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# Continuously compounded rate

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14. The world population of Ferrets has increased at a continuously compounded rate from 20 in 1970 to about 60 in 2000. Develop a mathematical model to forecast population growth in future years.

a) Graph the population against time and determine, identify or describe this relationship in your own words. What is the effective percentage rate of growth?

b) Find (or estimate) the number of births (instantaneous rate of change) in 2009 and use this value to estimate the population in 2010.

c) When will the world population double to 120 and what is the average increase per year over this period?

https://brainmass.com/math/calculus-and-analysis/continuously-compounded-rate-435384

#### Solution Preview

Please see the attached file for solution along with graph.

14. The world population of Ferrets has increased at a continuously compounded rate from 20 in 1970 to about 60 in 2000. Develop a mathematical model to forecast population growth in future years.

Mathematical Model: Nt= No e^rt
Where
Nt= Population at time t
N0= Population at time 0
t = Number of years since 1970 (Year 0)
r= rate of increase

We have to estimate the value of r using the data

For 1970, N0= 20, t=0
For 2000, t= 2000-1970 =30, N30= 60,

For 1970, 20= No e^(r x 0) = No
Thus No = 20
And Nt= 20 e^rt

For 2000, 60= 20 e^(r x 30)
Or, 3= e^(r x 30)
Taking natural logs of both sides,
ln 3 = ...

#### Solution Summary

The solution develops a mathematical model to forecast population growth in future years.

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## Continuous compounding, gain or loss, equivalent rate

1. An interest rate is 12.5% per annum expressed with continuous compounding. What is the equivalent rate with semiannual compounding?

2. A trader sells 100 European put options (1 contract) with a strike price of \$51 and a time to maturity of six months. The price received for each option is \$3. The price of the underlying asset is \$38 in six months. What is the trader's gain or loss? If loss, report a negative number.

3. An interest rate is 14% per annum when expressed with semiannual compounding. What is the equivalent rate with continuous compounding? Report the answer in % and round the number to the nearest 2 decimal percentage points such as 5.75%.

4. The three year zero rate is 5.1% and the four year zero rate is 6.5% (both continuously compounded). What is the forward rate (continuously compounded) for the fourth year? Report in % and round the number to the nearest 2 decimal percentage points such as 5.78%.

5. A company enters into a long futures contract to buy 1,000 barrels of oil for \$62 per barrel. The initial margin is \$10,000 and the maintenance margin is \$4,000. What oil futures price will trigger a margin call?

6. A trader sells 100 European put options (1 contract) with a strike price of \$51 and a time to maturity of six months. The price received for each option is \$4. The price of the underlying asset is \$38 in six months. What is the trader's gain or loss? If loss, report a negative number.

7. The three year zero rate is 5.6% and the four year zero rate is 6% (both continuously compounded). What is the forward rate (continuously compounded) for the fourth year? Report in % and round the number to the nearest 2 decimal percentage points such as 5.78%.

8. An interest rate is 13.01% per annum expressed with continuous compounding. What is the equivalent rate with semiannual compounding? (

9. An investor writes four naked put options contracts. The option price is \$3.2, the strike price is \$63, and the stock price is \$80. What is the initial margin requirement?

10. An interest rate is 13% per annum when expressed with semiannual compounding. What is the equivalent rate with continuous compounding? Report the answer in % and round the number to the nearest 2 decimal percentage points such as 5.75%.

11. A company enters into a long futures contract to buy 1,000 barrels of oil for \$61 per barrel. The initial margin is \$8,000 and the maintenance margin is \$4,000. What oil futures price will trigger a margin call?

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