# Money and Banking

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Money & Banking ­ MGT411
VU
Lesson 9
APPLICATION OF PRESENT VALUE CONCEPTS
Application of Present Value Concept
Compound Annual Rate
Interest Rates vs. Discount Rate
Internal Rate of Return
Bond Pricing
Important Properties of Present Value
Present Value is higher:
The higher the future value (FV) of the payment
The shorter the time period until payment (n)
The lower the interest rate (i)
The size of the payment (FVn)
Doubling the future value of the payment (without changing time of the payment or interest
rate), doubles the present value
At 5% interest rate, \$100 payment has a PV of \$90.70
Doubling it to \$200, doubles the PV to \$181.40
Increasing or decreasing FVn by any percentage will change PV by the same percentage in the
same direction
The time until the payment is made (n)
Continuing with the previous example of \$100 at 5%, we allow the time to go from 0 to 30
years.
This process shows us that the PV payment is worth \$100 if it is made immediately, but
Figure: Present value of \$100 at 5% interest rate
The rule of 72
For reasonable rates of return, the time it takes to double the money, is given approximately by
t = 72 / i%
If we have an interest rate of 10%, the time it takes for investment to double is:
t = 72 / 10 = 7.2 years
This rule is fairly applicable to discount rates in 5% to 20% range.
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Money & Banking ­ MGT411
VU
The Interest rate (i)
Higher interest rates are associated with lower present values, no matter what size or timing of
the payment
At any fixed interest rate, an increase in the time until a payment is made reduces its present
value
Table: Present Value of a \$100 payment
Payment due in
Interest rate
1 Year  5 Years
10 Years
20 Years
1%
\$99.01
\$95.15
\$90.53
\$81.95
2% \$3.77  \$98.04
\$90.57  \$29.14  \$82.03
\$67.30
3%  4%  \$97.09
\$86.26  32%
\$74.41
\$55.37
4%
\$96.15
\$82.19
\$67.56
\$45.64
5%
\$95.24
\$78.35
\$61.39
\$37.69
6%
\$94.34
\$74.73
\$55.84
\$31.18
7%
\$93.46
\$71.30
\$50.83
\$25.84
8%
\$92.59
\$68.06
\$46.32
\$21.45
9%
\$91.74
\$64.99
\$42.24
\$17.84
10%
\$90.91
\$62.09
\$38.55
\$14.86
11%
\$90.09
\$59.35
\$35.22
\$12.40
12%
\$89.29
\$56.74
\$32.20
\$10.37
13%
\$88.50
\$54.28
\$29.46
\$8.68
14%
\$87.72
\$51.94
\$26.97
\$7.28
15%
\$86.96
\$49.72
\$24.72
\$6.11
Figure: The relationship between Present value and Interest Rates
Compound Annual Rates
Comparing changes over days, months, years and decades can be very difficult.
The way to deal with such problems is to turn the monthly growth rate into compound-annual
rate.
An investment whose value grows 0.5% per month goes from 100 at the beginning of the month
to 100.5 at the end of the month:
We can verify this as following
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Money & Banking ­ MGT411
VU
100 (100.5 - 100) = [(100.5/100) ­ 1] = 0.5%
100
What if the investment's value continued to grow at 0.5% per month for next 12 months?
We cant simply multiply 0.5 by 12
Instead we need to compute a 12 month compound rate
So the future value of 100 at 0.5 %( 0.005) per month compounded for 12 months will be:
FVn = PV (1+i)  n = 100(1.005)12 = 106.17
An increase of 6.17% which is greater than 6%, had we multiplied 0.5% by 12
The difference between the two answers grows as the interest rate grows
At 1% monthly rate, 12 month compounded rate is12.68%
Another use for compounding is to compute the percentage change per year when we know how
much an investment has grown over a number of years
This rate is called average annual rate
If an investment has increased 20%, from 100 to 120 over 5 years
Is average annual rate is simply dividing 20% by 5?
This way we ignore compounding effect
Increase in 2nd year must be calculated as percentage of the investment worth at the end of 1st
year
To calculate the average annual rate, we revert to the same equation:
FVn = PV (1+i)  n
120 = 100(1 + i)  5
Solving for i
i = [(120/100)1/5 - 1] = 0.0371
5 consecutive annual increases of 3.71% will result in an overall increase of 20%
Interest Rate and Discount Rate
The interest rate used in the present value calculation is often referred to as the discount rate
because the calculation involves discounting or reducing future payments to their equivalent
value today.
Another term that is used for the interest rate is yield
Saving behavior can be considered in terms of a personal discount rate;
People with a low rate are more likely to save, while people with a high rate are more likely to
borrow
We all have a discount rate that describes the rate at which we need to be compensated for
postponing consumption and saving our income
If the market offers an interest rate higher than the individual's personal discount rate, we would
expect that person to save (and vice versa)
Higher interest rates mean higher saving
Applying Present Value
To use present value in practice we need to look at a sequence or stream of payments whose
present values must be summed. Present value is additive.
To see how this is applied we will look at internal rate of return and the valuation of bonds
Internal Rate of Return
The Internal Rate of Return is the interest rate that equates the present value of an investment
with it cost.
It is the interest rate at which the present value of the revenue stream equals the cost of the
investment project.
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Money & Banking ­ MGT411
VU
In the calculation we solve for the interest rate
A machine with a price of \$1,000,000 that generates \$150,000/year for 10 years
\$ 150 , 000
\$ 150 , 000
\$ 150 , 000
\$ 150 , 000
\$ 1, 000 , 000 =
+
+
+ ...... +
(1 + i )  1
(1 + i )  2
(1 + i )  3
(1 + i )  10
Solving for i, i=.0814 or 8.14%
The internal rate of return must be compared to a rate of interest that represents the cost of funds
to make the investment.
These funds could be obtained from retained earnings or borrowing. In either case there is an
interest cost
An investment will be profitable if its internal rate of return exceeds the cost of borrowing
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