Present Value/Underlying Present Value

Description: Measures the current value of a future cash flow

Keywords: Present Value, Time Value of Money, PV

 

  1. Note: In RiskServer 4.7 and above, the behavior of the Underlying PV statistic for Convertible Bond Option positions was changed. Previously the Underlying PV statistic returned the Present Value of the instrument. Now, Underlying PV returns the Present Value of the underlying Convertible Bond. This statistic is internally consistent with the CBO position type. It gives an indicative underlying PV but does not call on the CB pricing code.

Introduction

A given amount of money is more valuable today than at a future date. Present value is, therefore, the future value of a cash flow discounted by the time value of money. (Note that the Financial Instrument Technical Notes section of this guide provides details of valuing most instruments)

 

If one invests an amount of money, C, for t years at a rate of i% compounded annually, the present value of C, t years in the future is given by:

 

 

The expression  represents the present value of 1 and occurs in almost all present value calculations.

 

If interest rates are expected to change during t, it is also common to use these different interest rates for future time periods. For example, an investment over three years would have a present value (PV) of , where t=3 and i=the annual rate of interest in each year.

 

The choice of interest rate is particularly important and this is typically, the risk free interest rate which is obtained from the rate of return of highly liquid sovereign debt, for example, US T-Bills.  The risk free rate is used as a basis upon which to assess the risk of an investment: if the investment is “safe” it should perform at least as well as the risk-free rate, however if the investment contains some risk, this is reflected in a risk premium.

 

Importantly, present value is additive. The sum of the PV of each position in a portfolio equates to the PV of the portfolio as a whole.

Example

Assume the risk free rate to be 5.5% and a portfolio has two instruments within it

 

  1. An investment is expected to deliver a cash flow of $1,000,000 in 6 months (0.5 years) time.
  2. An investment which is expected to deliver a cash flow of $750,000 in 12 months (1 year) time.

 

The present value of cash flow 1 is:

 = 973,584.77

 

The present value of cash flow 2 is:

 = 710,900.50

 

The present value of the portfolio is PV1 + PV2 = 973,584.77 + 710,900.50 = 1,684,485

 

Underlying Present Value

Introduction

The Underlying Present Value statistic is very similar to Present Valuestatistic in that it defines a measure of worth for an instrument or portfolio. However, the current Present Valuestatistic for some two-legged instruments can often be zero or near zero. While this is correct, it does not always provide an intuitive sense of the “size” of a position. For some instruments, one leg may be much more important than the other; in which case we can take the “present value” of only that leg as indicative of the true size of the position. We call this the Underlying Present Value.

 

For example, an equity index future will have a present value equal to zero since the futures contract is marked-to-market daily.  However, the underlying present value will represent how much of the market you are invested in.  For the S&P 500 index futures, the contract specifications for the ‘big’ contract are 250 times the index.  This means that the underlying present value will show that you are controlling 250 times the current value of the S&P for each contract held.

 

For FX forwards, the underlying present value will show that you are controlling 50MM Yen while the present value reports the mark-to-market value which may be a small amount.

 

For single-legged instruments Underlying Present Value will simply be the same as Present Value. For some two-leggedinstruments. We cannot ascertain which leg is more important.

 

Underlying Present Value is useful as a standalone-statistic as well as in conjunction with the Underlying Duration statistic defined in a previous specification.  Note that Underlying Present Valueis not meant to be a sensitivity measure but rather a value measure.

 

For options there is always the question of what the Underlying Price represents. RiskMetrics will report the option value when using the present value statistic.  The underlying present value will report the delta adjusted amount of the underlying security you are controlling.  If one has an option with a premium value of $1,000, the present value statistic will report $1,000.  If the option was at-the-money on 1000 shares on a $20 stock, the underlying present value would be $10,000 (1000 x $20 * 0.5), where the ATM option delta is 0.5.

 

Definition of Underlying Present Value by Asset Type

 

Amortizing Bond

Same as PV.

Amortizing Swap

PV of the fixed leg only (including sign based on pay/receives fixed).

Bond

PV of the bond only.  Ignore the settlement payment leg if it is present.

Bond Future

PV of the bond leg only, ignoring the “futures” payment.  This can be computed exactly as market price*number of contracts*notional/100.0.  Note that sometimes the user specifies a market price and other times it is left blank in which case RiskServer computes one.

Bond Future Option

PV of the bond leg only, ignoring all optionality.  This can be computed exactly as Underlying Price*Number of Contracts*notional/100.0.  Preserve sign if option type=call, reverse sign if option type=put.    Note that sometimes the user specifies an underlying price and other times it is left blank in which case RiskServer computes one.  Also note that Underlying PV is not meant to be an “exposure” statistic, which is why delta is ignored for option positions. 

Bond Option

PV of the bond leg only, ignoring all optionality and settlement.  This can be computed exactly as (underlying price+accrued_interest)*notional/100.0.  Preserve sign if option type=call, reverse sign if option type=put.

Cap

Unknown at this time and waiting further research.  For now simply set to PV.

Floor

Unknown at this time and waiting further research.  For now simply set to PV.

Collar

Unknown at this time and waiting further research.  For now simply set to PV.

Cash

Same as PV.

Cash Flow

Same as PV.

Commodity 

PV of the commodity only.  Ignore the settlement payment leg if it is present.

Commodity Option

PV of the commodity leg only, ignoring all optionality and settlement. This can be computed exactly as underlying price*number of units. Preserve sign if option type=call, reverse sign if option type=put. Note: I assumed that underlying price is the spot price of the commodity. If this is not the case (i.e. It actually represents the forward price at option expiry date) then we need to discount the value above by the appropriate yield curve to get a true PV of the commodity leg.

Commodity Future

PV of the commodity leg only, ignoring the “futures” payment.  This can be computed exactly as market price*number of contracts*number of units.

Commodity Future Opt

PV of the commodity leg only, ignoring all optionality. This can be computed exactly as underlying price*number of contracts*number of units.  Preserve sign if option type=call, reverse sign if option type=put.

Convertible Bond

PV of the convertible bond only. Ignore the settlement payment leg if it is present.

Equity

PV of the equity only. Ignore the settlement payment leg if it is present.

Equity Option

PV of the equity leg only, ignoring all optionality and settlement. This can be computed exactly as underlying price*number of shares.  Preserve sign if option type=call, reverse sign if option type=put.   Note:  I assumed that underlying price is the spot price of the equity.  If this is not the case (i.e. It actually represents the forward price at option expiry date) then we need to discount the value above by the appropriate yield curve to get a true PV of the equity leg.

Equity Future

PV of the equity leg only, ignoring the “futures” payment. This can be computed exactly as market price*number of contracts*number of shares.

Equity Future Opt

PV of the equity leg only, ignoring all optionality. This can be computed exactly as underlying price*number of contracts*number of shares. Preserve sign if option type=call, reverse sign if option type=put.

FRA

Unknown at this time and waiting further research. For now simply set to pv.

FRN

PV of the bond only. Ignore the settlement payment leg if it is present.

FX Forward

PV of the forward currency leg only.  Ignore the settle currency leg.

FX Option

PV of the option currency leg only, ignoring all optionality, settlement currency, and settlement. This can be computed exactly as underlying price*notional. Preserve sign if option type=call, reverse sign if option type=put.  Note: I assumed that underlying price is the spot price of the option currency. If this is not the case (i.e. It actually represents the forward price at option expiry date) then we need to discount the value above by the appropriate yield curve to get a true PV of the option currency leg.

Inflation Indexed Bond

PV of the bond only. Ignore the settlement payment leg if it is present.

Interest Rate Future

PV of the interest rate leg only, ignoring the “futures” payment.  We do not have a market price field, but if we did the value we need would be computed as market price*number of contracts*notional/100.0*time_factor, where time_factor is a fraction defined as term(in months) divided by 12 months.  For example, if the term is 3M then the time_factor would be 0.25.

IR Future Option

PV of the interest rate leg only, ignoring all optionality. This can be computed exactly as underlying price*number of contracts*notional/100.0*time_factor, where time_factoris a fraction defined as term(in months) divided by 12 months.  Preserve sign if option type=call, reverse sign if option type=put.

Money Market Deposit

PV of the underlying “bond” leg only.  Ignore the initial deposit leg if it present.

Mortgage-Backed Sec

Same as PV.

Mutual Fund

PV of the mutual fund only.  Ignore the settlement payment leg if it is present.

Overnight Indexed Swap

PV of the fixed leg only (including sign based on pay/receives fixed).

Swap

Unknown at this time and waiting further research. For now simply set to PV.  If we knew the swap was fixed for float than we could use the same rules as for overnight indexed swaps and amortizing swaps.  But this would not help with fixed/fixed or float/float swaps.

Swaption

Unknown at this time and waiting further research. For now simply set to PV