## Interest Rate Implicit in the Lease
### What It Is
The Interest Rate Implicit in the Lease (IRIL) is the discount rate that makes the present value of all lease cash flows equal to the fair value of the leased asset at the inception of the lease.
Cash flows to be discounted:
- Annual Lease Rentals (LR) at end of each year
- Guaranteed Residual Value (GRV) — receivable/payable at end of lease
- Unguaranteed Residual Value (UGRV) — also included
> Key Rule (AS 19): PV (LR + GRV + UGRV) = Fair Value of Asset
---
### Step-by-Step Method: Trial & Error + Interpolation
Step 1 – Identify cash flows
- Years 1 to (n−1): Lease Rental only
- Year n (last year): Lease Rental + GRV + UGRV
Step 2 – Assume a lower discount rate (r₁)
- Compute PV of all cash flows at r₁
- If PV > Fair Value → r₁ is too low (need higher rate)
Step 3 – Assume a higher discount rate (r₂)
- Compute PV at r₂
- Confirm PV at r₂ < Fair Value
Step 4 – Interpolate
$$\text{IRR} = r_1 + \frac{\text{PV}_{r_1} - \text{FV}}{\text{PV}_{r_1} - \text{PV}_{r_2}} \times (r_2 - r_1)$$
> Numerator = how much PV at r₁ exceeds Fair Value
> Denominator = total spread between the two PVs
> This gives the fractional distance to travel from r₁ toward r₂
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### Why Two Rates Are Needed
IRR cannot be solved algebraically when there are multiple cash flows. Trial and error narrows the range; interpolation linearly estimates the exact rate within that range. Always choose r₁ and r₂ such that one gives PV above Fair Value and the other gives PV below it.
### Example 1
Example 1 – Basic IRR Calculation
Given:
- Annual Lease Rentals: ₹1,00,000 p.a. (end of year), 5 years
- GRV: ₹50,000 | UGRV: ₹30,000 (both at end of Year 5)
- Fair Value at inception: ₹4,20,000
Year 5 total cash flow = 1,00,000 + 50,000 + 30,000 = ₹1,80,000
Trial at 10%:
| Yr | Cash Flow | DF @10% | PV |
|---|
| 1 | 1,00,000 | 0.909 | 90,900 |
| 2 | 1,00,000 | 0.826 | 82,600 |
| 3 | 1,00,000 | 0.757 | 75,700 |
| 4 | 1,00,000 | 0.683 | 68,300 |
| 5 | 1,80,000 | 0.621 | 1,11,780 |
| Total | | | 4,29,280 ≈ 4,28,680 |
PV (4,28,680) > FV (4,20,000) → rate too low, try higher.
Trial at 12%:
| Yr | Cash Flow | DF @12% | PV |
|---|
| 1 | 1,00,000 | 0.893 | 89,300 |
| 2 | 1,00,000 | 0.797 | 79,700 |
| 3 | 1,00,000 | 0.712 | 71,200 |
| 4 | 1,00,000 | 0.636 | 63,600 |
| 5 | 1,80,000 | 0.567 | 1,02,060 |
| Total | | | 4,05,860 |
PV (4,05,860) < FV (4,20,000) ✓
Interpolation:
$$IRR = 10 + \frac{4{,}28{,}680 - 4{,}20{,}000}{4{,}28{,}680 - 4{,}05{,}860} \times (12-10) = 10 + \frac{8{,}680}{22{,}820} \times 2 = 10 + 0.76 = \textbf{10.76\%}$$
### Example 2
Example 2 – Smaller Values
Given:
- Annual Lease Rentals: ₹25,000 p.a., 5 years
- GRV: ₹12,500 | UGRV: ₹7,500 (Year 5)
- Fair Value: ₹1,00,000
Year 5 total = 25,000 + 12,500 + 7,500 = ₹45,000
Trial at 12%:
| Yr | Cash Flow | DF @12% | PV |
|---|
| 1–4 | 25,000 each | 0.893/0.797/0.712/0.636 | 22,325/19,925/17,800/15,900 |
| 5 | 45,000 | 0.567 | 25,515 |
| Total | | | 1,01,465 |
PV > FV → try 14%.
Trial at 14%:
| Yr | Cash Flow | DF @14% | PV |
|---|
| 1–4 | 25,000 each | 0.877/0.769/0.675/0.592 | 21,925/19,225/16,875/14,800 |
| 5 | 45,000 | 0.519 | 23,355 |
| Total | | | 96,180 |
PV < FV ✓
Interpolation:
$$IRR = 12 + \frac{1{,}01{,}465 - 1{,}00{,}000}{1{,}01{,}465 - 96{,}180} \times 2 = 12 + \frac{1{,}465}{5{,}285} \times 2 = 12 + 0.55 = \textbf{12.55\%}$$
### Example 3
Solved Example – Uneven Rentals
Given:
- Lease Rentals: ₹50,000 p.a. for Years 1–5, 5-year term
- GRV: ₹25,000 | UGRV: ₹15,000 (Year 5)
- Fair Value: ₹2,00,000
Year 5 total = 50,000 + 25,000 + 15,000 = ₹90,000
Trial at 10% → PV = ₹2,14,340 (above ₹2,00,000, so too low)
Trial at 12%:
| Yr | Cash Flow | DF @12% | PV |
|---|
| 1 | 50,000 | 0.893 | 44,650 |
| 2 | 50,000 | 0.797 | 39,850 |
| 3 | 50,000 | 0.712 | 35,600 |
| 4 | 50,000 | 0.636 | 31,800 |
| 5 | 90,000 | 0.567 | 51,030 |
| Total | | | 2,02,930 ≈ 2,02,780 |
Trial at 14% → PV = ₹1,92,360 (below ₹2,00,000) ✓
Interpolation:
$$IRR = 12 + \frac{2{,}02{,}780 - 2{,}00{,}000}{2{,}02{,}780 - 1{,}92{,}360} \times 2 = 12 + \frac{2{,}780}{10{,}420} \times 2 = 12 + 0.53 = \textbf{12.53\%}$$