Question

Problem 3 A) Pierluigi is trying to get a loan for $10,000 to start a business as a financial advisor and is trying to decide between several options. (15 points)

i) A $10,000 loan that needs to be paid back after 5 years with a 5% nominal annual interest rate, compounded monthly.

ii) A $10,000 loan that needs to be paid back after 6 years, the first 2 years there is no interest, and after the annual effective interest rate is 10%. Hint: When solving for the annual payment consider PV=A+A+A(P/A,i%,n) where A is the uniform annual payment.

Assuming that in each case he plans to make equal yearly payments for the full length of the loan, starting in year 1:

1. What is the amount of his yearly payment in each case? (2*5 points)
2. Now assume that the interest rate of the market is an effective annual rate of 3%, what is the present value of the annuity payments for each option? Based on this metric, which is better? (4+1 points

PLEASE BE DETAILED IN YOUR EXPLANATION, THANK YOU!

Answer 1

a.

i.   Equal yearly payments for the 5 years

Using PV of annuity formula,

10000=Pmt.*(1-(1+r)^-n)/r)

where, PV= the value of the loan amt.= $ 10000

Pmt.= Mthly. payment to be found out & converted to annual

r= mthly compounding rate of interest ,ie.(5%/12)

n=No.of compounding periods= 12*5 yrs.=60

Plugging in the above values,

10000=Pmt.*(1-(1+(0.05/12))^-60)/(0.05/12)

& solving for pmt. We get the monthly payment as

Pmt.= $ 188.71

So, yearly payment is

188.71*12=

2264.52

Or

2265

ii.As the cash outflows start at end of Yr. 3 only,first,we need to find the PV of the Cash flows

Year	1	2	3	4	5	6
Cash flows for this loan	0	0	1000	1000	1000	11000
PV of the above cash flows=	(1000/1.1^3)+(1000/1.1^4)+(1000/1.1^5)+(11000/1.1^6)=
	8264.46
	Or
	8264

Now with this PV we find the annual PMT. Given 10% interest rate ,using the above formula for PV
ie.8264=Pmt.(1-1.1^-6)/0.1
Solving for pmt., we get the uniform annual payment as
1897.48
or
1897

b..

Given the interest rate of the market is an effective annual rate of 3%,

PV of Option i=

188.71*(1-(1+(0.03/12))^-60)/(0.03/12)=

10502.15642

OR

10502

PV of Option ii=

1897*(1-1.03^-6)/0.03=

10276.41

OR

10276

Option ii is a better option as the PV of future cash flows is the lesser of the two.