Equivalence of cash flows Construction Engineering 221

Equivalence of cash flows Non- annual compounding (discounting) –A sum of money invested at 4% annual interest is compounded semi-annually. How many years will it take for the invested sum to double in value: i = 2%, F = 2P, n = ? Go to the 2% tables (page 100) Find n where F/P = 2 (n= 35) 35 semiannual periods equal 17.5 years

Equivalence of cash flows If a 5 year balloon lease has a 10% annual rate, but interest is charged monthly (like a credit card), what will be the buyout amount on a $2500 purchase? –F = 2500(1 +.10/12) 60 or 2500(1.6453) =

Equivalence of cash flows What if the lease caps are compounded annually: –F = 2500 (n=5, i=.10) or –Buyout is –Lender makes an additional $87 merely by compounding the interest monthly instead of annually

Equivalence of cash flows Cash flow factors and symbols take the formula and solve using table factors. Expressed as: F = P(F/P, i, n) or P = F(P/F, i, n) Other equivalencies use the same procedure

Equivalence of cash flows Uniform series equivalence –Repeating cash flow for a known number of periods is called and annual amount (like a mortgage payment). Can be estimated (like maintenance) or known (like rental income) –Future worth of a annual payment (called an annuity) is F = A(F/A, i, n) where A is the size of the annual amount

Equivalence of cash flows Sinking fund is an account that you must make deposits in each year in order to have F dollars at n periods in the future. Therefore, the A/F factor is called the sinking fund factor, and A is the sinking fund allocation

Equivalence of cash flows An annuity is a series of equal payments made over a period of time (for instance, a bond that pays interest annually). Therefore, the P/A factor is called the annuity factor and A is the annuity amount When the stream of payments is “perpetual” (like a building generating rents), the term capitalized cost is sometimes used. Capitalized cost is P = A/i

Equivalence of cash flows Assume you have a client who wants to net lease the project on a 25 year lease. They can afford $20,000 a month in lease payments ($240,000 per year using end of year convention) What should the capital budget be for the new building if you are the turnkey contractor and your cap rate is 8%? P = 240,000( {p. 124}), or $2,562,000

Equivalence of cash flows If the building could be re-leased for another 25 years at the same rate, the initial cost budget would be: –240,000 ( ) = 2,936,000 –An infinite series of 240,000 annual payments has a present worth of 240,000/.08 or 3,000,000, so you can see that for many buildings, the capitalized cost formula is a viable option

Equivalence of cash flows Can calculate “past worth” by setting t=o at some arbitrary point in the cash flow series. Sometimes this calculation is needed in lawsuits to award damages or determine remedies for loss of use Can also calculate periods needed to earn a multiple of investment. Table 2 on Page 16 give double and triple earnings. –n= log X/ log (1 + i), where X is the multiple desired on the initial investment

Equivalence of cash flows Variable (non-standard flows) –Gradient cash flows are better representations of many equipment maintenance schedules than uniform series (gets more expensive to maintain as the equipment gets older) –The gradient starts in year 2 because it is the INCREASE in annualized payments that are of interest

Equivalence of cash flows P/G factor calculates the present worth of the ESCALATING costs, not the base costs Base costs must be calculated using the P/A factor Maintenance costs are the most typical example of gradient series

Equivalence of cash flows Stepped cash flows are handled as two separate cash flows using the superpositioning technique (see example on page 19). This would apply to a rental contract with an escalation clause Missing or extra cash flows can be handled similarly (major overhaul example), as can beginning of period payments

Equivalence of cash flows