Regular payments of $405 every 6 months for the next 12 years. The annuity earns 3.75%/a compounded semi-annually, and the first payment will be in 6 months.

Question

Determine the current value of the investment.

Answer 1

Let’s start with some definitions

The Present value PV, of a stream of payments, A for T periods discounted at some rate,

I, is given as PV=A*K(I,T) where K is a multiplier which increases as the number of periods increases and decreases as the interest rate increases. Formally,

K=(1/i)(1-(1/(1+i)^T) = (1/i)(1/(1+i)^T)((1+i)^T-1) where I is the PERIODIC INTEREST RATE and T is the NUMBER of PERIODS So for a thirty year mortgage with a 12% rate we would have 360 monthly payments with a rate of 6%/15 or 0.5% monthly rate.

In your case A= 405 i=3.75%/2 T= 24 semi annual payments.

So K = (1/.01875)(1-(1/(1.01875))²⁴) =(1/.01875)(.35971)= 19.1845 os 1 dollar for every 6 months for the next 12 years is worth about 19.18 in PV so,

405*k = 7,769.72 dollars. That is the fair price of your annuity

Hope this helps

Answer 2

The concept of present value (PV) and future value (FV) are financial tools for calculating the value of money today. PV is used to calculate annuities that is not due. The present value is the amount required now to produce those future payments.

The formula for calculating PV:

PV = P * {1/ ((1 /(1+r) ^(-n))/ r]

PV = present value

r = interest rate

n - number of periods

In your question

r = 3.75%

n - 12 years

PMT = $405 (dollar amount of)

Semi-annual rate of 0.0375

Semi annual payments in 12 years

12*6 = 24

0.0375/2

=0.01875

PV = $405 *(1/(( 1 + 0.01875)^ 24))

PV =405 * (1/((1.01875)^24