The recently opened Grand Hyatt Wailea Resort and Spa on Maui cost $600mill abou
ID: 2701727 • Letter: T
Question
The recently opened Grand Hyatt Wailea Resort and Spa on Maui cost $600mill about $800,000 per room, to build. Daily operating expenses average $135 a room if occupied and $80 a room if unoccupied (much of the labor cost of running a hotel is fixed) at an average room rate of $500 a night. A marginal tax of 40%, and a cost of capital of 11%, what year round occupancy rate do the Japanese investors who financed the Grand Hyatt Wailea require to break even in economic terms on their investment over its estimated 40 yr. life? What is the likelihood that this investment will have a positive NPV? Assume that the $450 million expense of building the hotel can be written off straight line over a 30yr period (the other$150 million is for the land which is not depreciable) and that the present value of the hotels terminal value will be $200million
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Explanation / Answer
In finance, the net present value (NPV) or net present worth (NPW)[1] of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values (PVs) of the individual cash flows of the same entity.
In the case when all future cash flows are incoming (such as coupons and principal of a bond) and the only outflow of cash is the purchase price, the NPV is simply the PV of future cash flows minus the purchase price (which is its own PV). NPV is a central tool in discounted cash flow (DCF) analysis and is a standard method for using the time value of money to appraise long-term projects. Used for capital budgeting and widely used throughout economics, finance, and accounting, it measures the excess or shortfall of cash flows, in present value terms, above the cost of funds.
NPV can be described as the %u201Cdifference amount%u201D between the sums of discounted: cash inflows and cash outflows. It compares the present value of money today to the present value of money in the future, taking inflation and returns into account
The NPV of a sequence of cash flows takes as input the cash flows and a discount rate or discount curve and outputs a price; the converse process in DCF analysis %u2014 taking a sequence of cash flows and a price as input and inferring as output a discount rate (the discount rate which would yield the given price as NPV) %u2014 is called the yield and is more widely used in bond trading.Each cash inflow/outflow is discounted back to its present value (PV). Then they are summed. Therefore NPV is the sum of all terms,
rac{R_t}{(1+i)^{t}}
where
t - the time of the cash flow
i - the discount rate (the rate of return that could be earned on an investment in the financial markets with similar risk.); the opportunity cost of capital
R_t - the net cash flow i.e. cash inflow %u2013 cash outflow, at time t . For educational purposes, R_0 is commonly placed to the left of the sum to emphasize its role as (minus) the investment.
The result of this formula is multiplied with the Annual Net cash in-flows and reduced by Initial Cash outlay the present value but in cases where the cash flows are not equal in amount, then the previous formula will be used to determine the present value of each cash flow separately. Any cash flow within 12 months will not be discounted for NPV purpose, nevertheless the usual initial investments during the first year R0 are summed up a negative cash flow.[2]
Given the (period, cash flow) pairs (t, R_t) where N is the total number of periods, the net present value mathrm{NPV} is given by:
mathrm{NPV}(i, N) = sum_{t=0}^{N} rac{R_t}{(1+i)^{t}}
The rate used to discount future cash flows to the present value is a key variable of this process.
A firm's weighted average cost of capital (after tax) is often used, but many people believe that it is appropriate to use higher discount rates to adjust for risk, opportunity cost, or other factors. A variable discount rate with higher rates applied to cash flows occurring further along the time span might be used to reflect the yield curve premium for long-term debt.
Another approach to choosing the discount rate factor is to decide the rate which the capital needed for the project could return if invested in an alternative venture. If, for example, the capital required for Project A can earn 5% elsewhere, use this discount rate in the NPV calculation to allow a direct comparison to be made between Project A and the alternative. Related to this concept is to use the firm's reinvestment rate. Reinvestment rate can be defined as the rate of return for the firm's investments on average. When analyzing projects in a capital constrained environment, it may be appropriate to use the reinvestment rate rather than the firm's weighted average cost of capital as the discount factor. It reflects opportunity cost of investment, rather than the possibly lower cost of capital.
An NPV calculated using variable discount rates (if they are known for the duration of the investment) may better reflect the situation than one calculated from a constant discount rate for the entire investment duration. Refer to the tutorial article written by Samuel Baker[3] for more detailed relationship between the NPV value and the discount rate.
For some professional investors, their investment funds are committed to target a specified rate of return. In such cases, that rate of return should be selected as the discount rate for the NPV calculation. In this way, a direct comparison can be made between the profitability of the project and the desired rate of return.
To some extent, the selection of the discount rate is dependent on the use to which it will be put. If the intent is simply to determine whether a project will add value to the company, using the firm's weighted average cost of capital may be appropriate. If trying to decide between alternative investments in order to maximize the value of the firm, the corporate reinvestment rate would probably be a better choice.
Using variable rates over time, or discounting "guaranteed" cash flows differently from "at risk" cash flows, may be a superior methodology but is seldom used in practice. Using the discount rate to adjust for risk is often difficult to do in practice (especially internationally) and is difficult to do well. An alternative to using discount factor to adjust for risk is to explicitly correct the cash flows for the risk elements using rNPV or a similar method, then discount at the firm's rate.
NPV is an indicator of how much value an investment or project adds to the firm. With a particular project, if R_t is a positive value, the project is in the status of positive cash inflow in the time of t. If R_t is a negative value, the project is in the status of discounted cash outflow in the time of t. Appropriately risked projects with a positive NPV could be accepted. This does not necessarily mean that they should be undertaken since NPV at the cost of capital may not account for opportunity cost, i.e., comparison with other available investments. In financial theory, if there is a choice between two mutually exclusive alternatives, the one yielding the higher NPV should be selected.
The time-discrete formula of the net present value
mathrm{NPV}(i) = sum_{t=0}^{N} rac{R_t}{(1+i)^{t}}
can also be written in a continuous variation
mathrm{NPV}(i) = int_{t=0}^{infty} (1+i)^{-t} cdot r(t) , dt
where
r(t) is the rate of flowing cash given in money per time, and r(t) = 0 when the investment is over.
Net present value can be regarded as Laplace-[4][5] respectively Z-transformed cash flow with the integral operator including the complex number s which resembles to the interest rate i from the real number space or more precisely s = ln(1 + i).
F(s) = mathcal{L} left{f(t) ight}=int_0^{infty} e^{-st} f(t) ,dt
From this follow simplifications known from cybernetics, control theory and system dynamics. Imaginary parts of the complex number s describe the oscillating behaviour (compare with the pork cycle and phase shift between commodity price and supply offer) whereas real parts are responsible for representing the effect of compound interest (compare with damping).
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