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Chapter 3: Problem 1

Find the present value of \(\$ 100000\) in 10 years' time if the discount rate is \(6 \%\) compounded (a) annually (b) continuously

Short Answer

Expert verified
Annual compounding: \$55,839.62. Continuous compounding: \$54,881.12.

Step by step solution

01

Understanding Present Value

The present value (PV) is the current value of a future amount of money, given a specific rate of return (discount rate). It helps to understand what a future sum of money is worth today.
02

Present Value Formula for Annual Compounding

When compounding is done annually, the present value is calculated by the formula: \[ PV = \frac{FV}{(1 + r)^n} \] Where: - \(PV\) is the present value - \(FV\) is the future value (\$100,000 in this case) - \(r\) is the annual discount rate (6% or 0.06) - \(n\) is the number of years (10 years)
03

Calculate Present Value for Annual Compounding

Substitute the given values into the formula: \[ PV = \frac{100000}{(1 + 0.06)^{10}} \] First, calculate \( (1 + 0.06)^{10} \): \[ (1 + 0.06)^{10} = 1.06^{10} \ \ 1.06^{10} \ \approx 1.790847 \] Now, find the present value: \[ PV = \frac{100000}{1.790847} \approx 55839.62 \] So, the present value compounded annually is approximately \$55,839.62.
04

Present Value Formula for Continuous Compounding

When compounding is done continuously, the present value is calculated by the formula: \[ PV = FV \times e^{-rt} \] Where: - \(PV\) is the present value - \(FV\) is the future value (\$100,000) - \(r\) is the annual discount rate (6% or 0.06) - \(t\) is the time in years (10 years) - \(e\) is the base of the natural logarithm (approximately 2.71828)
05

Calculate Present Value for Continuous Compounding

Substitute the given values into the formula: \[ PV = 100000 \times e^{-0.06 \times 10} \] First, calculate the exponent: \[ -0.06 \times 10 = -0.6 \] Now, use the value of \(e^{-0.6}\): \[ e^{-0.6} \ \approx 0.548811 \] Then find the present value: \[ PV = 100000 \times 0.548811 \ \approx 54881.12 \] So, the present value compounded continuously is approximately \$54,881.12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Annual Compounding
When we talk about annual compounding, it means calculating interest or growth on a yearly basis. This is a common method in finance where interest accumulates once per year.
To calculate the present value using annual compounding, the future value is discounted back to the present. This is achieved using the formula: PV = \frac{FV}{(1 + r)^n} Here are key points to understand:
  • **Future Value (FV):** The amount you will receive in the future. In our case, it's \(100,000.
  • **Discount Rate (r):** The interest rate used for discounting future cash flows, expressed as a decimal. Here, it's 6% or 0.06.
  • **Number of Years (n):** The time period over which the money will grow. In this problem, it's 10 years.
As calculated, plugging the numbers into the formula gives a present value of approximately \)55,839.62, meaning that if you want to have \(100,000 in ten years, you would need to invest \)55,839.62 today, assuming 6% annual compounding interest.
Deciphering Continuous Compounding
In continuous compounding, interest is calculated and added to the principal constantly, at infinitesimally small intervals. This creates a scenario where interest is compounding continuously, and it relies on the mathematical constant **e** (approximately 2.71828).
The formula for calculating present value with continuous compounding is: 鈱� PV = FV \times e^{-rt} where:
  • **Future Value (FV):** Stays the same as the annual compounding scenario, \(100,000.
  • **Rate (r):** The annual discount rate, given as 6% or 0.06.
  • **Time (t):** The number of years, which is 10 in this problem.
  • **e:** The base of the natural logarithm, approximately 2.71828.
When the values are substituted into the formula, it gives a present value of approximately \)54,881.12. This means you would need to invest about \(54,881.12 today to have \)100,000 in ten years if interest continuously compounds at a 6% rate.
Understanding Discount Rate
The discount rate plays a crucial role in determining the present value of a future amount. It's essentially the interest rate used to discount future cash flows to their present value.
Here are some key points to know:
  • **Opportunity Cost:** The discount rate reflects the opportunity cost of investing money elsewhere. Select a rate that matches what you might earn in alternative investments.
  • **Risk and Inflation:** Higher discount rates are used for riskier investments or in periods of high inflation. This protects against potential loss of value or purchasing power.
  • **Present vs. Future:** A higher discount rate causes a lower present value of future cash flows. Conversely, a lower discount rate results in a higher present value.
In our calculations, we used a 6% discount rate. This percentage reflects the assumed annual return if the money were invested elsewhere.
Calculating Future Value
Future Value (FV) is the worth of an investment at a specific time in the future, assuming a certain rate of growth or interest. It helps to project what money invested today will grow to over time.
Key components influencing future value include:
  • **Present Value (PV):** The initial amount of money invested or loaned.
  • **Interest Rate (r):** The rate at which the money grows each period. Annual or continuous compounding can be used.
  • **Number of Periods (n or t):** Compounding can occur annually, semi-annually, quarterly, or continuously.
The formula: 鈱� FV = PV 脳 (1 + r)^n鈽� For continuous compounding, FV can be calculated as: 鈱� FV = PV 脳 e^(rt)鈽咲iscovering the future value helps you plan and make informed investment decisions. In our example, the future value given was $100,000. By understanding how different rates and compounding methods affect present value, we can better understand and project future worth.

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Most popular questions from this chapter

How long will it take for a sum of money to triple in value if invested at an annual rate of \(3 \%\) compounded continuously?

The value of an asset, currently priced at \(\$ 100000\), is expected to increase by \(20 \%\) a year. (a) Find its value in 10 years' time. (b) After how many years will it be worth \(\$ 1\) million?

At the beginning of a month, a customer owes a credit card company \(\$ 8480\). In the middle of the month, the customer repays \(\$ A\), where \(A<\$ 8480\), and at the end of the month the company adds interest at a rate of \(6 \%\) of the outstanding debt. This process is repeated with the customer continuing to pay off the same amount, \(\$ A\), each month. (a) Find the value of \(A\) for which the customer still owes \(\$ 8480\) at the start of each month. (b) If \(A=1000\), calculate the amount owing at the end of the eighth month. (c) Show that the value of \(A\) for which the whole amount owing is exactly paid off after the \(n\)th payment is given by $$ A=\frac{8480 R^{n-1}(R-1)}{R^{n}-1} \text { where } R=1.06 $$ (d) Find the value of \(A\) if the debt is to be paid off exactly after 2 years.

Monthly sales figures for January are 5600 . This is expected to fall for the following 9 months at a rate of \(2 \%\) each month. Thereafter sales are predicted to rise at a constant rate of \(4 \%\) each month. Estimate total sales for the next 2 years (including the first January).

An engineering company needs to decide whether or not to build a new factory. The costs of building the factory are \(\$ 150\) million initially, together with a further \(\$ 100\) million at the end of the next 2 years. Annual operating costs are \(\$ 5\) million commencing at the end of the third year. Annual revenue is predicted to be \(\$ 50\) million commencing at the end of the third year. If the interest rate is \(6 \%\) compounded annually, find (a) the present value of the building costs (b) the present value of the operating costs at the end of \(n\) years \((n>2)\) (c) the present value of the revenue after \(n\) years \((n>2)\) (d) the minimum value of \(n\) for which the net present value is positive
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