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Class 3 – The Time Value of Money

Suppose market interest rates are 10%/year.

That means, if you invest 1000 today, in 1 year, you will have 1000(1.10) = 1,100 =

1,000 (your original principal) + 100 interest. We could also ask, what is the present

value (PV) (the value NOW) of 1,100 to be received in 1 year, if market interest rates are

10%/year? Answer: PV = X such that X(1.10) = 1100 => X = 1100/1.10 = 1000.

What is the PV of 1000 in 1 year, if the interest rate is 10%? Answer: X such that X(1.10)

= 1000 => X = 1000/1.10 = 909.09 What would you rather have, 909.09 now or 1000 in

1 year?

Example: If interest rates are 12%/year, what does 1500 grow to in 5 years? In 1 year,

1500 => 1500(1.12) = 1680 = 1500 + 180 of interest. In the second year, 1680 =>

1680(1.12) = 1881.60. Note: 1881.60 –1500 = 381.60 and 180 x 2 = 360 and 381.60 –

360 = 21.60. 21.60 is the interest earned in the second year, on the interest earned in the

first year.

Benjamin Franklin: “Money makes money, and the money that money makes, makes

more money.”

In 5 years, 1500 => 1500(1.12)5 = 1500(1.7623) = 2643.51. We say, 2643.51 is the future

value (FV) of 1500 in 5 years if interest rates are 12% annually with annual

compounding. In general: FV = C0(1+r)T where C0 is the amount invested, r = the annual

rate and T = the number of years it is invested for. Similarly, What is the PV of 2643.51

in 5 years if r = 12%? PV = X such that X(1.12)5 = 2643.51 => X = 2643.51/(1.12)5 =

1500.

PV of an amount C1 in 1 year: PV = C1/(1+r)

PV of an amount CT in T years is PV = CT/(1+r)T

Suppose we invest C0 now in return for:

C1 in 1 year

C2 in 2 years

…

CT in T years

First note: If r is the appropriate annual interest rate, the PV of the future payments is

given by: PV= C1/(1+r) + C2/(1+r)2 + … + CT/(1+r)T

And the Net Present Value (NPV) = -C0 + C1/(1+r) + C2/(1+r)2 + … + CT/(1+r)T

A general goal is to be able to calculate the value at any given time of a series of

payments made at various times.

Example:

Find the value at time 2 of the following payments, if r = 8% annually:

50 100 50 100 300 200

-----------------------------------------------------

-3 -2 -1 0 1 2 3 4 5

Answer: 50(1.08)4 + 100(1.08)3 + 50(1.08) + 100 + 300/(1.08)2 + 200/(1.08)3 = 763.96

Stated or Nominal rates of Interest

If the rate of interest is 10% with annual compounding, then 100 => 100(1.1) = 110 in 1

year. Suppose the rate is a 10% nominal rate with semi annual compounding. This means

(by definition) a rate of 5% per ½ year. So, 100 => 100(1.05) in ½ year, and 105 =>

105(1.05) = 110.25 in 1 year. 100(1.05)2 = 110.25

Note: 100(1.1025) = 110.25 if the effective annual rate is 10.25% so a nominal rate of

10% with semi annual compounding is equivalent to an effective annual rate of 10.25%.

Similarly, 10% with quarterly compounding means: 100 => 100(1.025)4 = 110.3813 in 1

year => annual effective rate of: 10.3813%.

In general, if r is a nominal rate with compounding m times per year, means a rate of r/m

per mth of a year. We write “EAR” to represent the Effective Annual Rate.

(1+r/m)m = (1 +EAR) => EAR = (1+r/m)m – 1 or r = m[(1+EAR)1/m – 1]

Example:

What is the FV of 1500 in 4 years if the interest rate is a nominal rate of 18% with

monthly compounding? FV = 1500(1+.18/12)48 = 1500(1+.015)48 = 3065.22. What is the

PV of 5000 in 5 years if the interest rate is 18% with monthly compounding? PV = 5000/

(1.015)60 = 2046.48.

Suppose we invest 1000 for 1 year at 10% with compounding as follows. How much do

we have in each case?

i) annual compounding: 1000(1.10) = 1100

ii) semi annual compounding: 1000(1.05)2 = 1102.5

iii) monthly compounding: 1000(1+.10/12)12 = 1104.713

iv) daily compounding: 1000(1+.10/365)365 = 1105.156

v) continuous compounding: 1000e.10 = 1105.171