Financial management at Danmarks Nationalbank

The following reviews the key calculations in connection with the compilation and decomposition of performance. This supplements Chapter 13 on measuring performance.

Calculation of index

Both the old and the new performance calculations are based on the following type of return index:

The index is a multiplicative time-weighted index, cf. Chapter 13 on performance. The return on the portfolio is calculated on a daily basis and multiplied over time to give an index for the return over the entire period. The index is robust vis-à-vis flows of funds to and from the portfolio, cf. below. The percentage return r_t from t-1 to t can be re-written from (1) to the following expression:

On the basis of (2) it is possible to describe the calculation of the index by considering the following situations:

If no instruments are traded, ΔN^s_t=T^s_t=0, and the return stems solely from changes in the dirty price and any coupon payments on the instrument on a settlement date.

If ΔN^s_t=0, i.e. no change in any of the nominal holdings, but T^s_t≠ 0, the same paper has been bought and sold during the day and the contribution to the return is the result of price changes during the day.

If ΔN^s_t≠0 and T^s_t≠0, the nominal holdings have changed, and the return is affected if the closing price P^S_tdeviates from the bid or offer prices for the instrument during the day. This means that purely nominal changes in the holdings from time t-1 to t do not affect the return index. The index can therefore resist changes in the holdings over time.

If all holdings of an instrument s are removed from the portfolio, i.e. ΔN^s_t=-N^S_t-1, the numerator is reduced to -T^S_t-N^S_t-1P^S_t-1 for the instrument in question. In this case, the numerator equals the proceeds from the sale less the holding's market value on the previous day.

If a new instrument is bought for the portfolio, the numerator equals N^S_tP^S_t-T^S_t, which is the gain from the time of purchase until closing.

The last term in the numerator in (2) can be called the timing effect. It is positive if the dealer has bought or sold at the right time during the day.

It is important to note that the denominator in (1) and (2) assumes that changes in the portfolio size during day t cannot generate a return on the same day, i.e. interest or capital gains, and/or the portfolio change is not at the disposal of the dealer until time t+1.

This is best illustrated by means of an example. Assume that the holding doubles from DKK 100 million to DKK 200 million from time t-1 to t. The denominator equals MV_t-1=DKK 100 million. If the dealer can invest the increase in the holding during the day and generate a positive return, e.g. as a result of favourable price development, this return will be included in the numerator via the element P^S_tΔN^S_t-T^S_t. If this is the case, the denominator is too low, since the return on the portfolio is generated by invested capital exceeding DKK 100 million at time t-1, so the return on the portfolio is overestimated. Kjeldsen (1996) gives a detailed discussion of this problem and its solutions, cf. the literature reference list for Chapter 13.

Decomposition of return as direct return and market development

The following example considers a portfolio with no movement in the holding comprising one bond of the bullet issue type, i.e. the entire bond is redeemed on expiry. (2) can be reduced to the following formula (the index s is dropped initially, and it is explicitly stated that the market value equals the dirty price of the bond):

The return can be written as the change in the dirty price plus any coupon payments C_t as a ratio of the dirty price on the previous day. The focus in the following is first on the absolute return ΔV (in DKK) expressed as:

This return can be decomposed in various ways, initially as:

direct return "carry", ΔV_direct
market development, ΔV_market

ΔV_direct is here defined as the coupon payment with the addition of the mathematical price adjustments due to the reduction of maturity. ΔV_direct can be written as:

The first term is the change in accrued interest AI with the addition of any settlement payments C_t on day t. The last term is the change in the theoretical clean price P_tfrom t-1 to t under unchanged market conditions, the so-called maturity reduction. This term is determined by pricing the bond at time t and t-1 assuming unchanged yield curves and spreads via the function P^clean(M_t,t), where M_t represents the chosen input of market data and t the time of the pricing. Note that, maturity-reduction (also known as pull to par) as defined here contains both the effect of pure maturity reduction and a possible effect of changes in interest rates, if the yield curve is not flat (roll-down). For example, if the yield curve has a positive slope and the bond is priced under par, both the maturity reduction in isolation, and the effect of roll-down will draw towards a higher price^[1].

The part of the return attributable to the market development is calculated as the difference between the theoretical clean price at time t and t-1 less the effect of the maturity reduction, i.e.:

It should be noted that the prices in (5) are the observed stock-exchange prices (not the theoretical prices). That (4) + (5) equals (3) can be seen from the following:

Detailed decomposition of respectively direct return and market development

The coupon payment ΔV_coupon, the maturity reduction ΔV_{maturity reduction} and the return due to the market development ΔV_market can be broken down into several components, cf. below.

The coupon payment ΔV_coupon can be broken down as:

coupon according to the general yield structure ΔV_{yield structure, coupon}
coupon premium due to sectoral spread ΔV_{sectoral spread, coupon}
coupon premium due to instrument-specific spread ΔV_{instrument-specific spread, coupon}

The maturity reduction ΔV_{maturity reduction}can be broken down as:

maturity reduction across the general yield structure ΔV_{yield structure, maturity reduction}
a premium due to maturity reduction along the sectoral curve ΔV_{sectoral spread, maturity reduction}
a premium due to maturity reduction along the instrument-specific curve ΔV_{instrument-specific spread, maturity reduction}

The market development ΔV_market can be broken down as:

contribution from parallel shift in the general yield structure, ΔV_{yield structure, parallel shift}
contribution from change in the shape of the general yield structure, ΔV_{yield structure, change in shape}
contribution from development in the sectoral spread, ΔV_{sectoral spread, development}
contribution from development in the bond-specific spread, ΔV_{instrument-specific spread, development}

The breakdown of the coupon payment ΔV_coupon into a coupon according to a general yield structure, e.g. the government-yield curve, and subsequent coupon premiums attributable to a sectoral or instrument-specific spread is as follows. First the coupon k_{yield structure, derived} that, at the current general yield structure equals the theoretical stock-exchange price to the observed stock-exchange price, is calculated. This means that the coupon k_{yield structure, derived} is found by solving the following equation:

The calculation is then repeated using the relevant sectoral curve. For a corporate bond the yield structure for corporate bonds with the same rating is used. The difference between the two calculated coupons is the coupon premium attributable to the spread between the government and corporate bond curves:

The coupon premium due to the sectoral spread is most often positive and can be interpreted as the extra coupon payment to an investor as compensation for investing in e.g. a corporate bond with a higher credit risk rather than in a government bond. In addition, the corporate bond will include an instrument-specific coupon premium, since in practice the bond is never traded on the corporate-bond curve. This instrument-specific premium can be positive or negative and may be the result of company-specific factors such as gain or a possible liquidity premium on the bond. The instrument-specific premium is determined residually as:

where k is the actual coupon on the corporate bond. The two coupon premiums and the coupon according to the general yield structure add up by construction to the total coupon. The total coupon payment in absolute terms can therefore be written as:

where the individual payments are calculated on the basis of the calculated coupons. Besides the coupon payment the direct return also comprises the price development due to the reduction of maturity, cf. (4). The maturity reduction can be calculated along various yield curves. The starting point is again a general yield structure such as the zero-coupon yield curve for government bonds. The price development along this curve can be calculated as:

It should be noted that the calculated reduction of maturity comprises two effects. Firstly, the bond is brought closer to maturity, and secondly other discounting factors are applied if the curve is not flat. Viewed in isolation, the first effect will push the price towards par, while the price effect of changed discounting factors depends on the shape of the yield curve. The calculation in (11) can be repeated using a relevant sectoral curve, cf. above, and finally by using an instrument-specific curve. The instrument-specific curve can be determined residually by means of parallel shift of the sectoral curve whereby the bond is priced on the curve. The maturity reduction along the instrument-specific curve is an estimate of the bond's true price development as a consequence of the maturity reduction, and can likewise be written as a sum of 3 factors:

The only remaining element to be decomposed is the market development ΔV_market. The market development can first be written as follows, cf. (5) and (12) :

In (13) the market development is divided into the share attributable to the development in the general yield structure, the share attributable to the development in the sectoral spread and finally, the share attributable to the development in the instrument-specific spread. It should be noted that the individual contributions from the reduction of maturity, cf. (12) are deducted in the calculation of the contributions from the market development. That way the reduction of maturity is included in the calculation of the direct return on the bond.

The effect of the development in the general yield structure,

ΔV_{yield structure, development} can be further broken down as parallel shift and shape:

where P(M_t^parallel,t) is the notional stock-exchange price on a parallel shift in the general yield structure from t-1 to t. The parallel shift can be calculated in several ways. Box 13.3 in Chapter 13 shows an example of parallel shift of the curve by the average yield change in the 2-, 5- and 10-year segments from time t to t-1. V_shapeis finally calculated residually as the difference between the notional stock-exchange price at the actual yield curve and at the parallel-shifted curve.

The total decomposition of the return can be written as:

Written as a relative return, (15) is as follows:

The total result of the decomposition is presented as a table and described in further detail in Chapter 13.

The decomposition in (15) is additive in the individual components. Alternatively a multiplicative decomposition of the return can be applied. The multiplicative decomposition instead gives the following expression (the calculations are not described in further detail here)::

Decomposition of total return and performance

The decomposition of the return on the individual bond can now be used in a decomposition of the return on the total portfolio. The total return in DKK from t-1 to t can be written as:

where ΔV is now the return on the entire portfolio, and s refers to the individual bonds/instruments in the portfolio. (16) can be re-written as a relative return from t-1 to t:

where ω_S is now the weight of bond s in the portfolio at the time t-1. The weight is calculated as the bond's share of the market value of the portfolio. The relative return r on the portfolio can thus be calculated as a weighting together of the individual components of the return. The performance P, i.e. the difference between the return on the actual portfolio and on the benchmark portfolio, from t-1 to t can be calculated as follows:

where Δω_S is the difference between the weight of instrument s in respectively the actual portfolio and the benchmark portfolio. The performance from t-1 to t can in other words be calculated as a weighted average of the components of the return on the individual instruments. Performance is the result of differences in the weights of the individual instruments in respectively the actual portfolio and the benchmark portfolio. Furthermore, the timing effect is added in (18) for the sake of completion, cf. (2). As stated, the timing effect is the effect of gains/losses from intraday sales/purchases.

Aggregation of sub-components over time

The calculations above show how the total return from period t-1 to t can be broken down into various sub-components which together add up to the total return r_t in the period. However, the measurement of performance focuses on the return over a longer period, for which the return is measured via the time-weighted index. The time-weighted index can be written as follows, cf. (1):

In (1') the period returns are multiplied over time, leading to the return over the entire period, i.e. r_0,T. The question is now whether the total return for the period can be decomposed like the returns for individual periods. The return for an individual period from t-1 to t can be written as follows (17'):

where rc^s,t_j is the return component j on bond s in period t weighted using the bond's portfolio share, i.e. rc^s,t_j =ω_Sr^s,t_j.

A first attempt to decompose the total return over time r_0,Twould be to sum up the sub-components in (17') for the entire period. However, the problem is that the time-weighted index is a multiplicative index, whereby simply summing up the sub-components in (17') does not yield r_0,T. This problem can be solved by scaling the individual sub-components in the following way:

In (19) the sub-components from each period t are scaled using the return index up to and including the preceding period, i.e. (1+r_0,t-1). The following is shown to apply:

Where rc^0,T_j is now the total contribution to the return over the period from sub-component j. (20) shows that the total return for the entire period is the sum of the new scaled sub-components, which can also be summed up over the period. Scaling the original return contributions thus makes it possible to achieve a decomposition that sums up to the total return for the period over time.

[1] Unchanged market conditions are here defined as a situation where the yield curves are unchanged. Alternatively it could be defined as where the forward curves are realised. If the last definition is used the "maturity-reduction" could fx be calculated as P^clean(M_t-1,t , t) - P^clean(M_t-1, t), where M_t-1,t represent the forward curves at time t implied from the yield curves at time t-1. In such a context "maturity-reduction" contains both effects from pure maturity-reduction, roll-down on the curve and development in forward curves.


		Publication overview - Contents - Top/Bottom - Previous/Next
D. Index for Performance and Decomposition
	The following reviews the key calculations in connection with the compilation and decomposition of performance. This supplements Chapter 13 on measuring performance. Calculation of index Both the old and the new performance calculations are based on the following type of return index: The index is a multiplicative time-weighted index, cf. Chapter 13 on performance. The return on the portfolio is calculated on a daily basis and multiplied over time to give an index for the return over the entire period. The index is robust vis-à-vis flows of funds to and from the portfolio, cf. below. The percentage return r_t from t-1 to t can be re-written from (1) to the following expression: On the basis of (2) it is possible to describe the calculation of the index by considering the following situations: If no instruments are traded, ΔN^s_t=T^s_t=0, and the return stems solely from changes in the dirty price and any coupon payments on the instrument on a settlement date. If ΔN^s_t=0, i.e. no change in any of the nominal holdings, but T^s_t≠ 0, the same paper has been bought and sold during the day and the contribution to the return is the result of price changes during the day. If ΔN^s_t≠0 and T^s_t≠0, the nominal holdings have changed, and the return is affected if the closing price P^S_tdeviates from the bid or offer prices for the instrument during the day. This means that purely nominal changes in the holdings from time t-1 to t do not affect the return index. The index can therefore resist changes in the holdings over time. If all holdings of an instrument s are removed from the portfolio, i.e. ΔN^s_t=-N^S_t-1, the numerator is reduced to -T^S_t-N^S_t-1P^S_t-1 for the instrument in question. In this case, the numerator equals the proceeds from the sale less the holding's market value on the previous day. If a new instrument is bought for the portfolio, the numerator equals N^S_tP^S_t-T^S_t, which is the gain from the time of purchase until closing. The last term in the numerator in (2) can be called the timing effect. It is positive if the dealer has bought or sold at the right time during the day. It is important to note that the denominator in (1) and (2) assumes that changes in the portfolio size during day t cannot generate a return on the same day, i.e. interest or capital gains, and/or the portfolio change is not at the disposal of the dealer until time t+1. This is best illustrated by means of an example. Assume that the holding doubles from DKK 100 million to DKK 200 million from time t-1 to t. The denominator equals MV_t-1=DKK 100 million. If the dealer can invest the increase in the holding during the day and generate a positive return, e.g. as a result of favourable price development, this return will be included in the numerator via the element P^S_tΔN^S_t-T^S_t. If this is the case, the denominator is too low, since the return on the portfolio is generated by invested capital exceeding DKK 100 million at time t-1, so the return on the portfolio is overestimated. Kjeldsen (1996) gives a detailed discussion of this problem and its solutions, cf. the literature reference list for Chapter 13. Decomposition of return as direct return and market development The following example considers a portfolio with no movement in the holding comprising one bond of the bullet issue type, i.e. the entire bond is redeemed on expiry. (2) can be reduced to the following formula (the index s is dropped initially, and it is explicitly stated that the market value equals the dirty price of the bond): The return can be written as the change in the dirty price plus any coupon payments C_t as a ratio of the dirty price on the previous day. The focus in the following is first on the absolute return ΔV (in DKK) expressed as: This return can be decomposed in various ways, initially as: direct return "carry", ΔV_direct market development, ΔV_market ΔV_direct is here defined as the coupon payment with the addition of the mathematical price adjustments due to the reduction of maturity. ΔV_direct can be written as: The first term is the change in accrued interest AI with the addition of any settlement payments C_t on day t. The last term is the change in the theoretical clean price P_tfrom t-1 to t under unchanged market conditions, the so-called maturity reduction. This term is determined by pricing the bond at time t and t-1 assuming unchanged yield curves and spreads via the function P^clean(M_t,t), where M_t represents the chosen input of market data and t the time of the pricing. Note that, maturity-reduction (also known as pull to par) as defined here contains both the effect of pure maturity reduction and a possible effect of changes in interest rates, if the yield curve is not flat (roll-down). For example, if the yield curve has a positive slope and the bond is priced under par, both the maturity reduction in isolation, and the effect of roll-down will draw towards a higher price^[1]. The part of the return attributable to the market development is calculated as the difference between the theoretical clean price at time t and t-1 less the effect of the maturity reduction, i.e.: It should be noted that the prices in (5) are the observed stock-exchange prices (not the theoretical prices). That (4) + (5) equals (3) can be seen from the following: Detailed decomposition of respectively direct return and market development The coupon payment ΔV_coupon, the maturity reduction ΔV_{maturity reduction} and the return due to the market development ΔV_market can be broken down into several components, cf. below. The coupon payment ΔV_coupon can be broken down as: coupon according to the general yield structure ΔV_{yield structure, coupon} coupon premium due to sectoral spread ΔV_{sectoral spread, coupon} coupon premium due to instrument-specific spread ΔV_{instrument-specific spread, coupon} The maturity reduction ΔV_{maturity reduction}can be broken down as: maturity reduction across the general yield structure ΔV_{yield structure, maturity reduction} a premium due to maturity reduction along the sectoral curve ΔV_{sectoral spread, maturity reduction} a premium due to maturity reduction along the instrument-specific curve ΔV_{instrument-specific spread, maturity reduction} The market development ΔV_market can be broken down as: contribution from parallel shift in the general yield structure, ΔV_{yield structure, parallel shift} contribution from change in the shape of the general yield structure, ΔV_{yield structure, change in shape} contribution from development in the sectoral spread, ΔV_{sectoral spread, development} contribution from development in the bond-specific spread, ΔV_{instrument-specific spread, development} The breakdown of the coupon payment ΔV_coupon into a coupon according to a general yield structure, e.g. the government-yield curve, and subsequent coupon premiums attributable to a sectoral or instrument-specific spread is as follows. First the coupon k_{yield structure, derived} that, at the current general yield structure equals the theoretical stock-exchange price to the observed stock-exchange price, is calculated. This means that the coupon k_{yield structure, derived} is found by solving the following equation: The calculation is then repeated using the relevant sectoral curve. For a corporate bond the yield structure for corporate bonds with the same rating is used. The difference between the two calculated coupons is the coupon premium attributable to the spread between the government and corporate bond curves: The coupon premium due to the sectoral spread is most often positive and can be interpreted as the extra coupon payment to an investor as compensation for investing in e.g. a corporate bond with a higher credit risk rather than in a government bond. In addition, the corporate bond will include an instrument-specific coupon premium, since in practice the bond is never traded on the corporate-bond curve. This instrument-specific premium can be positive or negative and may be the result of company-specific factors such as gain or a possible liquidity premium on the bond. The instrument-specific premium is determined residually as: where k is the actual coupon on the corporate bond. The two coupon premiums and the coupon according to the general yield structure add up by construction to the total coupon. The total coupon payment in absolute terms can therefore be written as: where the individual payments are calculated on the basis of the calculated coupons. Besides the coupon payment the direct return also comprises the price development due to the reduction of maturity, cf. (4). The maturity reduction can be calculated along various yield curves. The starting point is again a general yield structure such as the zero-coupon yield curve for government bonds. The price development along this curve can be calculated as: It should be noted that the calculated reduction of maturity comprises two effects. Firstly, the bond is brought closer to maturity, and secondly other discounting factors are applied if the curve is not flat. Viewed in isolation, the first effect will push the price towards par, while the price effect of changed discounting factors depends on the shape of the yield curve. The calculation in (11) can be repeated using a relevant sectoral curve, cf. above, and finally by using an instrument-specific curve. The instrument-specific curve can be determined residually by means of parallel shift of the sectoral curve whereby the bond is priced on the curve. The maturity reduction along the instrument-specific curve is an estimate of the bond's true price development as a consequence of the maturity reduction, and can likewise be written as a sum of 3 factors: The only remaining element to be decomposed is the market development ΔV_market. The market development can first be written as follows, cf. (5) and (12) : In (13) the market development is divided into the share attributable to the development in the general yield structure, the share attributable to the development in the sectoral spread and finally, the share attributable to the development in the instrument-specific spread. It should be noted that the individual contributions from the reduction of maturity, cf. (12) are deducted in the calculation of the contributions from the market development. That way the reduction of maturity is included in the calculation of the direct return on the bond. The effect of the development in the general yield structure, ΔV_{yield structure, development} can be further broken down as parallel shift and shape: where P(M_t^parallel,t) is the notional stock-exchange price on a parallel shift in the general yield structure from t-1 to t. The parallel shift can be calculated in several ways. Box 13.3 in Chapter 13 shows an example of parallel shift of the curve by the average yield change in the 2-, 5- and 10-year segments from time t to t-1. V_shapeis finally calculated residually as the difference between the notional stock-exchange price at the actual yield curve and at the parallel-shifted curve. The total decomposition of the return can be written as: Written as a relative return, (15) is as follows: The total result of the decomposition is presented as a table and described in further detail in Chapter 13. The decomposition in (15) is additive in the individual components. Alternatively a multiplicative decomposition of the return can be applied. The multiplicative decomposition instead gives the following expression (the calculations are not described in further detail here):: Decomposition of total return and performance The decomposition of the return on the individual bond can now be used in a decomposition of the return on the total portfolio. The total return in DKK from t-1 to t can be written as: where ΔV is now the return on the entire portfolio, and s refers to the individual bonds/instruments in the portfolio. (16) can be re-written as a relative return from t-1 to t: where ω_S is now the weight of bond s in the portfolio at the time t-1. The weight is calculated as the bond's share of the market value of the portfolio. The relative return r on the portfolio can thus be calculated as a weighting together of the individual components of the return. The performance P, i.e. the difference between the return on the actual portfolio and on the benchmark portfolio, from t-1 to t can be calculated as follows: where Δω_S is the difference between the weight of instrument s in respectively the actual portfolio and the benchmark portfolio. The performance from t-1 to t can in other words be calculated as a weighted average of the components of the return on the individual instruments. Performance is the result of differences in the weights of the individual instruments in respectively the actual portfolio and the benchmark portfolio. Furthermore, the timing effect is added in (18) for the sake of completion, cf. (2). As stated, the timing effect is the effect of gains/losses from intraday sales/purchases. Aggregation of sub-components over time The calculations above show how the total return from period t-1 to t can be broken down into various sub-components which together add up to the total return r_t in the period. However, the measurement of performance focuses on the return over a longer period, for which the return is measured via the time-weighted index. The time-weighted index can be written as follows, cf. (1): In (1') the period returns are multiplied over time, leading to the return over the entire period, i.e. r_0,T. The question is now whether the total return for the period can be decomposed like the returns for individual periods. The return for an individual period from t-1 to t can be written as follows (17'): where rc^s,t_j is the return component j on bond s in period t weighted using the bond's portfolio share, i.e. rc^s,t_j =ω_Sr^s,t_j. A first attempt to decompose the total return over time r_0,Twould be to sum up the sub-components in (17') for the entire period. However, the problem is that the time-weighted index is a multiplicative index, whereby simply summing up the sub-components in (17') does not yield r_0,T. This problem can be solved by scaling the individual sub-components in the following way: In (19) the sub-components from each period t are scaled using the return index up to and including the preceding period, i.e. (1+r_0,t-1). The following is shown to apply: Where rc^0,T_j is now the total contribution to the return over the period from sub-component j. (20) shows that the total return for the entire period is the sum of the new scaled sub-components, which can also be summed up over the period. Scaling the original return contributions thus makes it possible to achieve a decomposition that sums up to the total return for the period over time. [1] Unchanged market conditions are here defined as a situation where the yield curves are unchanged. Alternatively it could be defined as where the forward curves are realised. If the last definition is used the "maturity-reduction" could fx be calculated as P^clean(M_t-1,t , t) - P^clean(M_t-1, t), where M_t-1,t represent the forward curves at time t implied from the yield curves at time t-1. In such a context "maturity-reduction" contains both effects from pure maturity-reduction, roll-down on the curve and development in forward curves.


	Publication overview - Contents - Top/Bottom - Previous/Next

D. Index for Performance and Decomposition

Calculation of index

Decomposition of return as direct return and market development

Detailed decomposition of respectively direct return and market development

Decomposition of total return and performance

Aggregation of sub-components over time