Questions:
How do you compute the present value of an annuity with a fixed number of periods?

I am considering purchasing a home on a 20 year mortgage. The house currently costs $1,500,000 but I can only contribute $200,000 right now. Three financing companies have provided me with their financing plans. I want to choose the most affordable financial plan and secure the lowest possible payment per month. 

Chase Bank has provided with the option to fund the remaining payment at 5%. 

Bank of America has offered me $1,000,000 at 5% and $300,000 at 4% 

PNC Bank has a plan of financing my shortfall through the market rate of 4% and it's plan includes increasing the interest rate 0.5% after every $400,000. This means that the first $400,000 will be provided at 4%, the next $400,000 will increase to 4.5%, and so on. 

Which should I choose?

Reference answers:
The following formula can be used to value these annuities:

PV = p x ((1 – (1 + r) ^{-n}) / r)

where:
p = payment per period
r = interest/discount rate
n = number of periods remaining

Since the payments will be made every month, they will be accounted for as annuities. Instead of computing PV, we can solve for p instead. After doing some algebra, we get:

p = (r x PV) / [1 - (1+r) ^ -n ]

Computing p will then tell us the estimate montly payment.

1. Chase Bank: 

Financing = $1,300,000 
Annual Rate = 5% 

Since the payment is going to be monthly, the rate needs to be converted to monthly rate  

Monthly rate = 5% / 12 = 0.42% 
Number of periods = 20 years 

Since the payments will be made monthly, the number of periods = 20*12 = 240 

Annuity = (r x PV) / [1 - (1+r) ^ -n ] 
Annuity = (0.42% * $1,300,000) / [1 - (1+0.42 %) ^ - 240 
Annuity = $8608 

2. Bank of America

Financing = $1,300,000 
Annual Rate = 5% for 1,000,000 and 4% for $300,000. 

A weighted average rate needs to be calculated as there are two different rates 
Weighted average rate = (5% * 1,000,000 / (1300,000) + (4% * 300,000 / (1300,000)  
Weighted average rate = 4.76%. 

Since the payment is going to be monthly, the rate needs to be converted to montly rate  
Monthly rate = 4.76% / 12 = 0.40% 
Number of periods = 20 years 

Since the payments will be made monthly, the number of periods = 20*12 = 240 

Annuity = (r x PV) / [1 - (1+r) ^ -n ] 
Annuity = (0.40% * $1,300,000) / [1 - (1+0.40 %) ^ - 240 
Annuity = $8436 

3. PNC Bank

Financing = $1,300,000 

Annual Rate = 4% for 400,000, 4.5% for the next 400,000, 5% for the next 400,000 and 5.5% for the remaining 100,000 

A weighted average rate needs to be calculated as there are different rates involved 

Weighted average rate = (4% x 400,000 / (1300000) + (4.5% x 400,000 / (1300000) + (5% x 400,000 / (1300000) + (5.5% x 100,000 / (1300000) 

Weighted average rate = 4.58%. 

Since the payment is going to be monthly, the rate needs to be converted to monthly rate  
Monthly rate = 4.58% / 12 = 0.38% 
Number of periods = 20 years 

Since the payments will be made monthly, the number of periods = 20*12 = 240 

Annuity = (r x PV) / [1 - (1+r) ^ -n ] 
Annuity = (0.38% * $1,300,000) / [1 - (1+0.38%) ^ - 240 
Annuity = $8266 



The monthly payments in each of the plans will be as follows:
 
Chase Bank: $8608 
Bank of America: $8436 
PNC: $8266 

PNC offers the lowest payment per month plan which suits your needs.