What is the proper discounting of PIK and non-compounding bullet loans?

Question

This question pertains to two types of loans. Pay-in-kind (PIK) and bullet loans with quarterly payments.

1. PIK Loans

A PIK loan is a loan where periodic interest is NOT paid, but added to the principal instead and interest for the next period is calculated based on the new principal. I.e. a PIK loan compounds interest on interest. At the end of any period $t$ the principal $P_t$ can be calculated as follows:

$$ \begin{aligned} P_t &= P_{t-1} + I_t \\ I_t &= P_{t-1} × rd/365 \end{aligned}$$

Where:

$I_t$ = is the interest for period $t$

$r$ = annual coupon rate

$d_t$ = number of days in period $t$

As you can see, there are no cash-flows until maturity where $t = M$. I want to calculate the present value (fair value of this loan). My question is, which is the appropriate formula for discounting? Assuming a yield $y$ the present value $PV$ can be generalized as follows: $$ PV =P_M/(1+y/m)^{mn} $$

Where $P_M$ is the final cash-flow of principal and all interest, $m$ is the number of compounding periods per year and $n$ is the number years until maturity. The question is:

Should m = 1 or m = 4?

I.e. should we compound quarterly or annually? Remember there are NO cash-flows until the maturity M. But the interest IS compounded quarterly. Please provide reason why and if possibly any reference to the appropriate literature if you have one.

NOTE: The yield $y$ is market yield for similar loans / bonds. I'm not sure if this qualifies is as the Effective Annual Rate, in which case $m$ should probably equal 1.

2. Bullet Loans

A bullet loan is a loan where all principal is paid at maturity and interest is paid periodically, typically semi-annually or quarterly. The interest does not compound and is paid on constant principal. The question is:

If loan has quarterly payments, should the $m = 4$ or $m = 1$?

$$PV =∑_{i=1}^n{CF_i/(1+r⁄m)^{n_i m}}$$

Answer 1

The direct answer to your question on the choice of m is, "It depends."

Your choice of m is dependent on the convention used by the source of your discount rate. Either may be appropriate.

If you are actually looking to estimate a "fair" value, then the following will be relevant: A market yield(-to-maturity) approach assumes coupon reinvestment at that yield-to-maturity.[1] Using a single discount rate for multiple payments assumes a flat yield curve, which is almost never the case in reality.

With any valuation, you can always start off by forgetting any short-cut formula and focusing on the cash flows: determine what the cash flows are for every relevant period and then discount each cash flow back individually.

In the PIK case, you have only one cash flow. With the loan (similar to most bonds), you have a series of coupons and a terminal payment.

This leaves you with the question of what discount rate to use. If you are looking to generate a fair value estimate, then you would probably want to generate (or use a pre-generated) a zero-coupon "risk free"[2] curve and add a credit spread extracted from a credit-similar bond that is actively traded, plus possibly a liquidity adjustment for less liquid instruments.

I've managed to find some notes from CFA Level 1 material dealing with bond valuation and specifically the issue around using multiple discount rates for multiple cash flows through time: http://www.investopedia.com/exam-guide/cfa-level-1/fixed-income-investments/arbitrage-free-valuation-approach.asp

Hope this helps.

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[1] See e.g. http://www.investopedia.com/exam-guide/cfa-level-1/fixed-income-investments/traditional-yield-curve-measures-assumptions.asp

[2] Derived from some proxy. There's been some debate about relevant choices over the last years. See for e.g. http://www.prmia.org/sites/default/files/references/HullPresentation.pdf