tailieunhanh - The annuity puzzle economic of annuities_4

Tham khảo tài liệu 'the annuity puzzle economic of annuities_4', khoa học xã hội, kinh tế chính trị phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 34 Chapter 5 Figure . Optimum nonannuitized assets. course Y 1 f0F z e pz dz. The dynamic budget constraint is b z Ya pb z - c z with solution b z ep z i -px ya - c x dx W - a 0 The amount of b z changes with age depending on the consumption path. The only constraint is that b z 0 for all z 0 z T. Hence W - a 0. For simplicity consider the special case Ơ 1 u c ln c 8 0 T X and F z e-az. For this case Y a p. Maximization of expected utility subject to yields optimum consumption c z c 0 e p-a z. Assume that p - a 0 implying that consumption rises with age. Solve for c 0 from setting limz . b z e-pz 0. Since b 0 0 it is optimum to set b 0 0 and a W or c 0 a p a p W. Substituting in we obtain the optimum path b z . It is now seen from that b 0 p - a p p a W 0. Nonannuitized assets accumulate and then decumulate to support the optimum consumption trajectory figure . Comparative Statics 35 Partial Annuitization Low Returns on Annuities Cannon and Tonks 2005 observe that the issuers of annuities insurance firms invest their assets for reasons of liquidity and risk mainly in bonds that yield a lower return than equities. While the reasons given for this policy are rather weak and as annuity markets grow insurance firms are expected to hold more balanced portfolios this may be another explanation why individuals annuitize only later in life holding nonannuitized assets at early ages. To see this let annuities have a rate of return of p0 r z while nonannuitized assets yield a return of p p p0. The budget constraint now becomes a. z p0 r a z pb z w z - c z - b z . Multiplying both sides of by e-p0zF z and integrating by parts yields e-p0zF z w z -c z dz-Ị r z - p - p0 e-p0zF z b z dz. Recall that b z 0 0 z T. If the hazard rate r z increases with age so that r z - p - p0 I 0 as z I zc then the individual s optimum policy is to invest all assets in b up to age zc switching to annuities afterward. Length of .