## Special Case 1: Annual Lease Rental Not Given

### When does this arise?

The question gives you the Fair Value of the asset but does not state the annual lease rental. The lease rentals are missing.

### Underlying Assumption

> The lessor prices the lease so that the Fair Value of the asset is fully recovered through the present value of future cash flows:

> `Fair Value = PV of MLP + PV of UGRY`

This is equivalent to saying: Net Investment = Fair Value on Day 1.

---

### Step-by-Step Method to Find Annual Lease Rent (LR)

Step 1 – Isolate PV of MLP

```

PV of MLP = Fair Value − PV of UGRY

```

Step 2 – Find Annual LR using Annuity Factor

If the lease rentals are a uniform annual amount:

```

PV of MLP = Annual LR × Annuity Factor (n years, r%)

∴ Annual LR = PV of MLP ÷ Annuity Factor

```

Step 3 – Compute Gross Investment, Net Investment, UFI as normal

```

GI = (Annual LR × n) + GRV + UGRY [face values]

NI = Fair Value (= PV of MLP + PV of UGRY)

UFI = GI − NI

```

---

### What is the Annuity Factor?

The annuity factor for n years at rate r% is simply the sum of individual discount factors:

```

Annuity Factor = DF₁ + DF₂ + … + DFₙ

= PV of ₹1 received each year for n years

```

Worked example

### Example 1

Example – Find Annual Lease Rent (3-year lease, 10% rate)

Given: Fair Value = ₹16,99,999.5 | Lease term = 3 years | UGRY = ₹1,33,500 | Rate = 10%

Step 1 – PV of UGRY

PV of UGRY = 1,33,500 × DF(3 yrs, 10%) = 1,33,500 × 0.751 = 1,00,258.5

Step 2 – PV of MLP

PV of MLP = 16,99,999.5 − 1,00,258.5 = 15,99,741

Step 3 – Annual LR

Annuity Factor (3 yrs, 10%) = 0.909 + 0.826 + 0.751 = 2.486

Annual LR = 15,99,741 ÷ 2.486 = ₹6,43,500.6

Step 4 – GI, NI, UFI

GI = (6,43,500.6 × 3) + 1,33,500 = 20,64,001.8

NI = 17,00,000 (≈ Fair Value)

UFI = 20,64,001.8 − 17,00,000 = ₹3,64,000.8

Verification (PV table check):

Yr	Cash Flow	DF 10%	PV
1	6,43,500.6	0.909	5,84,942
2	6,43,500.6	0.826	5,31,531.5
3	6,43,500.6 + 1,33,500	0.751	5,83,527.4
Total			17,00,000.9 ≈ 17,00,000 ✓

### Example 2

Example – Illustration 11 (3-year, 10%, Fair Value ₹10,00,000, UGRY ₹1,08,899)

PV of UGRY = 1,08,899 × 0.751 = 75,130 (≈ 1,00,000 × 0.7543 in some variants)

PV of MLP = 10,00,000 − 75,130 = 9,24,870

Annuity Factor (3 yrs, 10%) = 2.486

Annual LR = 9,24,870 ÷ 2.486 = ₹3,71,911.7

GI = (3,71,911.7 × 3) + 1,08,899 = 12,24,634 (approx 12,15,735)

NI = ₹10,00,000

UFI = ₹2,15,735

⚠️ Common exam mistakes

Forgetting to subtract PV of UGRY from Fair Value before dividing by the annuity factor – using Fair Value directly as PV of MLP overstates the rental.
Using UGRY's face value (not discounted) when computing PV of UGRY in Step 1.
Computing the annuity factor manually incorrectly – it is the algebraic sum of individual year discount factors, not multiplied.
Confusing annuity factor with the discount factor for year n only.
Rounding the annual LR too early, causing cumulative errors in GI and UFI calculations.

Reference:

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AS 19 Leases – Special Case 1: Annual Lease Rent Not Given (Derive from Fair Value)

Worked example

⚠️ Common exam mistakes