Time Value

• Interest Rates

• Compounding • Discounting

• Effective Rates

• Annuities • Perpetuities

2

Interest Rates

• Types

– Bank rate vs. Prime rate – Mortgage rates – Deposit, Loan, Credit rates

• Movement

– Demand / Supply – Inflation/ Deflation – Government intervention 3

Main Components

1. Real 2. Inflation

3. Risk

*Note:

- Risk Free (Rf) = Real + Inflation - Nominal = Rf + Risk Premium 4

Risk Free & Real Rate

• Risk Free (Rf) = Real + Inflation

• Real = [(1 + Rf) / (1 + Inflation)] - 1

• Given Rf = 10% & Inflation = 6% • Real Rate = [(1.1) / (1.06)] – 1 = 3.77% 5

Compounding

FV

r (APR) PV t

FV = PV×(1 + r)t

= Future Value

= Interest rate = Present Value = Time

Discounting

PV = FV×[1 / (1 + r)t]

6

Effective Interest Rates (EAR)

m APR EAR 1 1 m

Where APR = Annual Percentage Rate (Nominal) m = Rate of Compounding

Compounding “APR” for “m” times a year = Effective Rate once a year 7

Effective Interest Rate (EAR) = [1 + r/m]m -1

Compounding period (t) Number of times compounded (m) Effective annual rate (%)

Year Quarter Month Week Day Hour Minute

1 4 12 52 365 8,760 525,600

10.00 10.38 10.47 10.506 10.515 10.517 10.5171

8

Annuities & Perpetuities

11 + r )t (1 r

PV Annuity =

FV Annuity =

(1 + r)t – 1 r

C Perpetuity : PV r

C Growing Perpetuity : PV rg

9

Future Value Annuity

FV calculated by compounding forward one period at a time

0 1 2 3 4

5

Time (years)

$0 0

$0 2,000 x 1.1

$2,200 2,000 x 1.1 $4,200 x 1.1

$4,620 2,000 $6,620 x 1.1

$7,282 2,000 $9,282 x 1.1

$10,210.2 2,000 $12,210.2

$0

$2,000

FV calculated by compounding each cash flow separately

0 1 2 3 4 5 Time (years)

$2,000

$2,000

$2,000

$2,000 x 1.1

$2,000.0 2,200.0 2,420.0 2,662.0 2.928.2

x

1.12

Using Annuity formula: $2,000 x (1.10)5 .10 = $2,000 x 6.1051 = $12,210.2 -1 x 1.14

x

1.13

Total future value

$12,210.2 0

10

Present Value Annuity

Present value calculated by discounting each cash flow separately 0 1 2 3 4 5 $1,000 x 1/1.06 $ 943.40 890.00 839.62 x 1/1.064 x 1/1.062 x 1/1.063 $1,000 $1,000 $1,000 $1,000 Time (years)

Using Annuity formula: $1,000 x 1 – 1/(1.06)5 .06

792.09

747.26 $4,212.37 Total present value x 1/1.065

r = 6%

= $1,000 x 4.2123 = $4,212.37

Present value calculated by discounting back one period at a time 0 1 2 3 4 5

$4,212.37 0.00 $4,212.37

$3,465.11 1,000.00 $4,465.11

$2,673.01 1,000.00

$1,833.40 1,000.00

$ 943.40 1,000.00 $1,943.40

$

0.00

Time (years)

1,000.00 $1,000.0

$3,673.01

$2,833.40

0

Total present value = $4,212.37

r = 6% 11

Loan = $5,000;

Amortization Schedule – Fixed Payments

r = 9%; t = 5 years

Beginning Balance Balance Total Payment Interest Paid Principal Paid Ending

Year

1 $5,000.00 2 4,164.54 3 3,253.88 4 2,261.27 5 1,179.32 Totals

$1,285.46 $ 450.00 1,285.46 374.81 1,285.46 292.85 1,285.46 203.51 1,285.46 106.14 $6,427.30 $1,427.31

$ 835.46 $4,164.54 910.65 3,253.88 992.61 2,261.27 1,081.95 1,179.32 1,179.32 0.00 $5,000.00

12

Stealing with Interest Rates

Going out for business sale!

$ 1000 instant credit! 12% simple interest! 3 years to pay back! Very low monthly payments!

13

1. Borrow $1000 today at 12% per year for 3 years, you will owe: $1000 + $1000(.12)(3) = $1360 1. Make 36 (low) payments of $1360/36 = $37.78. 2. BUT, this is not a 12% loan $1,000 = $37.78 x [1 - 1/(1 + r)36] / r

r

= 1.766% per month

EAR = 1.0176612 - 1 =

APR = 12(1.766%) = 21.19% 14 23.38%!