The arithmetic takes one formula; deciding which yield to weight decides whether the answer means anything.
The weighted average yield of a bond or credit portfolio is each holding's yield multiplied by its share of the money invested, summed: =SUMPRODUCT(cost, yield)/SUM(cost). On an illustrative $500,000 portfolio of seventeen credit lines it is 7.46 per cent. Weight the expected yields instead, after default losses, and it is 6.25 per cent; the portfolio went on to earn 6.02. The arithmetic is easy. Choosing which yield to weight is the whole question.
Worked in full in The Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The number on the screen is the cost-weighted displayed yield: yield to maturity for bonds, the distribution rate for funds and BDCs. It mixes measures that do not mean the same thing, and it treats a CCC bond's 10.06 per cent as income. Weighting the right figure takes the same three columns.
| Line | Rating | Cost ($) | Weight | Displayed | Expected |
|---|---|---|---|---|---|
| Aldermoor Utilities | A | 25,000 | 5.00% | 4.80% | 4.77% |
| Carrow Rail | A | 25,000 | 5.00% | 4.95% | 4.92% |
| Hesketh Foods | BBB | 25,000 | 5.00% | 5.15% | 5.03% |
| Penrith Medical | BBB | 25,000 | 5.00% | 5.30% | 5.18% |
| Stanwick Networks | BBB | 25,000 | 5.00% | 5.45% | 5.33% |
| Blakeney Packaging | BB | 12,375 | 2.48% | 6.99% | 6.39% |
| Dunmore Leisure | BB | 12,312.5 | 2.46% | 7.36% | 6.76% |
| Fenwick Chemicals | B | 12,250 | 2.45% | 7.74% | 5.94% |
| Garside Auto Parts | B | 12,187.5 | 2.44% | 8.12% | 6.32% |
| Marlow Home Brands | B | 12,125 | 2.43% | 8.75% | 6.95% |
| Harrowgate Pneumatics | CCC | 11,750 | 2.35% | 10.06% | 1.06% |
| Brindle Floating Rate Fund | loan fund | 75,000 | 15.00% | 6.65% | 5.60% |
| Ledgerwood Capital | BDC | 25,000 | 5.00% | 10.11% | 9.80% |
| Tamsin Lending | BDC | 25,000 | 5.00% | 10.67% | 10.77% |
| Corlett Credit | BDC | 25,000 | 5.00% | 14.67% | 8.98% |
| Ashcombe Interval Fund | interval fund | 100,000 | 20.00% | 9.00% | 7.19% |
| Treasury bills | reserve | 52,000 | 10.40% | 4.00% | 4.00% |
| Portfolio | 500,000 | 100% | 7.46% | 6.25% |
Weighted average yield = Σ (costi × yieldi) ÷ Σ costi
Expected yield of a bond = yield to maturity − PD × (1 − recovery)
Excel, costs in C2:C18, yields in E2:E18: =SUMPRODUCT(C2:C18,E2:E18)/SUM(C2:C18). For a portfolio already held, weight by market value, not cost.
Step 1, weights. Each line's cost over $500,000. The high-yield bonds were bought below par, so their weight is their price, not their face: Harrowgate's $12,500 of face cost $11,750, 2.35 per cent.
Step 2, displayed. The products of cost and displayed yield sum to $37,292 a year; divided by $500,000 that is 7.46 per cent.
Step 3, expected. For the bonds, take off the annual default probability times the loss given default at a 40 per cent recovery: 0.03 points on an A bond, 1.80 on a B bond, 9.00 on the CCC. For the loan fund, take off 1.05 points of credit losses (a 3 per cent default rate at 35 per cent loss). For the BDCs and the interval fund use income less losses from their fund models. The products sum to $31,250: 6.25 per cent, $6,042 a year less than the screen.
| Sleeve | Weight | Displayed | Expected | Points of the portfolio gap |
|---|---|---|---|---|
| Investment grade | 25.0% | 5.13% | 5.05% | 0.02 |
| High yield | 14.6% | 8.15% | 5.60% | 0.37 |
| Loan fund | 15.0% | 6.65% | 5.60% | 0.16 |
| BDCs | 15.0% | 11.81% | 9.85% | 0.29 |
| Interval fund | 20.0% | 9.00% | 7.19% | 0.36 |
| Reserve | 10.4% | 4.00% | 4.00% | 0.00 |
| Portfolio | 100% | 7.46% | 6.25% | 1.21 |
Two lines, Harrowgate and Corlett, hold 7.35 per cent of the money and produce 41 per cent of the gap: 0.21 and 0.28 points. The highest yields on the screen are the ones that overstate most, which is why the case for each is worked separately in the yield after expected default losses and whether a BDC below NAV is cheap.
| Method | Portfolio yield |
|---|---|
| Simple average of the seventeen displayed yields | 7.63% |
| Weighted by effective annual yield (bonds semiannual, funds quarterly) | 7.68% |
| Weighted by face rather than cost | 7.46% |
| Weighted by cost, displayed | 7.46% |
| Weighted by cost, expected | 6.25% |
| Realised: the portfolio's three-year IRR, base case | 6.02% |
The simple average overweights small, high-yielding lines. Restating everything as an effective annual rate, the basis an IRR uses, adds 0.22 points: bond yields are quoted semiannually and fund distributions are paid quarterly, so 7.46 and an IRR are not on the same footing. Weighting by face barely matters here because the discounts are small.
Strictly, a portfolio's yield is the single rate that discounts all its cash flows to its cost. A value-weighted average of the line yields is the standard approximation; it is close when durations are similar and drifts when they are not. And IRRs do not add: the cost-weighted average of the seventeen realised line IRRs is 5.78 per cent, while the portfolio's IRR is 6.02.
Weighting the displayed yields and calling the result the portfolio's return. It is a weighted average of promises with different meanings: a yield to maturity that assumes no default, distribution rates that include return of capital and ignore NAV erosion, and a Treasury bill rate that is what it says. On this portfolio that overstates by 1.21 points on expected losses alone, and by 1.44 against what was earned.
The seventeen lines, their expected yields and the twelve-quarter cash flows are in the free workbook for this case.
By the money at stake: cost on the day of purchase, market value afterwards. Face value overweights bonds bought below par. In the illustrative portfolio the high-yield discounts are small, so face weighting and cost weighting both give 7.46 per cent, but on a book of distressed bonds the difference can be several points.
Because displayed yields assume no defaults and fund distribution rates ignore NAV erosion. Weighting expected yields, after default losses, takes the illustrative portfolio from 7.46 to 6.25 per cent; it actually earned 6.02. Two lines, a CCC bond and a high-yielding BDC, hold 7.35 per cent of the money and explain 41 per cent of the gap.
Only if every position is the same size. A simple average of the seventeen displayed yields gives 7.63 per cent against 7.46 weighted, because small high-yield lines count as much as the $100,000 interval fund. Use SUMPRODUCT of weights and yields instead.
This article is one calculation from The Credit Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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