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How to calculate Monthly payment and Present Value of Lease payment?

In previous post we had discussed about lease and its type and now we will discuss its methods.
Minimum payment: The guaranteed amount which lessee pays to lessor if he does not earn enough money from leased asset or not. It does not include executory cost that is insurance cost, taxes and maintenance cost.

Formula for present value of lease payment:
PV= C [(1-(1/ (1+i) n)) / i] + RV/ (1+i) n
Formula for monthly lease payment:
C= [PV - (RV/ (1+i) n)]/ (1-(1/ (1+i) n)/ i)
Formula for monthly lease payment with advance:
C= [PV - (RV/ (1+i) n)]/ [(1-(1/ (1+i) n - A)/ i) +A]
Formula for present value of minimum lease payment:
PV = (P / (1+i) n) + (RV / (1+i) n)

Where,
C = monthly payment
PV = present value of lease amount
RV = residual value of lease asset
i = interest rate on lease
n = number of years
A = number of advance payment
P = annual lease payment
  
Example: Suppose a company has taken a machine on lease for 6 years and pays Rs.20, 000 per month and the interest on it is 6% p.a. the residual value of lease is Rs.60, 000. Find out the present value of lease payment.

Solution: C = 20,000, i= 6/12 = 0.5, RV = 60, 000
PV= C [(1-(1/ (1+i) ^n)) / i] + RV/ (1+i) ^n
= 20, 000 [(1-(1/ (1+0.005^6)) / 0.005] + 60, 000/ (1+0.005) ^6
= 20, 000 [(1-(1/1.03) / 0.005] + 60, 000/ 1.03
= 20, 000[(1-0.97) / 0.005] + 60, 000 / 1.03
= 20, 000[0.03/0.005] + 60, 000 / 1.03
= 20, 000*6 + 58, 252.43
PV = 1, 78,252.43

Example: A company is willing to take a land on lease for 10 years and the lessor charge 7% interest on it and the residual value of that land is Rs. 10, 05,000. Find out the how much company will have to pay to lessor if the present value is Rs. 56, 00,000?

Solution: i= 7/12 = 0.58%,

C= [PV - (RV/ (1+i) ^n)]/ (1-(1/ (1+i) ^n)/ i)
= [56, 00,000 - (10, 05,000/ (1+0.0058) ^10)]/ (1-(1/ (1+0.0058) ^10)/ 0.0058)
= [56, 00,000 - (10, 05,000/ 1.06)]/ (1-(1/ 1.06)/ 0.0058)
= [56, 00,000 – 9, 48,116.21]/ ((1-0.94)/ 0.0058)
= 46, 51,883.79 / 0.06/0.0058
= 46, 51,883.79 / 10.34

C = Rs.4, 49,892.05

Example: Company has taken an asset on lease from lessor at 8% p.a. interest and the annual payment of lease is Rs.85, 000 for 5 years and the residual value of asset is Rs.98, 000. Find out the present value of minimum lease payment.

Solution: i= 8/12 = 0.67, P = 85,000, RV = 98, 000

PV = (P / (1+i) n) + (RV / (1+i) n)
= (85, 000 / (1+0.0067) 5) + (98, 000 / (1+0.0067) 5)
= (85, 000 / 1.03) + (98, 000 / 1.03)
= 82, 524.27 + 95, 145.63
PV= 1, 77,669.9

Example: Suppose a company has purchased a lease for 7 years by paying 2 months advance at the rate of 6% p.a. the present value of leased amount is Rs.50, 000 and the residual value is Rs.20, 000.  How much company will have to pay monthly to lessor?

Solution: i= 6/12 = 0.5, PV = 50,000, RV = 80, 000, A = 2

C= (PV - (RV/ (1+i) n))/ ((1-(1/ (1+i) n - 2)/ i) +A)
= (50,000 - (20,000/ (1+0.005) 7))/ ((1-(1/ (1+0.005) 7 -2)/ i) +2)
= (50,000 - (20,000/ 1.04))/ ((1-(1/ 1.03)/ 0.005) +2)
= [50,000 – 19, 230.77]/ ((1-0.97/ 0.005) + 2)
= 30, 769.23 / ((0.03/0.005) + 2)
= 30, 769.23 / (6 + 2)
 = 30, 769.23 / 8 
= 3, 846.15




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