F371 – Financial Management Part 3.2:

DISCOUNTED CASH FLOW VALUATION AND MULTIPLE CASH FLOWS

COMPARING RATES WITH DIFFERENT COMPOUNDING PERIODS The stated rate required on consumer loans is called the Annual Percentage Rate (A.P.R.). 𝐴𝑃𝑅 = 𝑅 × 𝑚 and 𝑅=

𝐴𝑃𝑅 𝑚

where m = number of compounding periods in a single year.

Examples: APR = 12% compounded monthly

R = 1% per month.

APR = 16% compounded semiannually R = 2% per quarter

R = 8% per semiannual period.

APR = 8% compounded quarterly.

Effective annual interest rate (E.A.R.):

The rate, on an annual basis, that reflects

compounding effects. Given some rate, the EAR is the annual rate which gives the same Future Value at the end of one year. Starting from the same PV, if the FV is the same at the end of one year, then the (1+r)m terms must be the same. In other words: (1 + 𝐸𝐴𝑅 𝑎𝑠 𝑎 𝑑𝑒𝑐𝑖𝑚𝑎𝑙)1 = (1 + 𝑔𝑖𝑣𝑒𝑛 𝑟𝑎𝑡𝑒)𝑚 F371 JCS

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Example: Consider a rate of 2% per quarter. What is the EAR that is equivalent to 2% per quarter? (1 + 𝐸𝐴𝑅)1 = (1 + .02)4 1 + 𝐸𝐴𝑅 = 1.024 = 1.0824 𝐸𝐴𝑅 = 8.24% 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟 NOTE: The EAR is NOT necessarily the same as the APR. This is because the APR does not include the effects of compounding.

To find the EAR when starting from an APR: Steps: 1) Divide the APR by m (the number of compounding periods in one year). 2) Convert this percentage to (1 + r) format (divide by 100, add 1). 3) Raise this term to the power of m (number of compounding periods). This gives you the EAR in (1 + r) format. 4) Convert from (1 + r) format back to a percent (minus 1, times 100).

The formal equation is: 𝐴𝑃𝑅 𝑎𝑠 𝑎 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑚 𝐸𝐴𝑅 = (1 + ) −1 𝑚 𝑄𝑢𝑜𝑡𝑒𝑑 𝑟𝑎𝑡𝑒 𝑚 = (1 + ) −1 𝑚 Example: Consider an APR of 10% compounded quarterly. What is the effective annual rate (EAR)? 𝐸𝐴𝑅 = (1 +

0.10 4 ) −1 4

= 1.0254 − 1 = 1.1038 − 1 𝐸𝐴𝑅 = 10.38% 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟 F371 JCS

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Example 2: Which option would you choose if you were getting a loan? A)

10% compounded monthly

B)

10% compounded quarterly

C)

10.25% compounded annually

D)

11% compounded annually

IMPLICATIONS FOR SOLVING FOR PV AND FV When we solve for present or future value, we must either use the E.A.R. with years or the periodic rate (quoted/m) and N periods (where N = m * number of years). =>

FV = PV * (1 

q mt ) ; m

PV = FV * [

1 ]; q mt (1  ) m

PV =

FV q (1  ) mt m

Example: What is the present value of $100 to be received in 2 years at 10% compounded quarterly? There is no periodic payment in this one (PV and FV only), so either of these two methods will work: Method 1: Use a quarterly rate and the total number of quarters.

Method 2: Use the EAR (yearly rate) and the number of years.

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RATE ADJUSTMENTS FOR MULTIPLE COMPOUNDING PERIODS--SUMMARY Whenever the compounding periods differ from once per year (e.g. semiannually, quarterly, monthly, etc.) you must make the appropriate adjustments to all formulas! t (n) r

=> =>

= the # of years * # compounding periods per year. = stated rate / # compounding periods per year.

With annuities, you must be sure to match the appropriate rate to your cash flows!

Example 1: Suppose a company has borrowed $500,000. The loan will be repaid in five annual payments starting at the end of the first year. The interest rate on the loan is 9% compounded monthly. What is the amount of the annual payment?

Example 2: You borrow $10,000 to purchase a car and agree to repay the loan over four years of monthly payments. The APR (quoted rate) is 10.58% compounded quarterly. What is the amount of your monthly payment?

Example 3: A commercial loan in the amount of $50,000 will be repaid with semiannual payments for four years. The quoted interest rate is 8.96% compounded quarterly. What will be the amount of the semiannual payment?

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PERPETUITIES Perpetuity: A series of level cash flows which continues forever. Perpetuity Present value: CF1 / r where CF1 is the amount of the cash flow per period, and r is the periodic interest rate. 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑇𝑖𝑚𝑒 𝑍𝑒𝑟𝑜 =

𝐶𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑎𝑡 𝑇𝑖𝑚𝑒 1 𝑟

Example: Suppose that starting in one year, you will receive a perpetuity of $100 each year, forever. What is the present value of the series of cash flows at a 10% yearly interest rate?

Example 2: Suppose that starting four years from now, you will start receiving a perpetuity cash flow of $5,000 per year. What is the present value of this perpetuity, assuming a 12% yearly interest rate?

Example 3: A new investment opportunity is forecast to produce a cash outflow next year of $8,000. In Year 2, a cash inflow of $2,000 is forecast. After that, starting in Year 3, it is expected to generate a cash inflow of $3,000 per year. At a discount rate of 6% per year, what should be the price of this investment today?

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AMORTIZATION SCHEDULES An amortization schedule is a repayment schedule for a typical consumer loan that shows: 1) the number of payments 2) the amount of each payment 3) the interest paid per period 4) the reduction in principal per period 5) the remaining loan balance

Example: Suppose you borrow $4,000 and agree to repay the loan in 5 equal installments over a 5-year period. The interest rate on the loan is 10% per year. Step 1: Solve for payment using the PVIFA formula:

Step 2: Complete the amortization table as follows: Calculations 

Period 1

Initial loan amount; then ending balance from prior period

PMT

Loan balance × periodic interest rate

Periodic payment – interest charge

Loan balance – reduction in principal

Loan Balance

Periodic Payment

Interest Charge

Reduction in Principal

Ending Balance

$4,000.00

2 3 4 5

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