- Application of Present Value Concept
- Compound Annual Rate
- Interest Rates vs. Discount Rate
- Internal Rate of Return
- Bond Pricing

- The higher the future value (FV) of the payment
- The shorter the time period until payment (n)
- The lower the interest rate (i)

- 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

- 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 gradually declines to $23 for a payment made in 30 years

- 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.

- 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

- 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

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%

- 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

- 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

- 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.
- 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

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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