VT Implied APY

Bullet Maturity Redemption

The implied Annual Percentage Yield (APY) for a bullet maturity redemption is derived from the compound growth rate that equates the current price to the future face value over the remaining term. It assumes no intermediate cash flows beyond interest (which is often separately accounted for in yield-to-maturity calculations) and focuses on the principal's appreciation or depreciation.

The implied APY is calculated as:

$$Implied APY= \text{Price}(t_0)^{\frac{1}{T_{\text{yearstoexpiry}}}} - 1$$

Where:

• $\text{Price}(t_0)$ : The current market price of VT.

• $T_{\text{years to expiry}}$: The time to maturity in years.

For example, if $( \text{Price}(t_0) = 4 )$ and $( T_{\text{yearstoexpiry}} = 3 )$

$$Implied APY= 4^{1/3} - 1 = 0.59$$

Amortizing Redemption

Amortizing redemption is redeemed in installments over a defined period, rather than in a single lump sum at maturity. This approach spreads out the redemption payments, which introduces complexity in yield calculations due to the time value of money.

For illustration, consider a hypothetical amortizing redemption schedule: the total face value (FV) is redeemed over 10 months, with 10% of the principal redeemed each month (i.e., equal installments totaling 100%). The present value (PV) of the investment is known upfront. The goal is to determine the APY, which accounts for compounding effects.

Precise Calculation: Iterative Internal Rate of Return (IRR) Method

Calculating the APY for amortizing redemptions requires solving for the monthly internal rate of return (IRR, denoted as ( r )) that equates the present value of the cash inflows (redemption payments) to the initial investment (PV). This involves the annuity present value formula, as the redemptions form an ordinary annuity.

Step 1: Solve (r) from Annuity Present Value Formula

The PV of an ordinary annuity (payments at the end of each period) is given by:

$$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$

Where:

• ( PV ): Present value of the investment (e.g., initial purchase price).

• ( PMT ): Periodic payment

• ( n ): Number of periods (e.g., 10 months).

• ( r ): Monthly IRR (to be solved for).

Step 2: Compute APY

Once ( r ) is obtained, annualize it assuming monthly compounding:

$$ImpliedAPY=(1+r)^{12}−1$$

This method is accurate but computationally intensive, especially for longer periods or variable payments.

Simplified Approximation: Adjusted Geometric Mean Method

Given the complexity of iteration, a practical approximation can be used. This leverages the geometric growth rate over an "average holding period" to estimate the APY without solving the full annuity equation.

The implied APY is calculated as:

$$Implied APY= \text{Price}(t_0)^{\frac{2}{T_{\text{yearstoexpiry}}}} - 1$$

Where:

• $\text{Price}(t_0)$ : The current market price of VT.

• $T_{\text{years to expiry}}$: The time to maturity in years.

For example, if $( \text{Price}(t_0) = 4 )$ and $( T_{\text{yearstoexpiry}} = 3 )$

$$Implied APY= 4^{2/3} - 1 = 1.52$$