Sums and Approximations 1.3 Return of the annuity problem Now we can solve the annuity pricing problem! The value of an annuity that pays m dollars at the start of each year for n years is (1+k) = m> 2 (where z 1+p We apply Theorem 1 on the second line, and undo the the earlier substitution z= 1/(1+p) on the last line The last expression is a closed form; it can be evaluated with a fixed number of ba- sic operations. For example, what is the real value of a winning lottery ticket that pays $40,000 per year for 50 years? Plugging in m =$40, 000, n= 50, and p=0.035 gives Va S971, 063. Youd be better off taking the million dollars today 1.4 Infinite Sums All right, would you prefer a million dollars today or $40,000 a year forever? This might seem like an easy choice- when infinite money is on offer, why worry about inflation? This is a question about an infinite sum. In general, the value of an infinite sum is defined as the limit of a finite sum as the number of terms goes to infinity means lim>Zk n→ So the value of an annuity with an infinite number of payments is given by our previ- ous answer in the limit as n goes to infinity 1 1+p m:一� � � � � � 4 Sums and Approximations 1.3 Return of the Annuity Problem Now we can solve the annuity pricing problem! The value of an annuity that pays m dollars at the start of each year for n years is: �n−1 m V = (1 + k)p k=0 �n−1 1 k = m z (where z = ) 1 + p k=0 n 1 − z = m · 1 − z n 1 1 − 1+p = m · � � 1 1 − 1+p We apply Theorem 1 on the second line, and undo the the earlier substitution z = 1/(1+p) on the last line. The last expression is a closed form; it can be evaluated with a fixed number of ba￾sic operations. For example, what is the real value of a winning lottery ticket that pays $40, 000 per year for 50 years? Plugging in m = $40, 000, n = 50, and p = 0.035 gives V ≈ $971, 063. You’d be better off taking the million dollars today! 1.4 Infinite Sums All right, would you prefer a million dollars today or $40,000 a year forever? This might seem like an easy choice— when infinite money is on offer, why worry about inflation? This is a question about an infinite sum. In general, the value of an infinite sum is defined as the limit of a finite sum as the number of terms goes to infinity: ∞ n zk means lim zk k=0 n→∞ k=0 So the value of an annuity with an infinite number of payments is given by our previ￾ous answer in the limit as n goes to infinity: n 1 1 − 1+p V = lim m · � � n→∞ 1 1 − 1+p 1 = m · � � 1 1 − 1+p 1 + p = m · p