Measuring Investment Performance in Peer Lending
Author: Emmanuel Marot - LendingRobot
Date: April 2014
Version: 20140415

# Measuring Investment Performance in Peer Lending

Preliminary - Please do not distribute - © LendingRobot / Emmanuel Marot

## Introduction

Calculating the return of Peer Lending investments is surprisingly confusing. Several different calculation methods can be used, each producing different numbers and each presenting its own weaknesses. This paper reviews the most common methods, and recommends a specific way to compute returns.

Our goal is to find a way to calculate returns suitable for both past, present and future investments. The ideal method should meet the following criteria:

• Make sense: the Return for fully paid loans should be close to their Annual Percentage Yield. For fully defaulting loans (no payments made whatsoever), the Return should be close or equal to –100%.

• Be computationally tractable: it should be always possible to calculate the Return, even when no payments have been made.

• Simple to aggregate amongst multiple loans, even when their maturities or durations differ.

• It should be compatible with the notion of Risk. The Return must be able to factor in the probability of defaults in the future, and the Return ex-ante should never exceed its value ex-post. In other words, since the younger the loan, the higher the risk, without new event the Return expected at issuance should never be above the realized Return once the loan has been fully paid.

• Be compatible with performance standards such as GIPS, an international effort to “establish a recognized standard for calculating and presenting investment performance around the world”. In the present case, it mostly means calculating returns net of fees and be annualized or time-weighted.

## Preamble

### Arithmetic Averaging and Net-of-Fees

Imagine investing in two loans, one of $20,000 with a return of 15% and one of$3,000 with a return of –40%. There are two different ways to calculate the aggregate returns.

The first one is simply to average them: $$\frac{+15\% - 40\%}{2} = -12.5\%$$.

The second method is to consider we have invested $23,000, the entirety of loans #1 and #2. The total return is then: $$\frac{\20,000 \times 15\% + \3,000 \times - 40\%}{\23,000} = 8\%$$ The first method is the arithmetic average, the second one is the dollar-weighted average. As one can see, they can give very different results. The dollar-weighted average is more indicative of the performance of the market in general. It answers a question like ‘how much money investors make in Peer Lending?’. However, this way of aggregating returns is only suitable for market overview and for institutional investors participating in the whole loan program. Individual investors are likely to invest small amounts (fractions) in multiple loans, which means the arithmetic average is more accurate. Unless otherwise specified, we’ll therefore use only arithmetic average. In accordance with GIPS that stipulate returns must always be presented net-of-fees, and unless otherwise specified, all the returns presented here are after the service fee (e.g. 1% for Lending Club). ### Examples Most of the calculations below will take 3 loans as examples: • Loan A has an amount of$5,000, a term of 36 months, an interest rate of 13% and is fully paid at maturity. A service fee of 1% is taken from payments to investors.
• Loan B has the same characteristics than A, except that payments stop after the 27th month, and the loan status is changed to ‘charged off’ on the 31st month.
• Loan C has the same characteristics than A, except that it defaults immediately, with no payments made at all. Therefore we expect the return to be –100%. The loan status is changed to ‘charged off’ on the 4th month.

### Detailed Calculations

An Excel spreadsheet accompanies this paper. It can be accessed here.

## Monthly Payment

Peer Lending loans rely on fully amortizing payments. It means that with a constant interest rate, the payment remains the same for the duration of the loan.
The formula to calculate monthly payment is:

$p=\frac { { \frac { r }{ 12 } } }{ 1-{ \left( 1+\frac { r }{ 12 } \right) }^{ -n } } \quad \times \quad A \tag{eq1}$

where $$r$$ is the annual interest rate (13%), $$n$$ the number of months (36), and $$A$$ the amount borrowed ($5,000). Therefore for loans A and B, the monthly payment is$168.47.

Each payment $$p$$ is split between the interest and the principal (paying back the debt). The interest equals the remaining principal times the interest rate, adjusted each month. For instance, the first interest payment of loan A is $$\5,000 \times \frac{13\%}{12} = \54.17$$. The remaining ($168.47 -$54.17) = $114.30 is the principal payment, and reduces the outstanding principal to$5,000 - $114.30 =$4,885.70 after the first month.

Over time, the interest part decreases since there is less of the principal left to pay interest for, although it doesn’t make much difference for Peer Lending investors, for whom the negative and positive cash-flows matter most.

## Return Calculations Methods

### Return on Investment

The ROI, or Return on Investment is the simplest and most widely used formula to calculate financial returns. It doesn’t take time into account, and simply divides the surplus by the initial amount. If one sells for $117 something that cost him$95, the ROI is $$\frac{\117-\95}{\95} = 23\%$$

For loans, we have: $ROI = \frac{\text{gain}}{\text{principal}} = \frac {\sum {p} - A}{A}$

where $$p$$ are the payments and $$A$$ the initial loan amount.

Since there is a 1% service fee, the monthly payment received by investors for loans A and B is $$\168.47 \times \left( 100\% - 1\% \right) = \166.79$$

For loan A, the ROI is $$\frac{36 \times \166.79 - \5,000}{\5,000} = 20.1\%$$
For loan B, the ROI is $$\frac{27 \times \166.79 - \5,000}{\5,000} = -9.9\%$$
For loan C, the ROI is $$\frac{0 \times \166.79 - \5,000}{\5,000} = -100\%$$

The surprising fact is how high the ROI is: a loan with 13% interest rate give a ROI of over 20%, after service fees! It looks too good to be true… and it is, in a way, because the ROI does not take time into account. Earning $22 in one day is better than earning$22 in 3 years, therefore future cash flows should be discounted.

In the Peer Lending case, ROI would be a fair indicator only if the borrower were to make all the payments at once, and at the same time the money was borrowed. Unlikely, indeed…

Using non-annualized ROIs is misleading, because financial performances are commonly expressed on an annual basis. A consistent time period is required to make apple-to-apple comparisons, and Peer Lending may be depicted too optimistically when not converted to annual terms. For instance, the S&P 500 grew by 27.7% in 2013, a formidable year, but still in the same realm than the 20.1% of loan A… until one realizes it took 3 years for loan A to generate that return. Another problem is that without taking time into account, 60-months loans will almost always have higher returns than 36-months loans.

### Alternate Return on Investment

In order to get more realistic numbers, some Peer Lending blogs use a method where the gain is divided by the principal plus the gain. Since the gain equals the sum of payments minus the principal, we have:
$ROI_{alt} = \frac{\text{gain}}{\text{principal + gain}} = \frac { \sum { p } - A }{ A + \left(\sum { p } - A \right)} = \frac { \sum { p } - A }{ \sum { p } }$

It gives slightly more conservative numbers, with a return of 16.7% for loan A and –11.0% for loan B. Unfortunately, this method cannot be used for loans that default immediately. It cannot be calculated for loan C since the denominator would be zero. At last, it returns meaningless numbers when few payments have been made. For instance, if only 2 payments were made, the result would be a surrealist $$\frac{2 \times \166.79 - \5,000}{2 \times \166.79} = -1,398.89\%$$.

It is possible to improve it by only taking the gain into account in the denominator if positive:
$ROI_{alt} = \frac { \sum { p } - A }{ A + \left( \left(\sum { p } - A \right)\wedge 0 \right)}$

It makes it always computable, but introduces a bias since positive and negative returns are not weighted consistently.

### Annualized and Compounded Returns

Annualizing Returns answers the questions “If I earned x dollars in n years, how much did I earn per year on average?”. In some cases, the money obtained through the investment (e.g. dividends for stocks, payments for loans) can be re-invested in the security itself, and contributes to better returns, this is called compounding. In other cases, the money cannot or is not re-invested in the same security, but can still be used for investing in something else. Since a common basis is necessary to compare investments of different horizons, it is customary to convert investment returns to yearly basis.

There are two fundamentally different ways to see this. The difference is in considering whether the money earned (or that could have been earned) contributed to the end results or not.

The simplest version, called ‘Average Annualized’, simply divides the return by the duration: 12% in 3 years equals $$\frac{12\%}{3} = 4\%$$ per year. $1,200,000 in 6 years equals $$\frac{\1,200,000}{6} = \20,000$$ per year. The second version considers the return to be ‘compounded’ ### Compounding effect If an investment produces 10% annual interest, and this interest in re-invested, after 2 years it will return $$(1 + 10\%) \times (1 + 10\%) - 1 = 21\%$$. After 3 years, $$1.1 \times 1.1 \times 1.1 - 1 = 33.1\%$$. This is called the law of compounding returns. As a matter of fact, the 13% interest loan A compounds at a higher pace than 13%, because it’s paid monthly. Therefore in one year, the loan compounds at $$(1+ \frac {13\%}{12})^{12}-1 = 13.8\%$$. This is also called the Annual Percentage Yield: $APY = { \left( 1+\frac { r }{ 12 } \right) }^{ 12 }-1$ Where $$r$$ is the annual interest rate, and payments are made monthly. The compounding effect is fundamental in finance, and can have huge effects. For instance, earning$2 out of an $100 investment in one day is not very exciting, but if one can re-invest it with the same return every day for an entire year, the$100 investment becomes an astounding $$\100 \times 1.02^{365} = \137,741$$. As mentioned above, even when an investment does not compound, the money earned along the way could be re-invested in something else and produce returns on its own, and therefore it us customary to compound returns.

Annualizing a compounded Return requires to reverse the compounding effect, which is done by calculating the root:

$R_a = ({1+R}) ^ {\frac {1} {n}}-1$

where $$R$$ is the end return, and $$n$$ the number of periods elapsed before payment. If one sells for $117 something that 3 years earlier cost her$95, the compounded annualized ROI is $$(1 + \frac {\117 - \95}{\95})^{\frac{1}{3}} -1 = (1 + 23\%)^{\frac{1}{3}} -1 = 7.14\%$$.

For loan A, the Compounded Annualized Return is $$(1+20.1\%)^\frac{1}{3} -1 = 6.3\%$$
For loan B, the Compounded Annualized Return is $$(1 - 0.9\%)^\frac{1}{3} -1 = -3.4\%$$
For loan C, the Compounded Annualized Return is $$(1 - 100\%)^\frac{1}{3} -1 = -100\%$$

As we can see, now it’s the other extreme: how come a fully paid loan at 13% interest rate would only return 6.3% annually?
Because the above calculation considers all payments have been made at the very end of the 36 months, and did not have time to generate returns on their own. In reality, monthly payments are done, well, every month after the loan issuance.

Hence another approach, that considers the cash-flows that occurred during the investment lifetime should also have generated returns.

### Internal Rate of Return

The idea to de-compound a series of future cash-flow was first mentioned in 1907 with the concept of Net Present Value. It is calculated by discounting each future cash flow to reflect how much it is worth at the present time. The first cash flow event is usually negative, being the investment itself. In the case of an initial investment A followed by a succession of payments p and discounted at a rate of r, we have:

$NPV = \frac {-A} {(1+r)^0} + \frac {p_1} {(1+r)^1} + \frac {p_2} {(1+r)^2} + ... + \frac {p_{36}} {(1+r)^{36}}$

When the NPV is 0, it means r the discount rate is such that the sum of the discounted payments equals the loan amount. This is the Internal Rate of Return

Unfortunately finding the IRR is a rather tedious process, since it cannot be obtained by calculation, but only through trial and error. A computer program will approximate it using subsequent iterations, until the NPV is close enough to zero. A simple algorithm to speed up calculations, called the secant method, is:

${ r }_{ n+1 }={ r }_{ n }-{ NPV }_{ n }\left( \frac { r_{ n }-r_{ { n-1 } } }{ NPV_{ n }-NPV_{ n-1 } } \right)$

The function XIRR() in Excel uses this principle.

The Internal Rate of Return for the Loans A is 12.30%. Consistently enough, if the service fees are 0%, and the 13% interest rate loan is fully paid, its Internal Rate of Return is…13%.
For the loan B, the IRR is –8.8%.

So far, the IRR is the most accurate way to calculate returns on Peer Lending investments. Unfortunately, it has some flaws. The first is that fully defaulting loans like loan C cannot be calculated.

The second flaw, also related to defaults, is the way IRR behaves when there are very few payments. If a loan D makes only 3 payments, the ROI is –90%. Out of $5,000 lent, only$500.36 were paid back. But the IRR is a staggering –757.3%, painting a much darker picture than reality.

Lastly, IRRs can’t be easily averaged . If we invest in 9 loans A and 1 loan D, the total ROI will be $$\frac{9 \times 20.1\% + 1 \times -90.0\%}{10} = 9.1\%$$, still a positive return. But if we take the arithmetic average of the IRRs, we have $$\frac{9 \times 12.3\% + 1 \times -757.3\%}{10} = -64.7\%$$, which will give the false impression that we lost a lot of money. Instead of averaging IRRs, we have to calculate one IRR, summing the cash flows for all the loans month. In this case, its value will be 5.8%. But that may be impossible when mixing loans of different terms, such as 36-months loans and 60-months loans.

IRR could be reserved to non time-critical, ex-post calculations. As mentioned on LendingMemo, a good option to analyze one’s account performance is to list all transactions as cash-flow events and compute the IRR from them.

### LendingClub’s NAR

To estimate Returns, LendingClub uses a method called the ‘Net Annualized Return’, which is the annualized ratio of the sum of the adjusted interest payments divided by the sum of the remaining principals. The exact calculation is described here

In a simplified form (yes, simplified!), their formula is:

$NAR = \left(1 + \frac{\sum{p^r_i + l_i - s_i - d_i}}{\sum{P_i}} \right)^{12}-1$

where $$i$$ is the iteration for each payment month, $$p^r$$ the interest paid, $$l$$ the late fees, $$s$$ the service fees, $$d$$ equals the principal if the loan is defaulting, and $$P$$ is the remaining principal.

The first month, Loan A payment of $168.47 is split between $$\5,000 \times \frac{13\%}{12} = \54.17$$ of interest, and ($168.47 - $54.17) =$114.30 of principal. If we remove 1% of service fees, the numerator in LendingClub’s equation is $52.5, which gives a NAR of $$\left(1 + \frac{\52.5}{\5,000}\right)^{12} - 1 = 13.35\%$$ The second month, the principal left is$5,000 - $114.30 =$4,885.7, and the interests paid, minus service charges are $$(\4,885.7 \times \frac{13\%}{12}) \times (1 - 1\%) = \52.4$$. The NAR for the second month is therefore $$\left(1 + \frac{\52.5 + \52.4}{\5,000 + \4,885.7} \right) ^{12} - 1 = 13.34\%$$

This shows a few problems with this method. First, it is complex to calculate because it needs to be done recursively. So much for computational tractability… Second, the NAR exceeds the interest rate, even with service fees. Lastly, the NAR decreases over time while it should be the opposite, since only one detrimental event can happen in a loan life: default. Said otherwise, a NAR of 13.35% on the first month is misleading, because if everything goes well, we’ll end up at 13% on the 36th month. Incidentally, if the service fees are waived, then the NAR is constant… at 13.8%, which is the monthly compounding rate.
Speaking of default, below is the NAR for the last 10 months of loan B:

Month Event NAR
Month #27 Last Payment 13.1%
Month #28 12.9%
Month #29 12.7%
Month #30 12.5%
Month #31 Loan is Charged off 6.0%
Month #32 –5.9%
Month #33 –5.8%
Month #34 –5.7%
Month #35 –5.7%
Month #36 –5.6%

First, while the NAR slightly decreases as payments stop, it does no reflect the return one can really expect. 12.5% of return for a loan that is 3 months late is very optimistic (more on this later). Second, even when the loan is charged off and the remaining principal is subtracted, the NAR remains too high. Remember from the ROI calculation that 9.9% of the lent money is missing…

In November 2013, Lending Club’s introduced a modified version of their NAR calculation, taking into account the status of the loan. If a loan is already late in payment, there’s quite a significant risk it may default, and a loss estimate is applied.

Although crude, this adjustment principle is a real and laudable improvement from Lending Club. The main drawback is that if a loan keeps paying, the NAR doesn’t take any probability of default into account, and therefore remains too optimistic. In Lending Club’s own words ‘NAR is not a forward looking projection of performance’.

It is also very coarse, as no distinction is made between still-paying loans with different maturities.

### Semi-Compounded Return

We have seen that the ROI is too high compared to annualized returns, but also that considering all the payments to compound at the beginning of the investment period is inaccurate. We can look for a solution to consider the payments to have been made somewhere ‘in the middle’.

As a reminder, the formula to de-compound an investment lasting n years is $$R' = (1+R)^\frac{1}{n} -1$$. We can also write the simple, non-annulalized ROI formula as: $$R = (1+R)^\frac{n}{n} -1$$

Therefore the exponent for value at half-life is:
$\frac{\frac{1}{n} + \frac{n}{n}}2 = \frac {n+1}{2n}$

And our formula to calculate the semi-compounded Return is:

$R^" = (1+R)^\frac {n+1}{2n} -1$

For loan A, the Semi-Compounded Return is $$(1+20.1\%)^\frac{2}{3} -1 =12.98\%$$
For loan B, the Semi-Compounded Return is $$(1 - 0.9\%)^\frac{2}{3} -1 = -6.74\%$$
For loan C, the Semi-Compounded Return is $$(1 - 100\%)^\frac{2}{3} -1 = -100\%$$

Results are somewhat consistent with the IRR, but still inexact.

### Modified Internal Rate of Return

We saw that the problem with de-compounding the total return is that it is only accurate when all the payments are made at maturity. Once way to solve that is to take into consideration the future value of loan payments by compounding them until the end of the loan. In other words, with an yearly interest rate r, a payment made on the first month of a 36-months loan will be worth $$\left((1+\frac{r}{12})^{35} - 1 \right)$$ times more at loan maturity.

The return R is the compounding rate of the initial investment A, such that its future value is equal to the sum of the future values of the n payments p, themselves compounded at an interest rate r:

$A \cdot (1+R)^n = p_1 \cdot (1+r)^{n-1} + p_2 \cdot (1+r)^{n-2} + ...+ p_n \cdot (1+r)^{1}$

And therefore the formula to calculate Return is:

$R = \left( \frac {p \sum _{1}^{n} {(1+r)^i}}{A} \right)^{\frac{1}{n}} - 1$

Incidentally, such such a principle is similar to the Modified Internet Rate of Retun, where the compounding rates can be different for cash-flows and the initial investment. This calculation is very sensitive to the interest rate r. With r=0, the future value of payments is equal to their present value, and the Return is too low, as seen previously. If r is too high, it will consider the future value of payments to be unreasonably high. However, the interest rate is already defined, in a way: it is the loan’s interest rate itself, minus service fees.

Since in our case the payments are made monthly, we have:

$R = \left( \left( \frac {p \sum _{1}^{n} {(1+\frac{r}{12})^i}}{A} \right)^{\frac{1}{n}} - 1\right) \times 12$

Where A is the initial investment, n the number of monthly payment, p their amount and r the annual interest rate.

For loan A, the Modified Compounded Return is 12.14%
For loan B, the Semi-Compounded Return is a surprisingly positive 0.92%, due to the compounding of the 27 payments made.
For loan C, the Semi-Compounded Return is –100%.

### Annualized Dietz

The Dietz method, invented in 1966, is a well-known and easy-to-calculate performance measurement for a portfolio that has cash inflows or outflows within the investment period. To take those cash-flow events into account, the idea is to compare the investment performance to the average capital over the period:
$R = \frac{\text{Gain}}{\text{Average Capital}}$

If $$V_0$$ and $$V_1$$ are respectively the investment value at start and at the end of the period, and $$c_i$$ is a cash-flow event at time i, then the gain equals the end value $$V_1$$ minus the start value $$V_0$$ and everything that was added during the life of the investment $$\sum c_i$$. As for the average capital, it’s the money put initially in the investment $$V_0$$ , plus the time-averaged sum of the money added $$\sum \left(c_i\times \frac{n-i}{n}\right)$$ :

$R = \frac{V_1 - \left( V_0 + \sum c_i \right)}{V_0 + \sum \left(c_i\times \frac{n-i}{n}\right)}$

For instance, if we have a $1,000 investment, in which we put an additional$150 three months later, and sell for $1,300 at the end of the year: $R_D = \frac{\1,300 - \left(\1,000 + \150 \right)}{\1,000 + \left(150\times \frac{12-3}{12}\right)}= 13.48\%$ The$150 cash injection is kind of ‘discounted’, because it contributed to the end gain only during 9 months out of 12.

In the case of Peer Lending, monthly payments should be considered as cash withdrawals, and are negative cash-flow events. Therefore the average of the cash-flows over n months, with monthly payments equal to p is:
$\sum {\left( - c_i \times \frac{n-i}{n}\right) } = - p \times \sum { \frac{n-i}{n} } = - p \times \frac{n}{2}$

With P being the sum of all the payments and $$A$$ the loan amount, the Dietz Method can be applied to Peer Lending as:
$R = \frac { P - A }{ A - \frac {P}{2} }$

Readers familiar with financial calculations will notice than in our case, the Modified Dietz Method and the Simple Dietz Method are indeed equivalent.
With such a calculation method, we get R = 50.27% for loan A, R = –18.1% for loan B and R=–100% for loan C.
The Dietz Method is also easy to calculate and is endorsed by GIPS.

Now we need to annualized those values. In order to be consistent with GIPS practices, we elect to NOT annualize Returns when they are less than a year old. Therefore our Annualized Dietz Return obtained over a period of n months is:

$R = \left( \left( \frac { P - A }{ A - \frac {P}{2} }+ 1 \right) ^{\frac{12}{\max ( n , 12 )}} - 1\right) \tag{eq2}$

For loan A, the Return is 14.5%
For loan B, the Return is –8.5%
For loan C, the Return is –100%.

## Comparing Returns methods

### Sample Portfolio

We define a sample portfolio based on 10 different loans with the following properties:

Loan Amount Interest Rate Term Number of Payments made Total Paid Status
$1,000 12.61% 36 months 31$1,200.34 fully paid
$7,000 13.98% 36 months 19$8,211.24 fully paid
$14,900 14.61% 36 months 19$17,602.90 fully paid
$2,982 14.84% 36 months 20$3,526.27 fully paid
$6,906 8.00% 36 months 15$7,477.07 fully paid
$2,816 12.09% 36 months 16$1,666.27 charged off
$12,889 10.95% 36 months 36$15,170.37 fully paid
$22,402 10.99% 36 months 36$26,324.90 fully paid
$5,591 10.20% 36 months 36$6,473.23 fully paid
$3,741 11.14% 36 months 36$4,308.51 fully paid

The results are:

10 loans Before Fees After Fees
Method Arithmetic Average Dollar-Weighted Arithmetic Average Dollar-Weighted
ROI 10.73% 14.63% 9.63% 13.48%
Alternative ROI 5.76% 12.76% 4.81% 11.88%
Average Annualized 3.91% 6.65% 3.36% 6.13%
Compounded Annualized 3.08% 4.66% 2.73% 4.31%
Semi-Compounded 6.66% 9.53% 5.95% 8.80%
IRR 6.03% 12.28% 5.02% 11.35%
IRR Modified 3.08% 4.12% 2.73% 3.78%
Annualized Dietz 10.61% 14.33% 9.32% 13.12%

### Historical data

Historical data is based on Lending Club’s publicly available statistics. As of March 2014, 261,481 loans have been issued, 199,446 of them with a term of 36 months. The calculations below are for ‘Mature Loans’, which are the 19,841 36-months loans issued before February 2011, that are old enough to have matured or could have matured if they did not default or were paid back early.

19,481 loans Before Fees After Fees
Method Arithmetic Average Dollar-Weighted Arithmetic Average Dollar-Weighted
ROI 6.21% 7.71% 5.15% 6.64%
Alternative ROI n/a 7.16% n/a 6.22%
Average Annualized –0.07% 3.55% –0.60% 3.05%
Compounded Annualized 0.78% 2.51% 0.44% 2.17%
Semi-Compounded 3.09% 5.08% 2.40% 4.38%
IRR n/a 6.68% n/a 5.77%
IRR Modified 0.72% 2.41% 0.38% 2.07%
Annualized Dietz 5.04% 7.37% 3.85% 6.31%

The ‘n/a’ number means it was impossible to compute the corresponding number, because at least one loan did not make any payments.

## Conclusion

The Annualized Dietz method is simple, accurate, always computable and compatible with GPIS standard. Therefore our conclusion is that Annualized Dietz is the best method to calculate Peer Lending returns.