When a PPA target liability is the basis of a Gain/Loss Analysis, the interest rates used to determine the expected end-of-period liability are the forward rates for the periods over which the Gain/Loss Analysis is being run. In a one year Gain/Loss Analysis this will be the first segment rate or, if the full yield curve is used, the second spot rate entered.

The use of forward rates is consistent with the manner in which corporate bonds are valued in the marketplace. The principles underlying these calculations are illustrated below.

Note that some actuaries wish to use the effective interest rate to approximate the gain/loss. While a roll-forward using the effective interest rate does not represent the true expected value of liabilities at a future date, a workaround for this approximation, as well as discussion of its limitations, may be found at the end of this article.

 

Interest Rates used for Bond Pricing

PPA interest rates are based on corporate bond yield curves. The target liability represents the total cost of corporate bonds that could be purchased to cover the cash flows of the plan. In order to determine the expected target liability one year or more from the valuation date, we must look at the expected value of these bonds at a future date. The expected value of these bonds one year out, for example, can be determined by rolling forward the current value with the holding period return in the first year.

Chapter 15 of Investments by Bodie, Kane, and Marcus states that, where future interest rates are known, “…all bonds must offer identical rates of return over any holding period…. In fact, despite their different yields to maturity, each bond will provide a rate of return over the coming year equal to this year’s short interest rate.” This concept is illustrated by example in the book: if short rates (the interest rate for a given time interval) are expected to be 8% and 10% in the first two years, respectively, then a zero-coupon bond of $1,000 maturing one year from now will sell today for $925.93 ($1,000 / 1.08) and a zero-coupon bond of $1,000 maturing two years from now will sell for $841.75 ($1,000 / [1.08 * 1.10]). Because the second short rate is expected to be 10%, by definition the second bond is expected to sell one year from now for $909.09 ($1,000 / 1.10), which is equivalent to $841.75 * 1.08. In other words, the expected value of the future bond can be determined by rolling today’s value forward with the first short rate.

The book defines the short rate as the interest rate for a given time interval. In practice, future short rates are unknown, as they are the interest rates for future points in time. Forward rates are defined as the future short rates inferred from the current bond prices. These rates are used to determine expected bond prices and, accordingly, will be used to determine expected liabilities in a gain/loss calculation.

 

Converting Spot Rates into Forward Rates

Spot rates and forward rates are different, but equivalent*, ways to express interest discount. This is illustrated by example in the section below. Spot rates are converted to discount rates, then to forward rates at the valuation date. The steps are reversed to convert forward rates back into spot rates one year later. The formulas used to derive forward rates from spot rates are:

Discount (t) = 1 / [1 + SpotRate (t)] ^t

ForwardRate (t) = [Discount (t) / Discount (t+1)] - 1

 *While theoretically equivalent, there is a small difference in annuity payment values, resulting from use of the standard “11/24” adjustment for monthly payments. Because this does not come into play with gain/loss roll-forwards, it is ignored in this discussion.

 

Implied Future Spot Rate Curves at Future Dates

Inherent in a spot rate curve at a valuation date are implied future spot rate curves at each future valuation date. These curves can be derived by converting the spot rate curve into forward rates at the valuation date and then converting the forward rates back into spot rates at a future date. This can be seen, in part, in the example above, taken from Investments. A more extensive illustration, using segment rates, is provided below.

This example converts segment rates of 4%, 5% and 6% into forward rates and then subsequently converts these into an implied spot rate curve one year after the valuation date.

 

Valuation Date					One Year Following Valuation Date
(a)	(b)	(c)	(d)		(e)	(f)	(g)	(h)
		1/((1+b)^a)	(c(t)/c(t+1))-1			D	g(u-1)/(1+f(u-1))	((1/g)^(1/e))-1
Payment	Spot	Discount	Forward		Payment	Forward	Discount	Implied Future
Year (t)	Rate	Factor	Rate		Year (u)	Rate	Factor	Spot Rates
0	N/A**	1.0000	4.00%
1	4.00%	0.9615	4.00%		0	4.00%	1.0000	N/A**
2	4.00%	0.9246	4.00%		1	4.00%	0.9615	4.00%
3	4.00%	0.8890	4.00%		2	4.00%	0.9246	4.00%
4	4.00%	0.8548	9.10%		3	9.10%	0.8890	4.00%
5	5.00%	0.7835	5.00%		4	5.00%	0.8149	5.25%
6	5.00%	0.7462	5.00%		5	5.00%	0.7761	5.20%
7	5.00%	0.7107	5.00%		6	5.00%	0.7391	5.17%
8	5.00%	0.6768	5.00%		7	5.00%	0.7039	5.14%
9	5.00%	0.6446	5.00%		8	5.00%	0.6704	5.13%
10	5.00%	0.6139	5.00%		9	5.00%	0.6385	5.11%
11	5.00%	0.5847	5.00%		10	5.00%	0.6081	5.10%
12	5.00%	0.5568	5.00%		11	5.00%	0.5791	5.09%
13	5.00%	0.5303	5.00%		12	5.00%	0.5515	5.08%
14	5.00%	0.5051	5.00%		13	5.00%	0.5253	5.08%
15	5.00%	0.4810	5.00%		14	5.00%	0.5003	5.07%
16	5.00%	0.4581	5.00%		15	5.00%	0.4764	5.07%
17	5.00%	0.4363	5.00%		16	5.00%	0.4537	5.06%
18	5.00%	0.4155	5.00%		17	5.00%	0.4321	5.06%
19	5.00%	0.3957	26.92%		18	26.92%	0.4116	5.06%
20	6.00%	0.3118	6.00%		19	6.00%	0.3243	6.11%

**  By definition, the discount factor at t=0 is 1, so the corresponding spot rate is not applicable for purposes of converting between spot rate and forward rate curves.

As is demonstrated above in columns (d) and (f), the forward rates that exist today are identical, no matter which year is being considered (given the fact that all analysis of future years is being done as of the current date). The implied future spot rate curve, however, changes in future years (as seen by comparing columns (b) and (h)). In fact, (h) no longer “breaks” at the fifth and twentieth year, as (b) does. A true one year gain or loss would reflect no change whatsoever from the beginning-of-period to end-of-period interest rate assumptions. Therefore, in order to truly value a one year gain/loss on these liabilities, the end-of-period valuation would require that column (h) be input as a full yield curve. The “implicit assumption change” interest rate calculations in ProVal are performed assuming the user had input the same segment rates at end-of-period as those input for the beginning-of-period valuation (4% for 5 years, 5% for 15 years, 6% thereafter). ProVal will calculate the expected end-of-period liability once using the future spot rate curve implied by the beginning-of-period interest rate assumption, and once as though the beginning-of-period interest rate assumption had been input as the end-of-period interest rates. The difference in liability between these two calculations is included in the “implicit assumption change” source bucket. For additional reference, see the discussion of gain/loss elements under the labels “End-of-period Valuation(s), re-executed using assumptions consistent with the beginning of period” and “Implicit assumption changes” in the Technical Reference article entitled “Gain and loss analysis: a conceptual framework.

 

Gain/Loss Approximation using Effective Interest Rate

As previously mentioned, some actuaries would prefer to approximate the gain/loss using the effective interest rate to determine expected target liabilities. This can be accomplished by calculating a pure unit credit liability, under the Actuarial Liability topic of Valuation Assumptions, entering the effective interest rate as the (single) interest rate value (Use alternative interest rate parameter); however, the following limitations should be noted:

1. The effective interest rate of the target liability is generally not the same as the effective interest rate of the target normal cost. Therefore, computing unit credit actuarial liabilities at the effective interest rate will result in a normal cost different from the one generated for funding purposes.

2. The effective interest rate at end-of-period might be different from the effective interest rate at beginning-of-period. Therefore, in order to properly analyze gain/loss results using the effective interest rate, the end-of-period Valuation(s) should also be run using the beginning-of-period effective interest rate. Where effective interest rates differ between the beginning and the end of the period, doing this will result in an end-of-period liability for gain/loss purposes that differs from the liability produced by the “actual” end-of-period Valuation(s) used for funding purposes.