(Present value) The present value of a promise to pay one dollar t years from no
ID: 2827541 • Letter: #
Question
(Present value) The present value of a promise to pay one dollar t years from now is g(t) dollars.
(a) What is g(0)?
(b) Why is it reasonable to assume that g(t) <= I and that g is a decreasing function of t?
(c) What is the present value of a promise to pay q dollars t years from now?
(d) Assume that an investment made now will result in an income now at the rate of f(t) dollars per year t years from now. (Assume that f is a continuous function.) Estimate informally the present value of the income to be earned between time t and time t + dt, where dt is a small positive number.
(e) On the basis of the local estimate made in (d), set up a definite integral for the present value of all the income to be earned from now to time b years in the future.
Explanation / Answer
a)g(0)=1
b)We pay 1 dollar after t years ,so
1=PV+interest for t years (PV is present value)
Interest is non negative so PV<=1
also as t increases interest increases .So for fixed value of FV, PV decreases as t increases (PV=1-interest)
c) present value of a promise to pay q dollars t years from now=q*g(t)
d)present value of the income to be earned between time t and time t + dt=g(t)*f(t)*t dt
e)present value of all the income to be earned from now to time b years in the future=integral{0 to b}(g(t)*f(t)*t dt)
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