FINS5513 Chap.8 Performance Evaluation and Bond Pricing
Performance Evaluation and Bond Pricing
Week 8 joins two applications of present and benchmark-relative value. Performance evaluation asks whether a realised return compensated investors for the risk that was taken. The Sharpe ratio uses total standard deviation and is suited to a complete portfolio. The Treynor ratio uses beta and is more natural when the evaluated portfolio is one diversified component of a broader portfolio.
Jensen's alpha compares realised or average return with a model-implied required return, while the information ratio scales active return by tracking error or residual risk. M-squared converts a Sharpe comparison into percentage-return units by levering or de-levering a portfolio to the benchmark's volatility. These measures can rank managers differently because their risk denominators and benchmarks differ.
Attribution then separates allocation choices from security selection relative to benchmark sector weights and returns. The fixed-income block treats a bond as a schedule of promised cash flows. Price is the present value of coupons and principal discounted at a rate consistent with timing and compounding.
When coupon rate exceeds yield to maturity, a conventional bond trades above par; when coupon rate is below yield, it trades below par. Yield to maturity is an internal rate based on promised cash flows, not a guaranteed realised return. Realised yield also depends on reinvestment rates, sale price, default and horizon.
What this chapter covers
- 01
Sharpe ratio for total risk and Treynor ratio for systematic risk
- 02
Jensen alpha and information ratio relative to stated benchmarks
- 03
M-squared as a volatility-matched return comparison
- 04
Benchmark-relative allocation and selection attribution
- 05
Bond cash-flow timelines with coupons and redemption value
- 06
Price as discounted value using consistent periodic units
- 07
Premium, par and discount bonds as coupon-yield relationships
- 08
YTM assumptions versus holding-period realised return
Worked example · free
Price a premium coupon bond
- +1The annual coupon is 6% times $1,000, which equals $60, and there are four annual discount periods.
- +1Discount the coupons: $60 times the four-period annuity factor at 5%, which is approximately $212.76.
- +1Discount principal: $1,000 divided by 1.05 to the fourth power, which is approximately $822.70.
- +1Total price is approximately $1,035.46. It is above par because the 6% coupon rate exceeds the 5% market yield.
Key terms
- Sharpe Ratio
- Excess portfolio return divided by total return standard deviation.
- Treynor Ratio
- Excess portfolio return divided by portfolio beta.
- Jensen Alpha
- Return above or below the amount implied by a specified asset-pricing benchmark.
- Tracking Error
- Standard deviation of portfolio return relative to its benchmark return.
- Yield to Maturity
- The single discount rate equating promised bond cash flows with current price.
- Coupon Rate
- Annual contractual coupon as a proportion of bond face value.
Performance Evaluation and Bond Pricing FAQ
Why can performance rankings disagree?
Measures use different definitions of risk and different benchmarks. A concentrated fund may look weak on total volatility but stronger on beta, while benchmark choice can materially change alpha and active-risk calculations.
When is the Sharpe ratio preferable?
Use it when the evaluated portfolio represents the investor's complete risky position because all volatility matters. It is also useful for comparing portfolios that may not be well diversified against idiosyncratic risk.
What does M-squared add?
It expresses a Sharpe-based comparison in return units. The portfolio is conceptually scaled to the benchmark's volatility, making the risk-adjusted difference easier to communicate than a unitless ratio alone.
How do allocation and selection differ?
Allocation reflects overweights and underweights across sectors relative to the benchmark, while selection reflects choosing securities whose returns differ from their sector benchmark. Exact interaction treatment depends on the stated attribution convention.
Why do bond prices fall when yields rise?
Existing promised cash flows become less valuable when discounted at a higher required return. The cash amounts have not increased, so their present values and therefore the bond price decline.
Is yield to maturity a promised holding return?
No. It is the discount rate embedded in price under the promised cash-flow schedule. Realising it generally requires holding to maturity, avoiding default and reinvesting coupons at the same yield.
Exam move
For performance questions, first identify the evaluated object and benchmark. If the portfolio is complete, total volatility and the Sharpe ratio deserve attention. If it is a diversified component, beta-based Treynor may be informative. If the question is explicitly benchmark-relative, calculate alpha or the information ratio and state the model or index used. Keep expected, average and realised returns distinct.
In attribution tables, record portfolio weight, benchmark weight, portfolio return and benchmark return before applying the course convention; then reconcile the effects to active return. For bonds, replace verbal descriptions with a dated cash-flow line. Convert coupon rate into a cash coupon, translate maturity into the correct number of periods, match periodic yield to cash-flow frequency, and discount every amount.
Check the answer against the coupon-rate versus yield relationship: premium, par or discount. When discussing YTM, list its reinvestment, holding-horizon and default assumptions. A complete response reports the number, explains the economic sign and states why the chosen evaluation measure or yield concept fits the decision.
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