Suppose that you take out a $250,000 house mortgage from your local savings bank. The bank requires you to repay the mortgage in equal annual installments over the next 30 years. It must therefore set the annual payments so that they have a present value of $250,000.Calculate the annual payment that the mortgage bank should claim to have the present value of $250000. Interest rate is 12 %.
The continuously compounded interest rate is 12%.
What is the PV of a continuous stream of cash flows, amounting to
$2,000 per year, starting immediately and continuing for 15 years?
The correct answer and explanation is :
Part 1: Annual Payment for Mortgage
To calculate the annual payment for a mortgage, we can use the formula for the present value of an annuity when the interest rate is continuously compounded.
The formula for the present value ( PV ) of a continuous annuity is:
[
PV = \frac{C}{r} \left( 1 – e^{-rT} \right)
]
Where:
- ( PV ) is the present value of the loan (in this case, $250,000),
- ( C ) is the annual payment (the amount we need to solve for),
- ( r ) is the interest rate (12% or 0.12),
- ( T ) is the duration of the loan (30 years),
- ( e ) is Euler’s number (approximately 2.71828).
Given:
- ( PV = 250,000 ),
- ( r = 0.12 ),
- ( T = 30 ).
Rearranging the formula to solve for ( C ) (the annual payment), we get:
[
C = \frac{PV \cdot r}{1 – e^{-rT}}
]
Substitute the given values:
[
C = \frac{250,000 \times 0.12}{1 – e^{-0.12 \times 30}}
]
First, calculate the exponent:
[
e^{-0.12 \times 30} = e^{-3.6} \approx 0.0273
]
Now substitute back into the equation:
[
C = \frac{250,000 \times 0.12}{1 – 0.0273} = \frac{30,000}{0.9727} \approx 30,847.19
]
Thus, the annual payment required is approximately $30,847.19.
Part 2: Present Value of Continuous Stream of Cash Flows
Next, to find the present value (PV) of a continuous stream of cash flows amounting to $2,000 per year, we use the same formula:
[
PV = \frac{C}{r} \left( 1 – e^{-rT} \right)
]
Where:
- ( C = 2,000 ) (the cash flow amount),
- ( r = 0.12 ) (the continuously compounded interest rate),
- ( T = 15 ) (the number of years).
Substitute the given values into the formula:
[
PV = \frac{2,000}{0.12} \left( 1 – e^{-0.12 \times 15} \right)
]
First, calculate the exponent:
[
e^{-0.12 \times 15} = e^{-1.8} \approx 0.1653
]
Now substitute back into the formula:
[
PV = \frac{2,000}{0.12} \left( 1 – 0.1653 \right)
]
[
PV = \frac{2,000}{0.12} \times 0.8347 = 16,694.67
]
Thus, the present value of the continuous stream of cash flows is approximately $16,694.67.
Explanation
The present value of continuous cash flows is derived from integrating the stream of cash flows over time and accounting for the effect of the continuously compounded interest rate. The formula used above assumes that cash flows are received continuously and that they are discounted at a continuous rate. The main concept here is the time value of money, which states that a dollar received in the future is worth less than a dollar received today due to the opportunity cost of capital.
In this case, for both the mortgage and the continuous stream of cash flows, the present value reflects the amount of money you would need today, considering the time value of money, to match those future payments under a continuous compounding interest rate.