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Two approaches to equivalence calculation under

1. Estimate all future cash flows in constant dollars Use inflation-free (i′) to find equivalent worth: an estimate of the true earning power of when the inflation effects have been removed (also known as ).

2. Estimate all future cash flows in actual dollars Use interest rate (i) to find equivalent worth: interest rate that takes into account the combined effects of the earning of capital & any anticipated changes in power (also known as inflation-adjusted interest rate). Constant dollar analysis

In the absence of inflation, need only to use i′, to account for the earning power of the money. All economic analyses up to this point is, in fact, have been constant dollar analyses. Constant dollar analysis is common in the evaluation of many long-term public projects, in part because governments do no pay taxes. For private sector, income taxes are levied based on taxable income in actual dollars, actual dollar analysis is more common. Actual dollar analysis

Two approaches to estimating equivalent present worth, i.e. converting actual dollars into equivalent present worth dollars:

1. method Step 1. Bring all cash flows to have common purchasing power. Step 2. Consider the earning power.

2. Adjusted-discount method Combine Steps 1 and 2 into one step. Example 4.6, deflation method. Step 1: Convert actual dollars to constant dollars

Cash flows in actual Multiplied by Cash flows in n dollars deflation factor constant dollars 0 -$75,000 1 -$75.000 1 32,000 (1+0.05)-1 30,476 2 35,700 (1+0.05)-2 32,381 3 32,800 (1+0.05)-3 28,334

4 29,000 (1+0.05)-4 23,858

5 58,000 (1+0.05)-5 45,445

An A′n = A′ = cash flow in constant dollars (1 +⎯ f)n n Deflation method. Step 2: Convert constant dollars to equivalent present worth

Cash flows in Multiplied by Equivalent n constant dollars factor present worth 0 -$75,000 1 -$75,000

1 30,476 (1+0.10)-1 27,706

2 32,381 (1+0.10)-2 26,761

3 28,334 (1+0.10)-3 21,288

4 23,858 (1+0.10)-4 16,295

5 45,445 (1+0.10)-5 28,218

$45,268

Present worth of constant dollar amount: A′ i′ = inflation-free interest rate P = n n (1 + i′)n Deflation method: Converting actual dollars to constant dollars & then to equivalent present worth

n = 1 n = 5 n = 0 n = 2 n = 3 n = 4

Actual -$75,000 $32,000 $35,700 $32,800 $29,000 $58,000 dollars

Constant -$75,000 $30,476 dollars $32,381 $28,334 $23,858 $45,455

$28,218 Present $27,706 $26,761 $21,288 $16,295 -$75,000 worth

$45,268 Adjusted-discount method: Performs deflation & discounting in one step

An Since market interest rate reflects both Step 1: A′ = earning power & purchasing power, it is n (1 +⎯ f)n also true that: A P = n A′ n n Step 2: P = n (1 + i) n (1 + i′)n An An equate: = n n An (1 + i) [(1 +⎯ f)(1 + i′)] (1 +⎯ f)n combine: P = simplify: (1 + i) = (1 +⎯ f)(1 + i′) n (1 + i′)n

simplify: 1 + i = 1 + i′ + ⎯ f +⎯ f i′ rearrange:

An with the result that: Pn = [(1 +⎯ f)(1 + i′)]n i = i′ +⎯ f +⎯ f i′ Example 4.7 Adjusted-discounted method

Cash flows in actual Multiplied Equivalent n dollars by present worth 0 -$75,000 1 -$75,000

1 32,000 (1+0.155)-1 27,706

2 35,700 (1+0.155)-2 26,761

3 32,800 (1+0.155)-3 21,288

4 29,000 (1+0.155)-4 16,296

5 58,000 (1+0.155)-5 28,217

$45,268

iif=+''+if A =+0..10 0 05 +(0.10)(0.05) P = n n n = 15.5% (1 + i) Example 4.8. Equivalence calculation w/ composite cash flow elements: for college

Approach: Convert any cash flow elements in constant dollars into actual dollars. Then use the market interest rate to find the equivalent .

College expenses College expenses Age (in today’s dollars) (in actual dollars) 18 (Freshman) $30,000 $30,000(F/P,6%,13) = $63,988

19 (Sophomore) 30,000 30,000(F/P,6%,14) = 67,827

20 (Junior) 30,000 30,000(F/P,6%,15) = 71,897

21 (senior) 30,000 30,000(F/P,6%,16) = 76,211

Note: saving begins when child is age 5, 13 years before entering college Required quarterly contributions to college funds

V1 = C(F/A, 2%, 48)

V2 = $229,211

Let V1 = V2 and solve for C:

C = $2,888.48 Summary of ch. 4

The Consumer (CPI) is a statistical measure of change, over time, of the of and services in major expenditure groups—such as food, housing, apparel, transportation, and medical care—typically purchased by urban consumers. Inflation is the term used to describe a decline in purchasing power evidenced in an economic environment of rising prices. Deflation is the opposite: An increase in purchasing power evidenced by falling prices. The general inflation rate (f) is an average inflation rate based on the CPI. An annual general inflation rate (⎯f n ) can be calculated as follows:

CPInn− CPI −1 f n = CPIn−1 Specific, individual commodities do not always reflect the general inflation rate in their price changes. We can calculate an average inflation rate for a specific commodity (j) if we have an index (that is, a record of historical costs) for that commodity. Project cash flows may be stated in one of two forms

– actual dollars (An): dollars that reflect the inflation or deflation rate.

– constant dollars (A′n): year 0 dollars

Interest rates for project evaluation may be stated in one of two forms: – Market interest rate (i): a rate that combines the effects of interest & inflation; used with actual dollar analysis – Inflation-free interest rate (i′): a rate from which the effects of inflation have been removed; this rate is used with constant dollar analysis To calculate the present worth of actual dollars, we can use a two-step or a one-step process:

Deflation method – two steps: 1. Convert actual dollars by deflating with the general inflation rate ⎯f 2. Calculate the PW of constant dollars by discounting at i′ Adjusted-discount method – one step 1. Compute the market interest rate. 2. Use the market interest rate directly to find the present value.