Question : In an investment project, the investor needs to deposit an amount at the beginning of a certain year, and will get a total payment of $\$12000$
(a) Suppose a discount rate of $8\%$ compounded continuously is used. Find, correct to the nearest cent, the present value of the payment if
(ii) the payment is in the form of a continuous stream at a constant rate.
(b) Now suppose the initial investment is $10000$ dollars and a fixed payment of $2000$ dollars is made at the beginning of each of the six subsequent years.
(i) Estimate the annualised internal rate of return (IRR) of the project. (You should work out an equation of the form $f(r) = 0$, where $r$ is the desired rate. Start by testing $r = 5.5\%.$ Use differentiation to determine whether the actual answer should be larger or smaller. Then work out the answer correct to the nearest basis point (i.e. $0.01\%$).)
I don't know what does a(ii) mean :( . Does it mean I should integrate $12000e^{-0.08t}$ from $0$ to $6$ or from $1$ to $7$ ?
For (b)(i), should I set up an equation : $$-10000 + {2000\over1+r} + {2000\over(1+r)^2} + ... + {2000\over(1+r)^6} = 0$$ and then solve this equation ? It seems lengthy to differentiate such equation. Besides, how does differentiation help determine whether the actual answer should be smaller or larger?
I don't see how to answer your question about a(ii) unless you tell us what a(i) says.
For part b) your equation is right. What they are suggesting is that you first assume that $r=5.5\%$ and see whether it is too big or too small. The point about differentiation is that if you set $f(r)$ equal to the left-hand side, and you find say that $f(r)<0$ you can calculate $f'(r)$ and see whether the function is increasing. That will tell you whether your next guess for $r$ should be higher of lower.
This is silly advice in this case. As $r$ increases, plainly each of the fractions decreases, so $f$ is a decreasing function and there is no need to take derivatives. If $f(r)>0$ then $r$ is too small and your next guess should be bigger.
This is why bond prices rise when interest rates fall.