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I am a beginner to finance, today I found a question looks very simple that I am not quite sure about it.
Question: Given I am paid \$50,000 now, growing at $6\%$ per year for a total of 10 years, but the discount rate is $4\%$, solve for the present value.
My thought is first calculating the nominal value in 10 years, which is $50000 \times (1+0.06)^{10}$, then discount this value using $4\%$ to get the present value.
Is there anything wrong with my idea? Thanks.
The proper way is to discount each and every single item of cash flow (the initial $50,000 "grant" as well as the 10 individual interest payments) each one at the discount rate of 4%.
In numbers, it's quite a big difference between the two methods, as shown in this table:
If you discounted just the ending capital - as suggested by you - you would erroneously conclude the NPV as $60,491.63, as shown in red.
However, if you discount each and every cash flow item, the initial inflow of USD 50,000 as well as each interest payment, the NPV will show as $81,474.88
Unfortunately I do not agree with the answer provided by FutForFut.
In my opinion answer of 60 491,63$ can be correct under certain circumstances.
There is also another correct answer, but I cannot agree with answer of 81 474,88$.
More specifically. Problem was “I am paid $50,000 now, growing at 6% per year for a total of 10 years, but the discount rate is 4%, solve for the present value.”
Which implies that person received 50 000 today but decides to invest the amount for next 10 years. Nothing ambiguous here but when we look further then it’s not entirely clear what kind of investment is chosen. Whether it’s “growing at 6%” or is it perhaps distributes 6% coupons/dividends annually.
In first case we are dealing with investment resembling zero-coupon bond.
In second case we have coupon bond or dividend stock kind of investment.
In both cases NPV-s are slightly different.
First zero-coupon bond:
As we can see, this is exactly the solution initially proposed by TrueWarrior09.
However, there could be alternative solution. If we assume that investment distributes cash flows annually (e.g. coupons or dividends) then cash flow pattern looks different.
So the 58 110,9 can be considered as alternative solution.
However, I cannot agree with solution proposed by FutForFut ($81 4747,88) because it erroneously assumes that 50 000 initially received qualifies as cash flow.
In our case received 50 000 is invested and therefore major cash flow appears only after 10 years.
Or alternately, if initially received 50 000 qualifies as cash flow then we don’t have capital to provide us any benefits (additional cash flows in our case) during next 10 years.
I interpret the question differently: $50,000k being the recurring and increasing cash flow itself, not initial capital.
If you have $50 000 today and you invest it for 10 years and earn 6% interest annually and your investment is growing year by year then the correct answer would be as follows:
