Risk factors in the look-through: methodology
The “Risk factors” tab in the look-through measures how sensitive a portfolio is to interest rates, credit spreads and the equity market, instead of simply following the rule “stock = beta 1, bond = duration”. This methodology for the risk factors shows how Portfolio Lab derives duration, key rates, spread duration and beta from holdings lists and returns. The basics are covered in the article Duration.
Duration per position
Duration measures roughly by what percentage the price of a bond falls when interest rates rise by one percentage point. The measure goes back to Macaulay (1938); Fisher and Weil (1971) showed how to use it to immunise bond portfolios against interest rate changes. Wherever possible, Portfolio Lab takes the duration of each bond directly from the fund's holdings list (providers usually report an effective or modified duration). If it is missing, Portfolio Lab calculates it from maturity and coupon, as for a bond with a fixed annual coupon:
modified duration = [Σ t · PV(payment_t) / Σ PV(payment_t)] / (1 + y)- t
- time of each payment in years (coupons, plus the redemption at the end)
- y
- yield from the file, otherwise equal to the coupon, or 3% if there is no coupon
Each duration is multiplied by the position's weight in the fund and summed by currency (euro, dollar, pound, yen, other). The result is contributions in years per unit of fund assets; a fund with 50% bonds of duration 8 contributes 4 years. In the portfolio, the contributions are added up using the fund weights. Cash and money market holdings in the fund carry interest rate duration (usually almost zero) but no credit risk.
Key rate durations
A single duration figure assumes that the whole yield curve shifts in parallel. In reality, short and long rates move differently. Litterman and Scheinkman (1991) found three patterns (level, slope and curvature) that explain almost all movements. Key rate durations following Ho (1992) therefore show the interest rate risk at each maturity point. Portfolio Lab uses the points 1, 2, 3, 5, 7, 10, 15, 20 and 30 years.
Each payment of a bond is assigned its share of the duration (time × present value, normalised) and distributed linearly between the two neighbouring points; payments before one year go to the 1-year point, and payments after 30 years to the 30-year point. This way, the key rate durations add up exactly to the duration. Without a maturity date, the code treats the bond like a zero-coupon bond with a maturity equal to its duration.
Spread duration by issuer
Besides the level of interest rates, the credit spread matters: how much does a bond fall when its spread over government bonds widens? Portfolio Lab assigns each bond to an issuer type based on its sector and name (covered bonds take precedence over agencies, and agencies over sovereigns; everything else counts as corporate):
| Issuer type | Counts as spread risk? |
|---|---|
| Government bonds (in home currency) | no, rate risk only |
| Government bonds in foreign currency (e.g. Mexico in dollars) | yes |
| Agencies & supranationals (development banks, EU, World Bank) | yes |
| Covered & secured bonds | yes |
| Corporate bonds | yes |
The spread duration is the duration of all bonds except government bonds in their home currency. For fixed coupons, it is almost equal to the interest rate duration; floating-rate notes have hardly any interest rate duration but do have spread duration. Without a separate figure in the file, the two cannot be told apart.
Estimates without holdings data
Funds without position-level data (such as asset classes, or funds for which only the largest holdings are known) receive an explicitly flagged estimate based on the maturity segment in the taxonomy:
| Maturity segment | Assumed duration |
|---|---|
| Money market | 0.2 |
| 0–1 y | 0.5 |
| 1–3 y | 1.9 |
| 3–5 y | 3.8 |
| 5–7 y | 5.6 |
| 7–10 y | 7.8 |
| 10–20 y | 12.5 |
| 20+ y | 17.0 |
| mixed / unknown | 6.5 |
Bond asset classes with a fixed maturity (such as ten-year German Bunds) are calculated from yields; for these, Portfolio Lab assumes a bond at par with a 3% coupon and exactly this maturity (ten years: duration 8.53). Mixed bond funds without a holdings list split their spread duration half and half between sovereigns and corporates. The display hides contributions below 0.005 years.
Equity beta from the regression
For equity risk, Portfolio Lab regresses the monthly returns of each position in the base currency on Global equities (the asset class @AKTIEN_WELT), over the last 60 common months (at least 24), with an intercept, using ordinary least squares. The beta is the slope, as in the CAPM of Sharpe (1964); in addition, the page shows the alpha per year ((1 + α)¹² − 1), R² and the t-statistic of the slope. The portfolio beta is the weighted sum of the betas across all positions that have a beta.
β = Cov(r_fund, r_market) / Var(r_market) (last 60 months, at least 24)For comparison, the page shows the equity share from the look-through, i.e. the simple rule “beta 1 per euro in equities”. If the regression beta deviates significantly, it is worth taking a closer look: an equity fund with a beta of 0.7 fluctuates less than the world market (defensive stocks, a different region), while a mixed fund with a beta above its equity share has hidden equity risks.
Limitations
- Options embedded in bonds (call features, mortgage-backed securities) make duration change with interest rates; without an effective duration from the file, the formula underestimates the risk.
- Key rates are distributed according to cash flows, not according to a term structure model. That is sufficient for comparisons between funds, but not for hedging.
- Beta fluctuates over time; 60 months smooth it out but react slowly to changes.
Frequently asked questions
What is the difference between duration and spread duration?
Duration measures the response to a change in the general level of interest rates, spread duration the response to a change in the credit spread. For government bonds in their home currency, only the former counts.
Why are key rates useful?
Because yield curves twist and bend. Two funds with the same duration can react quite differently when short rates fall and long rates rise.
How does Portfolio Lab know the duration of an ETF?
From the provider's holdings list, if it has a duration column; otherwise from the maturity and coupon of each bond. Without a list, the duration is estimated and flagged as such.
Why does the beta differ from the equity share?
The equity share counts every euro in equities as beta 1. The regression measures how strongly the fund actually moved with the world market. Region, sectors and currency all play a part.
Literature
- Fisher, Lawrence; Weil, Roman L. (1971): Coping with the Risk of Interest-Rate Fluctuations: Returns to Bondholders from Naive and Optimal Strategies. The Journal of Business 44 (4). doi:10.1086/295402
- Ho, Thomas S.Y. (1992): Key Rate Durations. The Journal of Fixed Income 2 (2), pp. 29–44. doi:10.3905/jfi.1992.408049
- Litterman, Robert B; Scheinkman, Josè (1991): Common Factors Affecting Bond Returns. The Journal of Fixed Income 1 (1), pp. 54–61. doi:10.3905/jfi.1991.692347
- Macaulay, Frederick R. (1938): Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856. National Bureau of Economic Research. Source
- Sharpe, William F. (1964): Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance 19 (3), pp. 425–442. doi:10.1111/j.1540-6261.1964.tb02865.x
This page describes how Portfolio Lab calculates; past results are not a forecast. All terms in the glossary.