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Continuous Compounding

Continuous compounding uses Euler's number, $e=2.71828$, when interest is compounded at every instant.

$$F=Pe^{rn}$$
$$i_{eff}=e^r-1$$

Problem: CE Board May 2017

A savings bank offers long-term savings certificates at 7.5% per year, compounded continuously. If a 10-year certificate costs $1000, what will be its value at maturity?

$$F=1000e^{0.075(10)}=2117$$

Final answer: $2,117.

Problem: CE Board May 2015

What is the effective annual interest rate if the nominal rate is 14% compounded continuously?

$$i_{eff}=e^{0.14}-1=0.1503$$

Final answer: $15.03\%$.

Problem: CE Board Nov. 2015

If money is invested at a nominal rate of 8% for 4 years, what is the compound amount factor if compounded continuously?

$$e^{rn}=e^{0.08(4)}=1.377$$

Final answer: $1.377$.

Problem: Continuous Interest

Compute the interest on P200,000 for 8 years at 16% compounded continuously.

$$F=200000e^{0.16(8)}=719,328$$
$$I=F-P=719,328-200,000=519,328$$

Final answer: P519,328.

Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q158

MSTE - Engineering Economy / Continuous Compounding / Engr. Janclyde Espinosa (Clidez)

CE Board May 2017
A savings bank offers long-term savings certificates at 7.5% per year, compounded continuously. If a 10-year certificate costs $1000, what will be its value at maturity?

Answer:

  1. 2117
  2. 2711
  3. 2217
  4. 2172
For continuous compounding:
$F=Pe^{rt}$
$F=1000e^{0.075(10)}=1000e^{0.75}$
$\boxed{F\approx2117}$

Question Bank: q159

MSTE - Engineering Economy / Continuous Compounding / Engr. Janclyde Espinosa (Clidez)

CE Board May 2015
What is the effective annual interest rate if the nominal rate of interest is 14% compounded continuously?

Answer:

  1. 15.03%
  2. 13.05%
  3. 15.30%
  4. 13.50%
For continuous compounding, effective annual rate is:
$i_{eff}=e^r-1$
$i_{eff}=e^{0.14}-1=0.1503$
$\boxed{15.03\%}$

Question Bank: q160

MSTE - Engineering Economy / Continuous Compounding / Engr. Janclyde Espinosa (Clidez)

CE Board November 2015
If money is invested at a nominal rate of interest of 8% for a period of 4 yrs, what is the value of the compound amount factor if its compounded continuously?

Answer:

  1. 1.377
  2. 1.737
  3. 1.337
  4. 1.773
The continuous compound amount factor is:
$F/P=e^{rt}$
$F/P=e^{0.08(4)}=e^{0.32}$
$\boxed{1.377}$

Question Bank: q161

MSTE - Engineering Economy / Continuous Compounding / Engr. Janclyde Espinosa (Clidez)

Compute the interest for an amount of P200,000 for a period of 8 years if it was made at 16% compounded continuously.

Answer:

  1. 519,328
  2. 513,298
  3. 518,329
  4. 531,829
For continuous compounding:
$F=Pe^{rt}=200{,}000e^{0.16(8)}$
Interest is $I=F-P$:
$I=200{,}000(e^{1.28}-1)$
$\boxed{I\approx519{,}328}$

Question Bank: t1055

MSTE - Engineering Economy / Engineering Economy / Gemini mapped Chapter 7 to 10

Find the present worth of perpetuity of P5200 payable monthly if the interest is 16% compounded monthly.

  1. P450,000
  2. P310,000
  3. P350,000
  4. P390,000

Solution pending in psadquestions/t1055.json.

Question Bank: t2022

MSTE - Engineering Economy / Simple Interest / BEMz

If you borrowed money from your friend with simple interest of 12%, find the present worth of P50 000, which is due at the end of 7 months.

  1. P46 200
  2. P44 893
  3. P46 729
  4. P45 789
Use simple interest present worth:
$F=P(1+rt)$
Here $F=50{,}000$, $r=0.12$, and $t=7/12$ yr.
$P=\frac{50{,}000}{1+0.12(7/12)}$
$P=\frac{50{,}000}{1.07}$
$\boxed{P\approx\text{P}46{,}729}$

Question Bank: t2103

MSTE - Engineering Economy / Continuous Compounding / Besavilla CE Pre-Board Math & Surveying

P100000 is deposited at a nominal rate of 7% compounded annually, for 5 years. What would be the difference in the sums at the end of 5 years if the interest were compounded continuously.

  1. P1652
  2. P1562
  3. P1748
  4. P1885
  5. P1451
Annual compounding:
$F_a=100{,}000(1.07)^5=\text{P}140{,}255$
Continuous compounding:
$F_c=100{,}000e^{0.07(5)}=100{,}000e^{0.35}=\text{P}141{,}907$
Difference:
$F_c-F_a\approx141{,}907-140{,}255$
$\boxed{\text{P}1652}$

Problem: Present Worth with Continuous Compounding

Find the present worth of P50,000 due in 4 years if money earns 7% compounded continuously.

$$P=Fe^{-rt}=50000e^{-0.07(4)}=37790$$

Answer: The present worth is about P37,790.

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