THE GOLDMAN SACHS GROUP, INC.  AND SUBSIDIARIES Management’s Discussion and Analysis Investing & Lending Investing & Lending includes our investing activities and the origination of loans, including our relationship lending activities, to provide financing to clients.
These investments and loans are typically longer-term in nature.
We make investments, some of which are consolidated, including through our merchant banking business and our special situations group, in debt securities and loans, public and private equity securities, infrastructure and real estate entities.
Some of these investments are made indirectly through funds that we manage.
We also make unsecured and secured loans to retail clients through our digital platforms, Marcus and Goldman Sachs Private Bank Select (GS Select), respectively.
The table below presents the operating results of our Investing & Lending segment.
## Table 0 ##
Operating Environment.
During 2017, generally higher global equity prices and tighter credit spreads contributed to a favorable environment for our equity and debt investments.
Results also reflected net gains from companyspecific events, including sales, and corporate performance.
This environment contrasts with 2016, where, in the first quarter of 2016, market conditions were difficult and corporate performance, particularly in the energy sector, was impacted by a challenging macroeconomic environment.
However, market conditions improved during the rest of 2016 as macroeconomic concerns moderated.
If macroeconomic concerns negatively affect company-specific events or corporate performance, or if global equity markets decline or credit spreads widen, net revenues in Investing & Lending would likely be negatively impacted.2017 versus 2016.
Net revenues in Investing & Lending were $6.58 billion for 2017, 61% higher than 2016.
Net revenues in equity securities were $4.58 billion, including $3.82 billion of net gains from private equities and $762 million in net gains from public equities.
Net revenues in equity securities were 78% higher than 2016, primarily reflecting a significant increase in net gains from private equities, which were positively impacted by companyspecific events and corporate performance.
In addition, net gains from public equities were significantly higher, as global equity prices increased during the year.
Of the $4.58 billion of net revenues in equity securities, approximately 60% was driven by net gains from company-specific events, such as sales, and public equities.
Net revenues in debt securities and loans were $2.00 billion, 33% higher than 2016, reflecting significantly higher net interest income (2017 included approximately $1.80 billion of net interest income).
Net revenues in debt securities and loans for 2017 also included an impairment of approximately $130 million on a secured loan.
Operating expenses were $2.80 billion for 2017, 17% higher than 2016, due to increased compensation and benefits expenses, reflecting higher net revenues, increased expenses related to consolidated investments, and increased expenses related to Marcus.
Pre-tax earnings were $3.79 billion in 2017 compared with $1.69 billion in 2016.2016 versus 2015.
Net revenues in Investing & Lending were $4.08 billion for 2016, 25% lower than 2015.
Net revenues in equity securities were $2.57 billion, including $2.17 billion of net gains from private equities and $402 million in net gains from public equities.
Net revenues in equity securities were 32% lower than 2015, primarily reflecting a significant decrease in net gains from private equities, driven by company-specific events and corporate performance.
Net revenues in debt securities and loans were $1.51 billion, 9% lower than 2015, reflecting significantly lower net revenues related to relationship lending activities, due to the impact of changes in credit spreads on economic hedges.
Losses related to these hedges were $596 million in 2016, compared with gains of $329 million in 2015.
This decrease was partially offset by higher net gains from investments in debt instruments and higher net interest income.
See Note 9 to the consolidated financial statements for further information about economic hedges related to our relationship lending activities.
Operating expenses were $2.39 billion for 2016, essentially unchanged compared with 2015.
Pre-tax earnings were $1.69 billion in 2016, 44% lower than 2015.
the 2006 Plan and the 1997 Plan generally vest over four years and expire no later than ten years from the date of grant.
For restricted stock awards issued under the 2006 Plan and the 1997 Plan, stock certificates are issued at the time of grant and recipients have dividend and voting rights.
In general, these grants vest over three years.
For deferred stock awards issued under the 2006 Plan and the 1997 Plan, no stock is issued at the time of grant.
Generally, these grants vest over two-, three- or four-year periods.
Performance awards granted under the 2006 Plan and the 1997 Plan are earned over a performance period based on achievement of goals, generally over twoto three-year periods.
Payment for performance awards is made in cash or in shares of our common stock equal to its fair market value per share, based on certain financial ratios, after the conclusion of each performance period.
We record compensation expense, equal to the estimated fair value of the options on the grant date, on a straight-line basis over the options’ vesting periods.
We use a Black-Scholes option-pricing model to estimate the fair value of the options granted.
The weighted-average assumptions used in connection with the option-pricing model were as follows for the years indicated:
## Table 1 ##
Compensation expense related to stock options, stock appreciation rights, restricted stock awards, deferred stock awards and performance awards, which we record as a component of salaries and employee benefits expense in our consolidated statement of income, was $126 million, $321 million and $272 million for the years ended December 31, 2009, 2008 and 2007, respectively.
The 2009 and 2008 expense excluded $13 million and $47 million, respectively, associated with accelerated vesting in connection with the restructuring plan described in note 6.
The aggregate income tax benefit recorded in our consolidated statement of income related to the above-described compensation expense was $50 million, $127 million and $109 million for 2009, 2008 and 2007, respectively.
Information about the 2006 Plan and 1997 Plan as of December 31, 2009, and activity during the years ended December 31, 2008 and 2009, is presented below:
## Table 2 ##
The weighted-average grant date fair value of options granted in 2009, 2008 and 2007 was $2.96, $21.06 and $22.44, respectively.
The total intrinsic value of options exercised during the years ended December 31, 2009, 2008 and 2007, was $5 million, $102 million and $129 million, respectively.
As of December 31, 2009, total unrecognized compensation cost, net of estimated forfeitures, related to stock options and stock appreciation rights was $7 million, which is expected to be recognized over a weighted-average period of 21 months.
value of our liquid assets, as defined, totaled $75.98 billion, compared to $85.81 billion as of December 31, 2008.
The decrease was mainly attributable to unusually high 2008 deposit balances, as we experienced a significant increase in customer deposits during the second half of 2008 as the credit markets worsened.
As customers accumulated liquidity, they placed cash with us, and these incremental customer deposits remained with State Street Bank at December 31, 2008.
During 2009, as markets normalized, deposit levels moderated as customers returned to investing their cash, and our liquid asset levels declined accordingly.
Due to the unusual size and volatile nature of these customer deposits, we maintained approximately $22.45 billion at central banks as of December 31, 2009, in excess of regulatory required minimums.
Securities carried at $40.96 billion as of December 31, 2009, compared to $42.74 billion as of December 31, 2008, were designated as pledged for public and trust deposits, borrowed funds and for other purposes as provided by law, and are excluded from the liquid assets calculation, unless pledged to the Federal Reserve Bank of Boston.
Liquid assets included securities pledged to the Federal Reserve Bank of Boston to secure State Street Bank’s ability to borrow from their discount window should the need arise.
This access to primary credit is an important source of back-up liquidity for State Street Bank.
As of December 31, 2009, we had no outstanding primary credit borrowings from the discount window.
Based upon our level of liquid assets and our ability to access the capital markets for additional funding when necessary, including our ability to issue debt and equity securities under our current universal shelf registration, management considers overall liquidity at December 31, 2009 to be sufficient to meet State Street’s current commitments and business needs, including supporting the liquidity of the commercial paper conduits and accommodating the transaction and cash management needs of our customers.
As referenced above, our ability to maintain consistent access to liquidity is fostered by the maintenance of high investment-grade ratings on our debt, as measured by the major independent credit rating agencies.
Factors essential to retaining high credit ratings include diverse and stable core earnings; strong risk management; strong capital ratios; diverse liquidity sources, including the global capital markets and customer deposits; and strong liquidity monitoring procedures.
High ratings on debt minimize borrowing costs and enhance our liquidity by ensuring the largest possible market for our debt.
A downgrade or reduction of these credit ratings could have an adverse impact to our ability to access funding at favorable interest rates.
The following table presents information about State Street’s and State Street Bank’s credit ratings as of February 22, 2010.

Tables referenced above, in order (Table 0 first):
<table><tr><td></td><td>Year Ended December</td></tr><tr><td><i>$ in millions</i></td><td>2017</td><td>2016</td><td>2015</td></tr><tr><td>Equity securities</td><td>$4,578</td><td>$2,573</td><td>$3,781</td></tr><tr><td>Debt securities and loans</td><td>2,003</td><td>1,507</td><td>1,655</td></tr><tr><td>Total net revenues</td><td>6,581</td><td>4,080</td><td>5,436</td></tr><tr><td>Operating expenses</td><td>2,796</td><td>2,386</td><td>2,402</td></tr><tr><td>Pre-taxearnings</td><td>$3,785</td><td>$1,694</td><td>$3,034</td></tr></table>
<table><tr><td></td><td>2009</td><td>2008</td><td> 2007</td></tr><tr><td>Dividend yield</td><td>4.82%</td><td>1.32%</td><td>1.34%</td></tr><tr><td>Expected volatility</td><td>26.70</td><td>21.00</td><td>23.30</td></tr><tr><td>Risk-free interest rate</td><td>2.49</td><td>3.17</td><td>4.69</td></tr><tr><td>Expected option lives (in years)</td><td>7.8</td><td>7.8</td><td>7.8</td></tr></table>
<table><tr><td></td><td>Shares (in thousands)</td><td> Weighted Average Exercise Price</td><td> Weighted Average Remaining Contractual Term (in years)</td><td> Aggregate Intrinsic Value (in millions)</td></tr><tr><td>Stock Options and Stock Appreciation Rights:</td><td></td><td></td><td></td><td></td></tr><tr><td>Outstanding at December 31, 2007</td><td>16,368</td><td>$48.94</td><td></td><td></td></tr><tr><td>Granted</td><td>921</td><td>81.71</td><td></td><td></td></tr><tr><td>Exercised</td><td>-2,926</td><td>44.99</td><td></td><td></td></tr><tr><td>Forfeited or expired</td><td>-47</td><td>47.40</td><td></td><td></td></tr><tr><td>Outstanding at December 31, 2008</td><td>14,316</td><td>51.86</td><td></td><td></td></tr><tr><td>Granted</td><td>516</td><td>19.31</td><td></td><td></td></tr><tr><td>Exercised</td><td>-832</td><td>40.57</td><td></td><td></td></tr><tr><td>Forfeited or expired</td><td>-833</td><td>46.32</td><td></td><td></td></tr><tr><td>Outstanding at December 31, 2009</td><td>13,167</td><td>$51.64</td><td> 4.10</td><td>$24.8</td></tr><tr><td>Exercisable at December 31, 2009</td><td>11,172</td><td>$50.12</td><td> 3.42</td><td>$12.0</td></tr></table>

Question: What was the average value of theRisk-free interest rate in the years where Expected volatility is positive?
Answer: 3.45
Reasoning program: add(2.49,3.17), add(#0,4.69), divide(#1,const_3)