them using the geometric summation formula 1 - pn+1 s pk = 1+r+p2 +p3 + ... + p = 1-p k=0 Hint: Factoring out the first term of a geometric sum helps to rewrite it in the standard form above. (a) vº + v10 + v11 + ... +223 = (b) vº + v10 + v11 + ...= (c) 19 +v11 + v13 + ... + 123 = (d) (1 + i)23 + (1+i)22 + (1+i)21 +...+(1+i)! = (e) (1 - i)(1+i)23 + (1 - i)?(1 + i)22 + (1 - i)(1+i)21 + ... + (1 - i)23(1+i) =

Answer 1

Answer

i=7%

v=1/(1+i)=1/(1+7%)=0.9345

a)

v^9+v^10+v^11+...+ v^23

v^9*(1+v+v^2 ...+ V^14)=v^9*(1-v^15)/(1-v)=0.9345^9*(1-0.9345^15)/(1-0.9345)=5.30

b)

v^9+v^10+v^11+...

v^9*(1+v+v^2 ...)=v^9/(1-v)=0.9345^9/(1-0.9345)=8.314

C)

v^9+v^11+v^13+...v^23

v^9*(1+v^2+v^4+...v^14) =v^9*(1+v^2+(v^2)^2+........(v^2)^7)

=v^9*(1-(v^2)^(7+1))/(1-v^2) = 0.9345^9*(1-0.9345^16)/(1-0.9345^2)=2.84198

D)

(1+i)^23+(1+i)^22+(1+i)^21+....+(1+i)^9

let (1+i)=a

a=1+7%=1.07

So

a^23+a^22+a^21+...a^9 =a^9*(1+a+a^2+.....a^14) =a^9*(1-a^15)/(1-a) =1.07^9*(1-1.07^15)/(1-1.07)=46.1986

e)

(1-i)(1+i)^23+(1-i)^2*(1+i)^22+(1+i)^21+....+(1-i)^23*(1+i)

let (1-i)/(1+i) =a

a=(1-7%)/(1+7%)=0.8691

(1+i)^24*{((1-i)/(1+i)) +((1-i)/(1+i))^2+((1-i)/(1+i))^3+((1-i)/(1+i))^4+..........+((1-i)/(1+i))^23}

(1+i)^24*{a+a^2+a^3+..........................a^23)=(1+i)^24*a*(1-a^23)/(1-a)=(1+7%)^24*0.8691*(1-0.8691^23)/(1-0.8691)=32.355