Time value of moneyPart 3

Mixed flows example

Sharif will receive the set of cash flows below. What is the Present Value Present Value at a

discount rate of 10%10%?

0 1 2 3 4 55

$600 $600 $400 $600 $600 $400 $400 $100$400 $100PVPV00

10%10%

How to solve?

1. Solve a “piece-at-a-time” by discounting each piece back to t=0.

2. Solve a “group-at-a-time” by first breaking problem into groups of annuity streams and any single cash flow group. Then discount each group back to t=0.

“Piece-at-a-time”

0 1 2 3 4 55

$600 $600 $400 $600 $600 $400 $400 $100$400 $100

10%

$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09

$1677.15 $1677.15 = = PVPV00 of the Mixed Flowof the Mixed Flow

“Group-at-a-time” (#1)

0 1 2 3 4 55

$600 $600 $400 $600 $600 $400 $400 $100$400 $100

10%

$1,041.60$1,041.60$ 573.57$ 573.57$ 62.10$ 62.10

$1,677.08$1,677.08 = = PVPV00 of Mixed Flow of Mixed Flow [Using Tables][Using Tables]

$600(PVIFA10%,2) = $600(1.7355) = $1,041.30

$400(PVIFA10%,2)(PVIF10%,2) = $400(1.7355)(0.8264) = $573.69

$100 (PVIF10%,5) = $100 (0.6209) = $62.09

“Group-at-a-time” (#2)

0 1 2 3 4

$400 $400 $400 $400$400 $400 $400 $400

PVPV00 equals$1677.15$1677.15

0 1 2

$200 $200$200 $200

0 1 2 3 4 5

$100$100

$1,267.96$1,267.96

$347.10$347.10

$62.09$62.09

PlusPlus

PlusPlus

Frequency of compounding

General Formula:FVn = PVPV00(1 + [i/m])mn

n: Number of Yearsm: Compounding Periods per

Yeari: Annual Interest Rate

FVn,m: FV at the end of Year n

PVPV00: PV of the Cash Flow today

Impact of Frequency

Tonni has $1,000$1,000 to invest for 2 years at an annual interest rate of 12%.

Annual FV2 = 1,0001,000(1+ [.12/1])(1)

(2) = 1,254.401,254.40Semi FV2 = 1,0001,000(1+ [.12/2])(2)

(2) = 1,262.481,262.48

Impact of Frequency

Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)

= 1,266.771,266.77Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)

= 1,269.731,269.73Daily FV2 = 1,0001,000(1+[.12/365])(365)(2)

= 1,271.201,271.20

Effective annual interest rate

Effective Annual Interest RateThe actual rate of interest earned (paid)

after adjusting the nominal rate for factors such as the number of

compounding periods per year.

(1 + [ i / m ] )m - 1

Rata’s annual interest rate

Rata has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAREAR)?

EAREAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1

= .0614 or 6.14%!6.14%!

Steps to amortizing a loan

1. Calculate the payment per period.2. Determine the interest in Period t.

(Loan balance at t-1) x (i% / m)3. Compute principal payment principal payment in Period t.

(Payment - interest from Step 2)4. Determine ending balance in Period t.

(Balance - principal payment principal payment from Step 3)5. Start again at Step 2 and repeat.

Amortizing a loan expense

Bristi is borrowing $10,000 $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years.

Step 1: Payment

PVPV00 = R (PVIFA i%,n)

$10,000 $10,000 = R (PVIFA 12%,5)

$10,000$10,000 = R (3.6048)

RR = $10,000$10,000 / 3.6048

RR = $2,774.08 ≈ $2,774$2,774.08 ≈ $2,774

Amortizing a loan expense

End of Year

Payment Interest Principal Ending Balance

0 --- --- --- $10,000

1 $2,774 $1,200 $1,574 8,426

2 2,774 1,011 1,763 6,663

3 2,774 800 1,975 4,688

4 2,774 563 2,211 2,477

5 2,774 297 2,478 0

$13,870 $3,871 $10,000

Usefulness of amortization

2.2.Calculate Debt Outstanding Calculate Debt Outstanding – The quantity of outstanding debt may be used in financing the day-to-day activities of the firm.

1.1.Determine Interest Expense Determine Interest Expense – Interest expenses may reduce taxable income of the firm.