The introduction to intertemporal budgeting in this chapter can be applied to thinking about the pricing of basic financial assets. The assets we will consider will differ in terms of when they pay income to the owner of the asset. In order to know how much such assets are worth, we have to determine their present value which is equal to how much current consumption such an asset would allow us to undertake.
A: Suppose you just won the lottery and your lottery ticket is transferable to someone else you designate — i.e. you can sell your ticket. In each case below, the lottery claims that you won $100,000. Since you can sell your ticket, it is a financial asset, but depending on how exactly the holder of the ticket received the $100,000, the asset is worth different amounts. Think about what you would be willing to actually sell this asset for by considering how much current consumption value the asset contains—assuming the annual interest rate is 10%.
(a) The holder of the ticket is given a $100,000 government bond that “matures” in 10 years. This means that in 10 years the owner of this bond can cash it for $100,000.
(b) The holder of the ticket will be awarded $50,000 now and $50,000 ten years fromnow.
(c) The holder of the ticket will receive 10 checks for $10,000 — one now, and one on the next 9 anniversaries of the day he/she won the lottery.
(d) How does you answer to part (c) change if the first of 10 checks arrived 1 year from now, with the second check arriving 2 years fromnow, the third 3 years fromnow, etc.?
(e) The holder of the ticket gets $100,000 the moment he/she presents the ticket.
B: More generally, suppose the lottery winnings are paid out in installments of x1,x2, ...,x10, with payment xi occurring (i −1) years fromnow. Suppose the annual interest rate is r .
(a) Determine a formula for how valuable such a stream of income is in present day consumption— i.e. how much present consumption could you undertake given that the bank is willing to lend youmoney on future income?
(b) Check to make sure that your formula works for each of the scenarios in part A.
(c) The scenario described in part A(c) is an example of a $10,000 payment followed by an annual “annuity” payment. Consider an annuity that promises to pay out $10,000 every year starting 1 year fromnow for n years. How much would you be willing to pay for such an annuity?
(d) How does your answer change if the annuity starts with its first payment now?
(e) What if the annuity from (c) is one that never ends? (To give the cleanest possible answer to this, you should recall fromyourmath classes that an infinite series of 1/(1+x)+1/(1+x)2 + 1/(1+x)3 +... = 1/x.) How much would this annuity be worth if the interest rate is 10%?