(Derivation of perpetuity formula) Using the geometric series formula: if −1 < r < 1, then
a+ar+ar2 +ar3 +···= a ,1−r
derive the PV of perpetuity formula.
annual payment = P
discount rate = r
PV = P/(1 + r) + P/(1 + r)2.........
(first term (a) = P/(1 + r) and common ratio (x) = 1/(1 + r))
(sum of infinite GP = a/(1 - x))
PV = P/(1 + r) x (1/(1 - 1/(1 + r))
PV = P/r
(Derivation of perpetuity formula) Using the geometric series formula: if −1 < r < 1, then...
Find the sum of the finite geometric series by using the formula for Sn: 1 1 1 1 1 1 1 1 + + + 3 9 27 81 243 729 2187 The sum of the finite geometric series is (Simplify your answer. Type a fraction.)
Find the sum of the finite geometric series using the formula for Sn Σ 2(105/-1 i- 1 The sum of the finite geometric series is Sn (Round to four decimal places.)
Derive the formula for pricing a perpetuity: P = C / r. 2. Derive the formula for pricing a coupon bond that pays a coupon at the end of each period. What will the formula be if the coupons are paid at the beginning of each period (i.e. you receive a coupon immediately after you acquired the bond and only receive the face value when it matures).
Q4 Derive the formula for the PV of an n- annual payment annuity with growth at rate g and a discount rate r using a geometric series approach. The first payment after 1 year is C and the payment after 2 years is C(1+g) etc. Show that: PV = C+ (1 - (*)") What happens when g=r and what are the implications of this if you are buying an annuity where growth matches inflation?
Prove the well-known formula for the sum of a geometric series. First show by cross-multiplying that 1 + r + r2 + · · · + r^n = 1−r^(n+1)/1-r . Then assume that |r| < 1 and find the limit as n → ∞.
Problem 1 Geometric Series. We will need to sum the geometric series to simplify some of the partition functions developed in class. Prove that the geometric series 7:0 for r| < 1. You may find it helpful to consider the partial sums Sj ?, xk 1+1+-.+4 and rSi =x+x2 + +z?+1 take the limit J ? 00, Can you see why the geomet ric series converges for r < 1 and diverges for ll 2 1 Explain. . You will...
Given the following formula (1) for Present Value. Setup the above formula and illustrate the derivation to Derive the Factor (A/P) Finding A given P. i(1+i)" A=P(1+i)" - 1]
1. Prove that log(1+r) = geometric series. (-1)-1 for <1. The simplest way to do this is to integrate the
use geometric series. !!!! Consider the AR(1) model it-onrt-1 + ur where ur ~ NID(0, σ.). Show that fol ol1, the auto-covariance is σ.ofl-o2t 7(h) (Note that for large t, this reduces to the formula given in the notes.)
12-1 + + 4. The series £9) .. is a geometric series. 4 n=1 Which of the following is true? (a) The series is convergent and its sum is less than 1/2. (b) The series is convergent and its sum is 1/2. (c) The series is convergent and its sum is 2/3. (d) The series is convergent and its sum is more than 2/3. IS 5. For positive numbers a and r, it is known that the geometric series divergent....