Present and future value tables of $1 at 3% are presented below: Rosie\'s Floris

Question

Present and future value tables of $1 at 3% are presented below:

  

Rosie's Florist borrows $430,000 to be paid off in eight years. The loan payments are semiannual with the first payment due in six months, and interest is at 6%. What is the amount of each payment?

N FV $1 PV $1 FVA $1 PVA $1 FVAD $1 PVAD $1 1 1.03000 0.97087 1.0000      0.97087      1.0300      1.00000      2 1.06090 0.94260 2.0300      1.91347      2.0909      1.97087      3 1.09273 0.91514 3.0909      2.82861      3.1836      2.91347      4 1.12551 0.88849 4.1836      3.71710      4.3091      3.82861      5 1.15927 0.86261 5.3091      4.57971      5.4684      4.71710      6 1.19405 0.83748 6.4684      5.41719      6.6625      5.57971      7 1.22987 0.81309 7.6625      6.23028      7.8923      6.41719      8 1.26677 0.78941 8.8923      7.01969      9.1591      7.23028      9 1.30477 0.76642 10.1591      7.78611      10.4639      8.01969      10 1.34392 0.74409 11.4639      8.53020      11.8078      8.78611      11 1.38423 0.72242 12.8078      9.25262      13.1920      9.53020      12 1.42576 0.70138 14.1920      9.95400      14.6178      10.25262      13 1.46853 0.68095 15.6178      10.63496      16.0863      10.95400      14 1.51259 0.66112 17.0863      11.29607      17.5989      11.63496      15 1.55797 0.64186 18.5989      11.93794      19.1569      12.29607      16 1.60471 0.62317 20.1569      12.56110      20.7616      12.93794     

Explanation / Answer

Answer:

Since the amount is paid semi-annually, it means 2 payment in 1 year (1 payment in 6 month). So, the No. of Period in which borrow money to be paid = 8 Years x 2 = 16

Interest Rate = 6% p.a. or 6%/2 = 3 % semi-annually

Borrow Amount = $430,000

Since the borrown amount installment will be paid of equal amount including interest in every six month. It is the question of ANNUITY.

Annuity is a series of payment/receipt of equal amount for specified period of time.

Here, installment amount is equal under each payment in 6 months, so it is a series of payment. Period is specified i.e. 16.

We need to understand that sum of all the payment which is to be paid in every six month should be equal to present value of borrowing ($430,000) by using discounting rate 3% since the payment is to be made semi-annually.

So, the factor which convert all future equal amount installment is called Present Value Interest Factor for Annuity at 3% for 16 periods i.e. PVIFA (3%, 16)

From table (Column Number 5 -- PVA $1) the value of PVIFA (3%, 16) is 12.56110

Amount of Each Payment = Loan Amount / PVIFA (3%, 16) = $430,000 / 12.56110 = $34,232.67

More Explanation:

First we need to understand the question. Present Value of Loan Amount is given $430,000 that means all the future installment which are including interest must be equal to the Present Value of Loan i.e

Let Assume Amount of Each Payment = P

So the value of P for 16 period must be equal to $430,000 by using annuity factor at 3%

$430,000 = P x PVIFA (3%, 16)

P = $430,000 / PVIFA (3%, 6) = $430,000 / 12.56110 = $34,232.67

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