# Calculating the Expected Return of a Single Note

## Introduction

While monitoring the performance of an investment is a universal requirement in finance, Peer Lending creates unique challenges. We previously introduced general principles and how one can aggregate returns at the portfolio level. This article focuses on the performance calculation of a single asset, and how we update it over time.

## General Principle

As mentioned previously, we use the Internal Rate of Return (IRR) to calculate financial performance. IRR takes into account the time-value of money and is an industry-wide standard. The internal rate of return is the rate of compounding of a stream of cash flows. For instance, it is the discounting rate *r* such that the sum of *n* monthly payments of amount *p* equals the initial investment *A* in a loan:

\[ A = \sum_{i=1}^n{\frac{p_i}{(1+r)^i}} \]

In Peer Lending each loan is paid on a monthly basis, therefore the IRR calculated based on loan payments is also monthly. To present meaningful numbers, we ‘annualize’ it, by compounding it over 12 months:

\[ R = (1+r)^{12} – 1 \]

Please note that annualizing returns that occurred over short periods may suggest inflated performance. For example, annualizing a one-day return of 2% gives an unrealistic 137,640% over a year. It is customary to only annualize returns once they exceed one year. However, notes in a portfolio were issued at different times. To be able to compared loans returns with various maturities in a portfolio, we do need to put them on the same time basis and annualize each of them indiscriminately. To counter inflated returns we may decide to discard returns that are suspiciously high as non-computable when averaging returns over an entire portfolio.

## Monthly Cash-flows

The cash-flow array of a loan starts with a big negative number, the loan amount funded by the lenders, followed by a series of monthly payments back to those same lenders. If everything goes according to plan, each payment is identical and equals the installment calculated at loan issuance. We subtract the marketplace fees from those payments to ensure more accurate results. For instance, on Lending Club, service fees equal 1%, therefore a \$273.97 monthly installment equals 273.97 x (1.00 – 0.01) = \$271.23 net monthly payment distributed amongst lenders.

## Pre-payment

A borrower may decide to repay the loan earlier than term, which will result in a larger payment than expected.

In case of full early repayment, the last cash-flow event must equal the outstanding principal and accrued interest. The IRR in this case is identical to a fully paid loan at term. If the principal was not completely paid, it would have continued to generate interest, a portion of which would have been paid by each installment.

## Discounting Payments

A bigger ‘risk’ is that a loan will default, which is when a borrower stops paying before the installments reach term.

If a loan has a probability *g* to miss a payment at time *i*, the probability it makes a payment of installment amount *a*, as schedule, is *(1-g)*:

\[ p_i = \left\{ \begin{array}{ll}

0 & P( g_i)\\

a & P(1-g_i)\end{array} \right. \]

For simplicity reason, we’ll ignore the cases of the loan making a payment lower or greater than the installment.

Therefore:

\[ p_i = a \times (1-g_i) \]

In other words, we discount the installment by the probability that the loan miss the payment.

## Occurence of Default

The probability for a loan to default is not necessarily constant over time. Based on the historical data provided by each marketplace, we can measure when defaulting loans have stopped paying, and graph what is called a ‘hazard rate’. The cumulative hazard rate shows the probability that a loan will have stopped paying for any given month. Obviously the probability is zero at issuance (since no payments have had to be made, it is impossible to default), and reaches the overall average default rate at term.

The analysis of loans with varying features shows that loan hazard rates are approximately isomorphic. While the height of the curve is dependent of the default rate for a given characteristic (for instance the loan grade or the credit score of the borrower), the shape remains more or less identical.

Further analysis shows that the hazard rate is mostly dependent upon only 2 factors: the marketplace (e.g. Lending Club vs Prosper) and the term of the loans.

We can therefore calculate the probability *g* of a loan *l* of term *t* issued on marketplace *m* to miss a payment at time *i* based on 2 components: its lifetime probability of default *d* and a cumulative hazard rate *h*:

We can calculate the probability *g* of missing payment at any given time *i* based on 2 components: its lifetime probability of default *d* for a given vector of loan properties *l*, and a hazard rate *h* based on the marketplace *m*, with term *t* and the time *i*:

\[ g(l,i) = d(l) \times h(m,t,i) \]

## Calculating default rate

To identify the loans offering the best return opportunities, LendingRobot has designed a machine-learning algorithm that takes into account features such as debt-to-income ratio, loan purpose, or credit history.

In order to rely on something more ‘neutral’ to estimate returns, and avoid self-fulfilling prophecies (using the same algorithm to both select loans and predict their returns), we use a simpler, yet still relatively accurate estimator provided by each marketplace: the loan grade. This way, the accuracy of our return predictions is independent from our own selection model. To estimate the probability of default for a loan of a given grade, we consider the loans of that grade that are old enough to have reached maturity and take the ratio of defaulting ones to the total number of loans. However, such measurement is susceptible to noise, especially when going to a granular level like sub-grades. We smooth those measurements to eliminate noise by fitting a double exponential function, such as *y* the default rate for a loan of grade number *x* is:

\[ y = a + b \cdot e^{-c \cdot x} + d \cdot e^{-f \cdot x} \]

## Discounting future cash-flow

Once the hazard rate and the lifetime default probability have been estimated, we can discount future cash-flow events to take into account the upcoming and cumulative risks of default.

With the loan amount *A*, the monthly installment *a*, the overall default rate *d* and the hazard rates *h* for each month, the monthly return *r* is such that:

\[ A = \sum_{i=1}^n{\frac{a \; (1- d \times \sum_{j=1}^i h_j)}{(1+r)^i}} \]

## Taking account payments already made

The cumulative hazard rate shifts down as the borrower makes payments. This is because payments already made have, naturally, a probability of default of zero, and the probability of future payments becomes increasingly more likely.

To calculate expected returns for on-going loans, we combine the payments already made with the discounted future payments.

As a consequence, the expected return of a consistently paying loan increases over time and while the loan itself becomes less risky. If *k* payments have been made already, out of a total of *n* expected payments, we have:

\[ A = \sum_{i=1}^k{\frac{a}{(1+r)^i}} + \sum_{i=k+1}^n{\frac{a \; (1- d \times \sum_{j=k+1}^i h_j)}{(1+r)^i}} \]

An example of calculation is shown is the following Google spreadsheet: https://docs.google.com/spreadsheets/d/10Fuen2LDW6CIHRC3XNs7uqU15PUSnOznW1OxeXvoLtE/edit?usp=sharing

We consider a \$7,500 loan with term 36 months. The interest rate is 18.75%, which gives a monthly installment of \$273.97 (or \$271.23 net after deducting 1% of service fees). Historically, such a loan had a probability of default of 19.57%.

The hazard rates defined on row 7 are non-cumulative. To discount future payment, we sum the probabilities of default from the first month of future payment.

At issuance, when 0 payments have been made, all the 36 payments are discounted, which gives an annualized expected return of 9.64%. As more payments are made the discount on future payments is lowered and expected return rises. For instance, after 2 years (row 35), when 24 payments of \$271.33 have been made, the 25th payment is expected to be worth \$270.19 (it was only \$226.15 initially, due to the significant risk of default). The expected return is then 19.16%.

The maximum value is reached when the loan is fully paid. In that case, the expected return reaches 19.59%.

## Expected Return higher than Interest Rate

It may seem surprising at first that the expected return is higher than the interest rate, especially as the marketplace keeps 1% of servicing fees.

The reason is that the expected return is compounded (raising 1 plus the month return to the power of 12), while the monthly interest charged to the borrower is simply the annual interest divided by 12. Taking the time value of money into account, the borrower pays more than the nominal rate. For instance, 18.75% of annual interest divided by 12 gives 1.56% per month, but if annualized (raising 1.0156 to the power of 12 ), the rate paid by the borrower is 20.45%. Incidentally, is consistent with 19.16% expected return, once the 1% servicing fees are deducted.

## Incomplete Information

If we lack payment history for a given loan, or if this payment history is irrelevant because the note was purchased on the secondary market, we can discard any previous payments, and start with the outstanding principal or price paid for the note. In that case, the age of the loan is still needed, as it impacts the cumulative hazard rate.

## Late status

When a borrower missed a payment several days after it was due, the loan becomes late. The loan becomes increasingly late as the days go by, until, after 120 days, it is considered charged-off. Naturally, a loan becoming late is a negative predictor, and should affect the expected return adversely.

Marketplace statistics show the probably that a loan will default is based on the number of days the note is late. We can therefore estimate a function *F(l)* that gives the probability of a loan to stop paying, ex-ante of its overall probability of default, based on the number of days *l* it is late in payment. We further discount the installment by such probability.

\[ A = \sum_{i=1}^k{\frac{a}{(1+r)^i}} + \sum_{i=k+1}^n{\frac{a \; (1- d \times \sum_{j=k+1}^i h_j) \times ( 1 – F(l)) }{(1+r)^i}} \]

In the Google spreadsheet mentioned above, specifying a number of days a loan is late in payment in cell L1 shows the corresponding probability of default in the cell below. The future payments will automatically be updated when that value is changed. As mentioned previously, if the example loan is current and has made 24 payments already, the expected return is 19.16%. If the loan becomes late, for instance by 9 days, the expected return drops to 12.74%.

- Alex Neporozhniy
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